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93 cosine, 345 definition of, 94 finding, 97 graph of, 95 Horizontal Line Test, 96 sine, 343, 345 tangent, 345 Inverse of a matrix, 602 finding an, 604 Inverse properties of logarithms, 230 333200_Index_SE.qxd 12/8/05 11:32 AM Page A216 A216 Index of natural logarithms, 234 of trigonometric functions, 347 Inverse trigonometric functions, 345 Inverse variation, 107 Inversely proportional, 107 Invertible matrix, 603 Irrational number, A1 Irreducible over the integers, A26 over the rationals, 174 over the reals, 174 J Joint variation, 108 Jointly proportional, 108 K Kepler’s Laws, 796 Key points of the graph of a trigonometric function, 322 intercepts, 322 maximum points, 322 minimum points, 322 L Latus rectum of an ellipse, 752 of a parabola, 738 Law of Cosines, 439, 490 alternative form, 439, 490 standard form, 439, 490 Law of Sines, 430, 489 Law of Trichotomy, A3 Leading coefficient of a polynomial, A23 Leading Coefficient Test, 141 Least squares regression line, 104 Lemniscate, 789 Length of a directed line segment, 447 Length of a vector, 448 Like radicals, A17 Like terms of a polynomial, A24 Limaçon, 786, 789 convex, 789 dimpled, 789 with inner loop, 789 Line(s) in the plane graph of, 25 horizontal, 33 inclination of, 728 least squares regression, 104 parallel, 30 perpendicular, 30 slope of, 25, 27 vertical, 33 Linear combination of vectors, 452 Linear depreciation, 32 Linear equation, 16 general form, 33 in one variable, A46 intercept form, 36 point-slope form, 29, 33 slope-intercept form, 25, 33 summary of, 33 two-point form, 29, 182, 624 in two variables, 25 Linear extrapolation, 33 Linear Factorization Theorem, 169, 214 Linear function, 66 Linear inequality, 542, A62 Linear interpolation, 33 Linear programming, 552 problem, solving, 553 Linear speed, 287 Local maximum, 58 Local minimum, 58 Locus, 735 Logarithm(s) change- |
of-base formula, 239 natural, properties of, 234, 240, 278 inverse, 234 one-to-one, 234 power, 240, 278 product, 240, 278 quotient, 240, 278 properties of, 230, 240, 278 inverse, 230 one-to-one, 230 power, 240, 278 product, 240, 278 quotient, 240, 278 Logarithmic equation, solving, 246 Logarithmic function, 229 with base a, 229 common, 230 natural, 233 Logarithmic model, 257 Logistic curve, 262 growth model, 257 Long division, 153 Lower bound, 177 Lower limit of summation, 646 M Magnitude of a directed line segment, 447 of a vector, 448 Main diagonal of a square matrix, 572 Major axis of an ellipse, 744 Marginal cost, 31 Mathematical induction, 673 extended principle of, 675 Principle of, 674 Matrix (matrices), 572 addition, 588 properties of, 590 additive identity, 591 adjoining, 604 augmented, 573 coded row, 625 coefficient, 573 cofactor of, 613 column, 572 determinant of, 606, 611, 614 diagonal, 601, 618 elementary row operations, 574 entry of a, 572 equal, 587 idempotent, 639 identity, 594 inverse of, 602 invertible, 603 minor of, 613 multiplication, 592 properties of, 594 nonsingular, 603 order of a, 572 in reduced row-echelon form, 576 representation of, 587 row, 572 in row-echelon form, 576 row-equivalent, 574 scalar identity, 590 scalar multiplication, 588 singular, 603 square, 572 stochastic, 599 transpose of, 640 uncoded row, 625 zero, 591 Measure of an angle, 283 degree, 285 radian, 283 Method of elimination, 507, 508 of substitution, 496 Midpoint Formula, 5, 124 Midpoint of a line segment, 5 Minor axis of an ellipse, 744 Minor of a matrix, 613 Minors and cofactors of a square matrix, 613 333200_Index_SE.qxd 12/8/05 11:32 AM Page A217 Modulus of a complex number, 471 Monomial, A23 Multiplication of fractions, |
A7 of matrices, 592 scalar, 588 nth term of an arithmetic sequence, 654 recursion form, 654 of a geometric sequence, 664 of a sequence, finding a formula for, 678 Multiplicative identity of a real number, Number(s) A6 Multiplicative inverse, A5 for a matrix, 602 of a real number, A6 Multiplicity, 143 Multiplier effect, 671 Mutually exclusive events, 705 N n factorial, 644 Name of a function, 42, 47 Natural base, 222 Natural exponential function, 222 Natural logarithm properties of, 234, 240, 278 inverse, 234 one-to-one, 234 power, 240, 278 product, 240, 278 quotient, 240, 278 Natural logarithmic function, 233 Near point, 216 Negation, properties of, A6 Negative angle, 282 infinity, A3 of a vector, 449 Newton’s Law of Cooling, 268 Nonnegative number, A1 Nonrigid transformation, 78 Nonsingular matrix, 603 Nonsquare system of linear equations, 524 Normally distributed, 261 Notation exponential, A11 function, 42, 47 scientific, A13 sigma, 646 summation, 646 nth partial sum, 647 of an arithmetic sequence, 657 nth root(s) of a, A14 of a complex number, 475, 476 generalizations about, A15 principal, A14 of unity, 477 complex, 162 composite, A7 critical, 197, 201 imaginary, 162 pure, 162 irrational, A1 nonnegative, A1 prime, A7 rational, A1 real, A1 Number of permutations of n elements, 693 taken r at a time, 694 Number of solutions of a linear system, 522 Numerator, A5 O Objective function, 552 Oblique asymptote, 190 Oblique triangles, 430 area of, 434 Obtuse angle, 283 Odd function, 60 trigonometric functions, 298 One cycle of a sine curve, 321 One-to-one correspondence, A1 One-to-one function, 96 One-to-one property of exponential functions, 220 of logarithms, 230 of natural logarithms, 234 Operations of fractions, A7 Operations that produce equivalent systems, 520 Opposite side of a right triangle, 301 Optimal solution of a linear programming |
problem, 552 Optimization, 552 Order of a matrix, 572 on the real number line, A2 Ordered pair, 2 Ordered triple, 519 Orientation of a curve, 772 Origin, 2 of polar coordinate system, 779 of the real number line, A1 Index A217 symmetry, 18 Orthogonal vectors, 462 Outcome, 701 P Parabola, 128, 736, 793 axis of, 129, 736 classifying by discriminant, 767 by general equation, 759 directrix of, 736 eccentricity of, 793 focal chord of, 738 focus of, 736 latus rectum of, 738 reflective property, 738 standard form of the equation of, 736, 807 tangent line, 738 vertex of, 129, 133, 736 Parallel lines, 30 Parallelogram law, 449 Parameter, 771 eliminating the, 773 Parametric equation, 771 Partial fraction, 533 decomposition, 533 Pascal’s Triangle, 685 Perfect cube, A15 square, A15 square trinomial, A27, A28 Perihelion distance, 798 Perimeter, common formulas for, 7 Period of a function, 297 of sine and cosine functions, 324 Periodic function, 297 Permutation, 693 distinguishable, 695 of n elements, 693 taken r at a time, 694 Perpendicular lines, 30 Phase shift, 325 Piecewise-defined function, 43 Plane curve, 771 orientation of, 772 Point of diminishing returns, 151 equilibrium, 514, 546 Point-plotting method, 15 Point-slope form, 29, 33 Points of intersection, 500 Polar axis, 779 Polar coordinate system, 779 333200_Index_SE.qxd 12/8/05 11:32 AM Page A218 A218 Index origin of, 779 pole, 779 Polar coordinates, 779 conversion to rectangular, 780 quick tests for symmetry in, 787 test for symmetry in, 786 Polar equations of conics, 793, 808 Polar form of a complex number, 471 Pole, 779 Polynomial(s), A23 coefficient of, A23 completely factored, A26 constant term, A23 degree of, A23 equation, second-degree, A49 factors of, 173, 214 finding test intervals for, 197 guidelines for factoring, A30 irred |
ucible, A26 leading coefficient of, A23 like terms, A24 long division of, 153 prime, A26 prime factor, 174 standard form of, A23 synthetic division, 156 test intervals for, 144 Polynomial function, 128 real zeros of, 143 standard form, 142 test intervals, 197 of x with degree n, 128 Position equation, 525 Positive angle, 282 infinity, A3 Power, A11 Power function, 140 Power property of logarithms, 240, 278 of natural logarithms, 240, 278 Power-reducing formulas, 409, 425 Prime factor of a polynomial, 174 factorization, A7 number, A7 polynomial, A26 Principal nth root of a, A14 of a number, A14 Principal square root of a negative number, 166 of an event, 702 of independent events, 707 of the union of two events, 705 Producer surplus, 546 Product of functions, 84 of trigonometric functions, 407 of two complex numbers, 472 Product property of logarithms, 240, 278 of natural logarithms, 240, 278 Product-to-sum formulas, 411 Projection, of a vector, 464 Proof, 124 by contradiction, 568 indirect, 568 without words, 638 Proper rational expression, 154 Properties of absolute value, A4 of the dot product, 460, 492 of equality, A6 of exponents, A11 of fractions, A7 of inequalities, A61 of inverse trigonometric functions, 347 of logarithms, 230, 240, 278 inverse, 230 one-to-one, 230 power, 240, 278 product, 240, 278 quotient, 240, 278 of matrix addition and scalar multiplication, 590 of matrix multiplication, 594 of natural logarithms, 234, 240, 278 inverse, 234 one-to-one, 234 power, 240, 278 product, 240, 278 quotient, 240, 278 of negation, A6 one-to-one, exponential functions, 220 of radicals, A15 reflective, 738 of sums, 646, 722 of vector addition and scalar multiplication, 451 of zero, A7 Principle of Mathematical Induction, 674 Probability of a complement, 708 Pure imaginary number, 162 Pythagorean identities, 304, 374 Pythagorean Theorem, 4, 370 Q Quadrant, 2 Quadratic equation, 16, A49 solving by |
completing the square, A49 by extracting square roots, A49 by factoring, A49 using Quadratic Formula, A49 using Square Root Principle, A49 Quadratic Formula, A49 Quadratic function, 128 standard form, 131 Quick tests for symmetry in polar coordinates, 787 Quotient difference, 46 of functions, 84 of two complex numbers, 472 Quotient identities, 304, 374 Quotient property of logarithms, 240, 278 of natural logarithms, 240, 278 R Radian, 283 conversion to degrees, 286 Radical(s) index of, A14 like, A17 properties of, A15 simplest form, A16 symbol, A14 Radicand, A14 Radius of a circle, 20 Random selection with replacement, 691 without replacement, 691 Range of a function, 40, 47 Rate, 31 Rate of change, 31 average, 59 Ratio, 31 Rational exponent, A18 Rational expression(s), A36 improper, 154 proper, 154 Rational function, 184 asymptotes of, 186 domain of, 184 graph of, guidelines for analyzing, 187 test intervals for, 187 Rational inequality, test intervals, 201 Rational number, A1 333200_Index_SE.qxd 12/8/05 11:32 AM Page A219 Rational Zero Test, 170 Rationalizing a denominator, 384, A16, Rose curve, 788, 789 Rotation A17 Real axis of the complex plane, 470 Real number(s), A1 absolute value of, A4 division of, A5 subset of, A1 subtraction of, A5 Real number line, A1 bounded intervals on, A2 distance between two points, A4 interval on, A2 order on, A2 origin, A1 unbounded intervals on, A3 Real part of a complex number, 162 Real zeros of polynomial functions, 143 Reciprocal function, 68 Reciprocal identities, 304, 374 Rectangular coordinate system, 2 Rectangular coordinates, conversion to polar, 780 Recursion form of the nth term of an arithmetic sequence, 654 Recursion formula, 655 Recursive sequence, 644 Reduced row-echelon form of a matrix, 576 Reducible over the reals, 174 Reduction formulas, 402 Reference angle, 314 Reflection, 76 of a trigonometric function, 324 Reflective property of a parabola, 738 Relation, 40 Relative maximum, 58 |
Relative minimum, 58 Remainder Theorem, 157, 213 Repeated zero, 143 Representation of matrices, 587 Resultant of vector addition, 449 Right triangle definitions of trigonometric functions, 301 hypotenuse, 301 opposite side, 301 right side of, 301 solving, 306 Rigid transformation, 78 Root(s) of a complex number, 475, 476 cube, A14 principal nth, A14 square, A14 of axes, 763 to eliminate an xy-term, 763 invariants, 767 Row-echelon form, 519 of a matrix, 576 reduced, 576 Row-equivalent, 574 Row matrix, 572 Row operations, 520 Rules of signs for fractions, A7 S Sample space, 701 Satisfy the inequality, A60 Scalar, 588 identity, 590 multiple, 588 Scalar multiplication, 588 properties of, 590 of a vector, 449 properties of, 451 Scatter plot, 3 Scientific notation, A13 Scribner Log Rule, 505 Secant function, 295, 301 of any angle, 312 graph of, 335, 338 Secant line, 59 Second-degree polynomial equation, A49 Second differences, 680 Sector of a circle, 289 area of, 289 Sequence, 642 arithmetic, 653 finite, 642 first differences of, 680 geometric, 663 infinite, 642 nth partial sum, 647 recursive, 644 second differences of, 680 terms of, 642 Series, 647 finite, 647 geometric, 667 infinite, 647 geometric, 667 Sierpinski Triangle, 726 Sigma notation, 646 Sigmoidal curve, 262 Simple harmonic motion, 356, 357 frequency, 356 Index A219 Simplest form, A16 Sine curve, 321 amplitude of, 323 one cycle of, 321 Sine function, 295, 301 of any angle, 312 common angles, 315 curve, 321 domain of, 297 graph of, 325, 338 inverse, 343, 345 period of, 324 range of, 297 special angles, 303 Sines, cosines, and tangents of special angles, 303 Singular matrix, 603 Sketching the graph of an equation by point plotting, 15 Sketching the graph of an inequality in two variables, 541 Slant asymptote, 190 Slope inclination, 728, 806 of a line, 25, 27 Slope-intercept form, 25, 33 Solution(s) |
of an absolute value inequality, A63 of an equation, 14, A46 extraneous, A48, A54 of an inequality, 541, A60 of a system of equations, 496 graphical interpretations, 510 of a system of inequalities, 543 Solution point, 14 Solution set, A60 Solving an absolute value inequality, A63 an equation, A46 exponential and logarithmic equations, 246 an inequality, A60 a linear programming problem, 553 right triangles, 306 a system of equations, 496 Cramer’s Rule, 619, 620 Gaussian elimination with back-substitution, 578 Gauss-Jordan elimination, 579 graphical method, 500 method of elimination, 507, 508 method of substitution, 496 a system of linear equations, Gaussian elimination, 520 Special products, A25 333200_Index_SE.qxd 12/8/05 11:32 AM Page A220 A220 Index Square of a binomial, A25 of trigonometric functions, 407 Square matrix, 572 determinant of, 614 idempotent, 639 main diagonal of, 572 minors and cofactors of, 613 Square root(s), A14 extracting, A49 function, 68 of a negative number, 166 Square Root Principle, A49 Square system of linear equations, 524 Squaring function, 67 Standard form of a complex number, 162 of the equation of a circle, 20 of the equation of an ellipse, 745 of the equation of a hyperbola, 753 of the equation of a parabola, 736, 807 of Law of Cosines, 439, 490 of a polynomial, A23 of a polynomial function, 142 of a quadratic function, 131 Standard position of an angle, 282 of a vector, 448 Standard unit vector, 452 Step function, 69 Stochastic matrix, 599 Straight-line depreciation, 32 Strategies for solving exponential and logarithmic equations, 246 Strophoid, 810 Subset, A1 Substitution, method of, 496 Substitution Principle, A5 Subtraction of a complex number, 163 of fractions with like denominators, A7 with unlike denominators, A7 of real numbers, A5 Sum(s) of a finite arithmetic sequence, 656, 723 of a finite geometric sequence, 666, 723 of functions, 84 of an infinite geometric series, 667 |
nth partial, 647 of powers of integers, 679 properties of, 646, 722 of square differences, 104 Sum and difference formulas, 400, 424 Sum and difference of same terms, A25 Sum or difference of two cubes, A27 Summary of a sequence, 642 variable, A5 Terminal point, 447 Terminal side of an angle, 282 Test of equations of lines, 33 of function terminology, 47 for collinear points, 623 for symmetry in polar coordinates, Summation index of, 646 lower limit of, 646 notation, 646 upper limit of, 646 Sum-to-product formulas, 412, 426 Supplementary angles, 285 Surplus consumer, 546 producer, 546 Symmetry, 18 algebraic tests for, 19 graphical tests for, 18 quick tests for, in polar coordinates, 787 test for, in polar coordinates, 786 with respect to the origin, 18 with respect to the x-axis, 18 with respect to the y-axis, 18 Synthetic division, 156 uses of the remainder, 158 System of equations, 496 equivalent, 509 solution of, 496 with a unique solution, 607 System of inequalities, solution of, 543 System of linear equations consistent, 510 dependent, 510 inconsistent, 510 independent, 510 nonsquare, 524 number of solutions, 522 row-echelon form, 519 row operations, 520 square, 524 T Tangent function, 295, 301 of any angle, 312 common angles, 315 graph of, 332, 338 inverse, 345 special angles, 303 Tangent line to a parabola, 738 Term of an algebraic expression, A5 constant, A5, A23 786 Test intervals for a polynomial, 144 for a polynomial inequality, 197 for a rational function, 187 for a rational inequality, 201 Transcendental function, 218 Transformation nonrigid, 78 rigid, 78 Transpose of a matrix, 640 Transverse axis of a hyperbola, 753 Triangle area of, 622 oblique, 430 area of, 434 Trigonometric form of a complex number, 471 argument of, 471 modulus of, 471 Trigonometric function of any angle, 312 evaluating, 315 cosecant, 295, 301 cosine, 295, 301 cotangent, 295, 301 even and odd, 298 horizontal shrink of, 324 horizontal stretch of, 324 horizontal translation of, 325 inverse properties of |
, 347 key points, 322 intercepts, 322 maximum points, 322 minimum points, 322 product of, 407 reflection of, 324 right triangle definitions of, 301 secant, 295, 301 sine, 295, 301 square of, 407 tangent, 295, 301 unit circle definitions of, 295 Trigonometric identities cofunction identities, 374 even/odd identities, 374 fundamental identities, 304, 374 guidelines for verifying, 382 333200_Index_SE.qxd 12/8/05 11:32 AM Page A221 Pythagorean identities, 304, 374 quotient identities, 304, 374 reciprocal identities, 304, 374 Trigonometric values of common angles, 315 Trigonometry, 282 Trinomial, A23 perfect square, A27, A28 inverse, 107 joint, 108 in sign, 176 Vary directly, 105 as nth power, 106 Vary inversely, 107 Vary jointly, 108 Vector(s) Two-point form of the equation of a line, addition, 449 29, 33, 624 U Unbounded, A60 Unbounded intervals, A3 Uncoded row matrices, 625 Undefined, 47 Unit circle, 294 definitions of trigonometric functions, 295 Unit vector, 448, 621 in the direction of v, 451 standard, 452 Upper bound, 177 Upper limit of summation, 646 Upper and Lower Bound Rules, 177 Uses of the remainder in synthetic division, 158 V Value of a function, 42, 47 Variable, A5 dependent, 42, 47 independent, 42, 47 Variable term, A5 Variation combined, 107 constant of, 105 direct, 105 as an nth power, 106 properties of, 451 resultant of, 449 angle between two, 461, 492 component form of, 448 components, 463, 464 difference of, 449 directed line segment of, 447 direction angle of, 453 dot product of, 460 properties of, 460, 492 equality of, 448 horizontal component of, 452 length of, 448 linear combination of, 452 magnitude of, 448 negative of, 449 orthogonal, 462 parallelogram law, 449 projection, 464 resultant, 449 scalar multiplication of, 449, properties of, 451 standard position of, 448 unit, 448, 621 in the direction of v, 451 standard, 452 v in the plane, 447 vertical component of, 452 zero, 448 Vertex (vertices) of an |
angle, 282 of an ellipse, 744 Index A221 of a hyperbola, 753 of a parabola, 129, 133, 736 Vertical asymptote, 185 Vertical components of v, 452 Vertical line, 33 Vertical Line Test, 55 Vertical shift, 74 Vertical shrink, 78 Vertical stretch, 78 Volume, common formulas for, 7 W With replacement, 691 Without replacement, 691 Work, 466 x-axis, 2 symmetry, 18 x-coordinate, 2 y-axis, 2 symmetry, 18 y-coordinate, 2 X Y Z Zero(s) of a function, 56 matrix, 591 multiplicity of, 143 of a polynomial function, 143 bounds for, 177 real, 143 properties of, A7 repeated, 143 vector, 448 Zero-Factor Property, A7 333201_AP_FES.qxd 12/5/05 11:46 AM Page ES1 Definition of the Six Trigonometric Functions Right triangle definitions, where Adjacent e t i s o p p O sin opp. hyp. cos adj. hyp. tan opp. adj. csc hyp. opp. sec hyp. adj. cot adj. opp. Circular function definitions, where is any angle 2 2 y θ y r x x sin y r cos x r tan, csc r y sec r x cot 120° 4 135° 150° − ( 1, 0) π 180° y (0, 1) 90° 601, 0) π 3 45° π 4 30° 0° 360° 330 210° 225° π 5 240° π 4 4 3 270° − ) 3 2 π 11 6 315° π 7 300° π 4 5 π 3 3 2 (0, 1)−, ( Reciprocal Identities sin u 1 csc u csc u 1 sin u cos u 1 sec u sec u 1 cos u Quotient Identities tan u sin u cos u cot u cos u sin u tan u 1 cot u cot u 1 tan u Pythagorean Identities sin2 u cos2 u 1 1 tan2 u sec2 u Cofunction Identities u cos u sin 2 cos u sin u 2 tan u cot u 2 Even/Odd Identities sinu sin u cosu cos u tanu tan u 1 cot2 u csc2 u cot 2 sec 2 c |
sc 2 u tan u u csc u u sec u cotu cot u secu sec u cscu csc u Sum and Difference Formulas sinu ± v sin u cos v ± cos u sin v cosu ± v cos u cos v sin u sin v tanu ± v tan u ± tan v 1 tan u tan v Double-Angle Formulas sin 2u 2 sin u cos u cos 2u cos2 u sin2 u tan 2u 2 tan u 1 tan2 u 2 cos2 u 1 1 2 sin2 u 2 Power-Reducing Formulas sin2 u 1 cos 2u cos2 u 1 cos 2u tan2 u 1 cos 2u 1 cos 2u 2 Sum-to-Product Formulas sin u sin v 2 sinu v 2 sin u sin v 2 cosu v 2 cos u cos v 2 cosu v 2 cos u cos v 2 sinu v 2 cosu v 2 sinu v 2 cosu v 2 sinu v 2 cosu v cosu v Product-to-Sum Formulas sin u sin v 1 2 cos u cos v 1 2 sin u cos v 1 2 cos u sin v 1 2 sinu v sinu v sinu v sinu v cosu v cosu v 333201_AP_FES.qxd 12/5/05 11:46 AM Page ES2 FORMULAS FROM GEOMETRY Triangle: h a sin Area 1 2 bh h a θ c b c 2 a2 b2 2ab cos (Law of Cosines) Sector of Circular Ring: Area pw p average radius, w width of ring, in radians Ellipse: Area ab Circumference 2a2 b2 2 Cone: Volume Ah 3 A area of base Right Circular Cone: Volume r2h 3 Lateral Surface Area rr2 h2 Frustum of Right Circular Cone: Volume r 2 rR R2h 3 Lateral Surface Area sR Right Circular Cylinder: r2h Volume Lateral Surface Area 2rh r h s θ r r R p w Sphere: Volume 4 3 Surface Area 4r 2 r3 Wedge: A B sec A area of upper face, B area of base Right Triangle: Pythagorean Theorem c2 a2 b2 Equilateral Triangle: h 3s 2 Area 3s2 4 Parallelogram: Area bh Trapezoid: Area h 2 |
a b Circle: Area r 2 Circumference 2r Sector of Circle: r2 2 Area s r in radians Circular Ring: Area R2 r2 2pw p average radius, w width of ring 333201_AP_BES.qxd 12/5/05 11:45 AM Page ES1 Linear Function f x mx b y (0, b) ( m( b −, 0 ( m( b −, 0 x f(x) = mx + b, m > 0 f(x) = mx + b, m < 0 GRAPHS OF PARENT FUNCTIONS Absolute Value Function f x x x, x ≥ 0 x, x < 0 y 2 1 −2 −1 (0, 0) f(x) = x x 2 −1 −2 Square Root Function fx x y 4 3 2 1 f(x) = x −1 (0, 0) 2 3 4 −1,, Domain: Range: bm, 0 x-intercept: 0, b y-intercept: Increasing when Decreasing when m > 0 m < 0, Domain: 0, Range: 0, 0 Intercept: Decreasing on Increasing on Even function y-axis symmetry, 0 0, 0, Domain: 0, Range: 0, 0 Intercept: Increasing on 0, Greatest Integer Function fx x Quadratic (Squaring) Function fx ax2 Cubic Function fx x3 y f(x) = x[[ ]] 3 2 1 −3 −2 −1 1 2 3 x −2 −1 −3, Domain: Range: the set of integers x-intercepts: in the interval 0, 0 y-intercept: Constant between each pair of 0, 1 consecutive integers Jumps vertically one unit at each integer value y 3 2 1 −1 −2 −3 f(x) = ax, (x) = ax, a < 0 2 −3 −2 y 3 2 (0, 0) −1 −2 −3 3 2 1 f(x) = x3, Domain:, Range: 0, 0 Intercept: Increasing on Odd function Origin symmetry, 0,, 0, Domain: a > 0 : Range a < 0 Range : 0, 0 Intercept: Decreasing on Increasing on Increasing on Decreasing on Even function y-axis symmetry Relative minimum, 0 0, for, 0 0, a > 0 for a > 0 |
for a < 0 for a < 0 a > 0, a < 0, relative maximum or vertex: 0, 0 x x 333201_AP_BES.qxd 12/5/05 11:45 AM Page ES2 Rational (Reciprocal) Function fx 1 x Exponential Function Logarithmic Function fx ax, a > 0, a 1 fx loga x, a > 0, a 1 y 3 2 1 f(x) = 1 x −1 1 2 3 x y f(x) = ax f(x) = a−x (0, 1) y 1 f(x) = loga x (1, 0) 1 2 x x −1, 0 0, ), 0 0, ) Domain: Range: No intercepts Decreasing on Odd function Origin symmetry Vertical asymptote: y-axis Horizontal asymptote: x-axis, 0 and 0,, Domain: 0, Range: 0, 1 Intercept: Increasing on for fx ax Decreasing on for fx ax,, 0, Domain:, Range: 1, 0 Intercept: Increasing on Vertical asymptote: y-axis Continuous Reflection of graph of 0, fx ax Horizontal asymptote: x-axis Continuous in the line y x Sine Function fx sin x y 3 2 1 f(x) = sin x Cosine Function fx cos x y 3 2 f(x) = cos x Tangent Function fx tan x y f(x) = tan x 3 2 1 π 3π 2 x −2 −3 −2 −3, Domain: 1, 1 Range: 2 Period: x-intercepts: y-intercept: Odd function Origin symmetry n, 0 0, 0, Domain: Range: Period: 1, 1 2 n, 0 2 0, 1 x-intercepts: y-intercept: Even function y-axis symmetry n Domain: all x 2, Range: Period: x-intercepts: y-intercept: Vertical asymptotes: n, 0 0, 0 x n 2 Odd function Origin symmetry 333201_AP_BES.qxd 12/5/05 11:45 AM Page ES3 Cosecant Function fx csc x Secant Function fx sec x Cotangent Function fx cot x y f(x) = csc x = 1 sin x y f( |
x) = sec x = 1 cos x 3 2 1 3 2 f(x) = cot x = 1 tan x y 3 2 1 − π x π 2π 3π 2π 2π −2 −3 x n, 1 1, 2 Domain: all Range: Period: No intercepts Vertical asymptotes: Odd function Origin symmetry x n Domain: all x n 2, 1 1, Range: 2 Period: y-intercept: Vertical asymptotes: 0, 1 x n 2 Even function y-axis symmetry x n Domain: all Range: Period:, x-intercepts: 2 n, 0 Vertical asymptotes: Odd function Origin symmetry x n Inverse Sine Function fx arcsin x Inverse Cosine Function fx arccos x Inverse Tangent Function fx arctan x y π 2 −1 x 1 f(x) = arcsin x π 2 − y π f(x) = arccos x −1 x 1 y π 2 π− 2 −2 −1 x 1 2 f(x) = arctan x Range: Domain: 1, 1, 2 2 0, 0 Intercept: Odd function Origin symmetry 1, 1 Domain: 0, Range: y-intercept: 0, 2 Range:, Domain:, 2 2 0, 0 Intercept: Horizontal asymptotes: y ± 2 Odd function Origin symmetryparticipate. The variable could be the amount of extracurricular activities by one high school student. Let X = the amount of extracurricular activities by one high school student. The data are the number of extracurricular activities in which the high school students participate. Examples of the data are 2, 5, 7. 1.1 Find an article online or in a newspaper or magazine that refers to a statistical study or poll. Identify what each of the key terms—population, sample, parameter, statistic, variable, and data—refers to in the study mentioned in the article. Does the article use the key terms correctly? Example 1.2 Determine what the key terms refer to in the following study. A study was conducted at a local high school to analyze the average cumulative GPAs of students who graduated last year. Fill in the letter of the phrase that best describes each of the items below. 1. Population ____ 2. Statistic ____ 3. Parameter ____ 4. Sample |
____ 5. Variable ____ 6. Data ____ a) all students who attended the high school last year b) the cumulative GPA of one student who graduated from the high school last year c) 3.65, 2.80, 1.50, 3.90 d) a group of students who graduated from the high school last year, randomly selected e) the average cumulative GPA of students who graduated from the high school last year f) all students who graduated from the high school last year g) the average cumulative GPA of students in the study who graduated from the high school last year This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 9 Solution 1.2 1. f; 2. g; 3. e; 4. d; 5. b; 6. c Example 1.3 Determine what the population, sample, parameter, statistic, variable, and data referred to in the following study. As part of a study designed to test the safety of automobiles, the National Transportation Safety Board collected and reviewed data about the effects of an automobile crash on test dummies (The Data and Story Library, n.d.). Here is the criterion they used. Speed at which Cars Crashed Location of Driver (i.e., dummies) 35 miles/hour Front seat Table 1.1 Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars. Solution 1.3 The population is all cars containing dummies in the front seat. The sample is the 75 cars, selected by a simple random sample. The parameter is the proportion of driver dummies—if they had been real people—who would have suffered head injuries in the population. The statistic is proportion of driver dummies—if they had been real people—who would have suffered head injuries in the sample. The variable X = the number of driver dummies—if they had been real people—who would have suffered head injuries. The data are either: yes, had head injury, or no, did not. Example 1.4 Determine what the population, sample, parameter, statistic, variable, and data referred to in the following study. An insurance company would like to determine the proportion of |
all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit. Solution 1.4 The population is all medical doctors listed in the professional directory. The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population. The sample is the 500 doctors selected at random from the professional directory. The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample. The variable X = the number of medical doctors who have been involved in one or more malpractice suits. 10 Chapter 1 | Sampling and Data The data are either: yes, was involved in one or more malpractice lawsuits; or no, was not. Do the following exercise collaboratively with up to four people per group. Find a population, a sample, the parameter, the statistic, a variable, and data for the following study: You want to determine the average—mean—number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before. The answers were 1, 0, 1, 3, and 4 glasses of milk. 1.2 | Data, Sampling, and Variation in Data and Sampling Data may come from a population or from a sample. Lowercase letters like x or y generally are used to represent data values. Most data can be put into the following categories: • Qualitative • Quantitative Qualitative data are the result of categorizing or describing attributes of a population. Qualitative data are also often called categorical data. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O–, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type. Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics |
are examples of quantitative data. Quantitative data may be either discrete or continuous. All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three. Data that are not only made up of counting numbers, but that may include fractions, decimals, or irrational numbers, are called quantitative continuous data. Continuous data are often the results of measurements like lengths, weights, or times. A list of the lengths in minutes for all the phone calls that you make in a week, with numbers like 2.4, 7.5, or 11.0, would be quantitative continuous data. Example 1.5 Data Sample of Quantitative Discrete Data The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. The numbers of books, 3, 4, 2, and 1, are the quantitative discrete data. 1.5 The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has 10 machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 11 Example 1.6 Data Sample of Quantitative Continuous Data The data are the weights of backpacks with books in them. You sample the same five students. The weights, in pounds, of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data. 1.6 The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. ft., 160 sq. ft., 190 sq. ft., 180 sq. ft., and 210 sq. ft. What type of data is this? Example 1.7 You go to the supermarket and purchase three cans of soup (19 ounces tomato bisque, 14.1 ounces lentil, and 19 ounces Italian wedding), two packages of nuts (walnuts and peanuts |
), four different kinds of vegetable (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces pistachio ice cream and 32 ounces chocolate chip cookies). Name data sets that are quantitative discrete, quantitative continuous, and qualitative. Solution 1.7 A possible solution • One example of a quantitative discrete data set would be three cans of soup, two packages of nuts, four kinds of vegetables, and two desserts because you count them. • The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure weights as precisely as possible. • Types of soups, nuts, vegetables, and desserts are qualitative data because they are categorical. Try to identify additional data sets in this example. Example 1.8 The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data. 1.8 The data are the colors of houses. You sample five houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this? NOTE You may collect data as numbers and report it categorically. For example, the quiz scores for each student are recorded throughout the term. At the end of the term, the quiz scores are reported as A, B, C, D, or F. 12 Chapter 1 | Sampling and Data Example 1.9 Work collaboratively to determine the correct data type: quantitative or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words the number of. • • • • • the number of pairs of shoes you own the type of car you drive the distance from your home to the nearest grocery store the number of classes you take per school year the type of calculator you use • weights of sumo wrestlers • number of correct answers on a quiz • IQ scores (This may cause some discussion.) Solution 1.9 Items a, d, and g are quantitative discrete; items c, f, and h are quantitative continuous; items b and e are qualitative or categorical. 1.9 Determine the correct data type, quantitative or qualitative, for the number of cars in a parking lot. Indicate whether quantitative data are continuous or discrete. Example 1.10 A statistics professor |
collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart Figure 1.2. What type of data does this graph show? Figure 1.3 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 13 Solution 1.10 This pie chart shows the students in each year, which is qualitative or categorical data. 1.10 A large school district keeps data of the number of students who receive test scores on an end of the year standardized exam. The data he collects are summarized in the histogram. The class boundaries are 50 to less than 60, 60 to less than 70, 70 to less than 80, 80 to less than 90, and 90 to less than 100. Figure 1.4 Qualitative Data Discussion Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts, frequencies, and percentages or proportions, relative frequencies. For instance, to calculate the percentage of part time students at De Anza College, divide 9,200/22,496 to get.4089. Round to the nearest thousandth—third decimal place and then multiply by 100 to get the percentage, which is 40.9 percent. So, the percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College. De Anza College Foothill College Number Percent Number Percent Full-time 9,200 40.90% Full-time 4,059 28.60% Part-time 13,296 59.10% Part-time 10,124 71.40% Table 1.2 Fall Term 2007 (Census day) 14 Chapter 1 | Sampling and Data De Anza College Foothill College Total 22,496 100% Total 14,183 100% Table 1.2 Fall Term 2007 (Census day) Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. Two graphs that are used to display qualitative data are pie charts |
and bar graphs. In a pie chart, categories of data are shown by wedges in a circle that represent the percent of individuals/items in each category. We use pie charts when we want to show parts of a whole. In a bar graph, the length of the bar for each category represents the number or percent of individuals in each category. Bars may be vertical or horizontal. We use bar graphs when we want to compare categories or show changes over tim A Pareto chart consists of bars that are sorted into order by category size (largest to smallest). Look at Figure 1.5 and Figure 1.6 and determine which graph (pie or bar) you think displays the comparisons better. It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the best graph depending on the data and the context. Our choice also depends on what we are using the data for. Figure 1.5 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 15 Figure 1.6 Percentages That Add to More (or Less) Than 100 Percent Sometimes percentages add up to be more than 100 percent (or less than 100 percent). In the graph, the percentages add to more than 100 percent because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100 percent. Characteristic/Category Students studying technical subjects Students studying non-technical subjects Percent 40.9% 48.6% Students who intend to transfer to a four-year educational institutional 61.0% TOTAL 150.5% Table 1.3 De Anza College Year 2010 16 Chapter 1 | Sampling and Data Figure 1.7 Omitting Categories/Missing Data The table displays Ethnicity of Students but is missing the Other/Unknown category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart. Frequency Percent Asian Black Filipino Hispanic 8,794 1,412 1,298 4,180 Native American 146 Pacific Islander 236 5,978 White TOTAL 36.1% 5.8% 5.3% 17. |
1%.6% 1.0% 24.5% 22,044 out of 24,382 90.4% out of 100% Table 1.4 Ethnicity of Students at De Anza College Fall Term 2007 (Census Day) This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 17 Figure 1.8 The following graph is the same as the previous graph but the Other/Unknown percent (9.6 percent) has been included. The Other/Unknown category is large compared to some of the other categories (Native American,.6 percent, Pacific Islander 1.0 percent). This is important to know when we think about what the data are telling us. This particular bar graph in Figure 1.9 can be difficult to understand visually. The graph in Figure 1.10 is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret. Figure 1.9 Bar Graph with Other/Unknown Category 18 Chapter 1 | Sampling and Data Figure 1.10 Pareto Chart With Bars Sorted by Size Pie Charts: No Missing Data The following pie charts have the Other/Unknown category included since the percentages must add to 100 percent. The chart in Figure 1.11b is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure 1.11a. Figure 1.11 Marginal Distributions in Two-Way Tables Below is a two-way table, also called a contingency table, showing the favorite sports for 50 adults: 20 women and 30 men. Football Basketball Tennis Total Men Women Total Table 1.5 20 5 25 8 7 15 2 8 10 30 20 50 This is a two-way table because it displays information about two categorical variables, in this case, gender and sports. Data This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 19 of this type (two variable data) are referred to as bivariate data. Because the data represent a count, or tally, of choices, it is a two-way frequency table. The entries in the total row and the total column represent marginal frequencies or marginal distributions. Note—The term marginal distributions gets its name from the fact that the distributions are found in the margins of |
frequency distribution tables. Marginal distributions may be given as a fraction or decimal: For example, the total for men could be given as.6 or 3/5 since 30 / 50 =.6 = 3 / 5. Marginal distributions require bivariate data and only focus on one of the variables represented in the table. In other words, the reason 20 is a marginal frequency in this two-way table is because it represents the margin or portion of the total population that is women (20/50). The reason 25 is a marginal frequency is because it represents the portion of those sampled who favor football (25/50). Note: The values that make up the body of the table (e.g., 20, 8, 2) are called joint frequencies. Conditional Distributions in Two-Way Tables The distinction between a marginal distribution and a conditional distribution is that the focus is on only a particular subset of the population (not the entire population). For example, in the table, if we focused only on the subpopulation of women who prefer football, then we could calculate the conditional distributions as shown in the two-way table below. Football Basketball Tennis Total Men Women Total Table 1.6 20 5 25 8 7 15 2 8 10 30 20 50 To find the first sub-population of women who prefer football, read the value at the intersection of the Women row and Football column which is 5. Then, divide this by the total population of football players which is 25. So, the subpopulation of football players who are women is 5/25 which is.2. Similarly, to find the subpopulation of women who play football, use the value of 5 which is the number of women who play football. Then, divide this by the total population of women which is 20. So, the subpopulation of women who play football is 5/20 which is.25. Presenting Data After deciding which graph best represents your data, you may need to present your statistical data to a class or other group in an oral report or multimedia presentation. When giving an oral presentation, you must be prepared to explain exactly how you collected or calculated the data, as well as why you chose the categories, scales, and types of graphs that you are showing. Although you may have made numerous graphs of your data, be sure to use only those that actually demonstrate the stated intentions of your statistical study. While preparing your presentation, be sure that all colors, text, and scales are visible to the entire audience. Finally, make sure to allow time |
for your audience to ask questions and be prepared to answer them. Example 1.11 Suppose the guidance counselors at De Anza and Foothill need to make an oral presentation of the student data presented in Figures 1.5 and 1.6. Under what context should they choose to display the pie graph? When might they choose the bar graph? For each graph, explain which features they should point out and the potential display problems that might exist. Solution 1.11 The guidance counselors should use the pie graph if the desired information is the percentage of each school’s enrollment. They should use the bar graph if knowing the exact numbers of students and the relative sizes of each category at each school are important points to be made. For the pie graph, they should point out which color represents part-time students and which represents full-time students. They should also be sure that the numbers and colors are visible when displayed. For the bar graph, they should point out the scale and the total numbers for each category, and they should be sure that the numbers, colors, and scale marks are all displayed clearly. 20 Chapter 1 | Sampling and Data 1.11 Suppose you were asked to give an oral presentation of the data graphed in the pie chart in Figure 1.11(b). What features would you point out on the graph? What potential display problems with the graph should you check before giving your presentation? Sampling Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Each method has pros and cons. In a simple random sample, each group has the same chance of being selected. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 |
names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in Table 1.7. ID Name ID Name ID Name 00 01 02 03 Anselmo Bautista Bayani 11 12 13 King Legeny Lisa 22 23 24 Roquero Roth Rowell Cheng 14 Lundquist 25 Salangsang 04 Cuarismo 15 Macierz 26 Slade 05 Cuningham 16 Motogawa 27 Stratcher 06 07 08 09 10 Fontecha 17 Okimoto Hong Hoobler Jiao Khan 18 19 20 21 Patel Price Quizon Reyes 28 29 30 31 Tallai Tran Wai Wood Table 1.7 Class Roster Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. The most common random number generators are five digit numbers where each digit is a unique number from 0 to 9. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows:.94360,.99832,.14669,.51470,.40581,.73381,.04399. Lisa reads two-digit groups until she has chosen three class members (That is, she reads.94360 as the groups 94, 43, 36, 60.) Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers. The table below shows how Lisa reads two-digit numbers form each random number. Each two-digit number in the table would represent each student in the roster above in Table 1.7. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 21 Random number Numbers read by Lisa.94360.99832.14669.51470.40581.73381.04399 94 99 14 51 40 73 04 43 98 46 14 05 33 39 36 83 66 47 58 38 39 60 32 69 70 81 81 99 Table 1.8 Lisa randomly generated the decimals in the Random Number column. She then used each consecutive number in each decimal to make the numbers she read. Some of the read numbers correspond with the ID numbers given to the students in her class (e.g., 14 = Lundquist in Table |
1.7) The random numbers.94360 and.99832 do not contain appropriate two digit numbers. However the third random number,.14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Lundquist, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Lundquist, Cuningham, and Cuarismo. To generate random numbers perform the following steps: • Press MATH. • Arrow over to PRB. • Press 5:randInt(0, 30). • Press ENTER for the first random number. • Press ENTER two more times for the other two random numbers. If there is a repeat press ENTER again. Note—randInt(0, 30, 3) will generate three random numbers. Figure 1.12 Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample. To choose a stratified sample, divide the population into groups called strata and then the sample is selected by picking the 22 Chapter 1 | Sampling and Data same number of values from each strata until the desired sample size is reached. For example, you could stratify (group) your high school student population by year (freshmen, sophomore, juniors, and seniors) and then choose a proportionate simple random sample from each stratum (each year) to get a stratified random sample. To choose a simple random sample from each year, number each student of the first year, number each student of the second year, and do the same for the remaining years. Then use simple random sampling to choose proportionate numbers of students from the first year and do the same for each of the remaining years. Those numbers picked from the first year, picked from the second year, and so on represent the students who make up the stratified sample. To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four homeroom classes from your student population, the four classes make up the cluster sample. Each class is a cluster. Number each cluster, and then choose four different numbers using random sampling. All the students |
of the four classes with those numbers are the cluster sample. So, unlike a stratified example, a cluster sample may not contain an equal number of randomly chosen students from each class. A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others. Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased. They may favor a certain group. It is better for the person conducting the survey to select the sample respondents. When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error. In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error. In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others. Remember, each member of the population should have an equally likely chance of being chosen. When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied. For instance, if a survey of all students is conducted only during noon lunchtime hours is biased. This is because the students who do not have a noon lunchtime would not be included. Critical Evaluation We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include the following: • Problems with samples: —A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not reliable. Reliability in statistical measures must also be considered when analyzing data. Reliability refers to the consistency of a measure. A measure is reliable when the same results are produced given the same circumstances. • Self-selected samples—Responses only by people who choose to respond |
, such as internet surveys, are often unreliable. • Sample size issues—: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples include crash testing cars or medical testing for rare conditions. • Undue influence—: collecting data or asking questions in a way that influences the response. • Non-response or refusal of subject to participate: —The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can affect the results. • Causality: —A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable. • Self-funded or self-interest studies—: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done. • Misleading use of data—: These can be improperly displayed graphs, incomplete data, or lack of context. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 23 As a class, determine whether or not the following samples are representative. If they are not, discuss the reasons. 1. To find the average GPA of all students in a high school, use all honor students at the university as the sample. 2. To find out the most popular cereal among young people under the age of 10, stand outside a large supermarket for three hours and speak to every twentieth child under age 10 who enters the supermarket. 3. To find the average annual income of all adults in the United States, sample U.S. congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every U.S. congressman in the cluster. 4. To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you. 5. To determine the average cost of a two-day stay in a hospital in Massachusetts |
, survey 100 hospitals across the state using simple random sampling. Example 1.12 A study is done to determine the average tuition that private high school students pay per semester. Each student in the following samples is asked how much tuition he or she paid for the fall semester. What is the type of sampling in each case? a. A sample of 100 high school students is taken by organizing the students’ names by classification (freshman, sophomore, junior, or senior) and then selecting 25 students from each. b. A random number generator is used to select a student from the alphabetical listing of all high school students in the fall semester. Starting with that student, every 50th student is chosen until 75 students are included in the sample. c. A completely random method is used to select 75 students. Each high school student in the fall semester has the same probability of being chosen at any stage of the sampling process. d. The freshman, sophomore, junior, and senior years are numbered one, two, three, and four, respectively. A random number generator is used to pick two of those years. All students in those two years are in the sample. e. An administrative assistant is asked to stand in front of the library one Wednesday and to ask the first 100 undergraduate students he encounters what they paid for tuition the fall semester. Those 100 students are the sample. Solution 1.12 a. stratified, b. systematic, c. simple random, d. cluster, e. convenience 1.12 You are going to use the random number generator to generate different types of samples from the data. This table displays six sets of quiz scores (each quiz counts 10 points) for an elementary statistics class. 24 Chapter 1 | Sampling and Data #1 #2 #3 #4 #5 #6 5 10 10 10 8 9 7 8 7 8 10 9 8 10 9 9 10 9 8 10 9 8 6 9 5 10 9 10 Table 1.9 Scores for quizzes #1-6 for 10 students in a statistics class. Each quiz is out of 10 points. Instructions: Use the Random Number Generator to pick samples. 1. Create a stratified sample by column. Pick three quiz scores randomly from each column. a. Number each row one through 10. b. On your calculator, press Math and arrow over to PRB. c. For column 1, Press 5:randInt( and enter 1,10). Press ENTER. Record the number. Press ENTER 2 more times (even the repeats). Record |
these numbers. Record the three quiz scores in column one that correspond to these three numbers. d. Repeat for columns two through six. e. These 18 quiz scores are a stratified sample. 2. Create a cluster sample by picking two of the columns. Use the column numbers: one through six. a. Press MATH and arrow over to the PRB function. b. Press 5:randInt (“and then enter “1,6). Press ENTER. c. Record the number the calculator displays into the first column. Then, press ENTER. d. Record the next number the calculator displays into the second column. e. Repeat steps (c) and (d) nine more times until there are a total of 20 quiz scores for the cluster sample. 3. Create a simple random sample of 15 quiz scores. a. Use the numbering one through 60. b. Press MATH. Arrow over to PRB. Press 5:randInt(1, 60). c. Press ENTER 15 times and record the numbers. d. Record the quiz scores that correspond to these numbers. e. These 15 quiz scores are the systematic sample. 4. Create a systematic sample of 12 quiz scores. a. Use the numbering one through 60. b. Press MATH. Arrow over to PRB. Press 5:randInt(1, 60). c. Press ENTER. Record the number and the first quiz score. From that number, count ten quiz scores and This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 25 record that quiz score. Keep counting ten quiz scores and recording the quiz score until you have a sample of 12 quiz scores. You may wrap around (go back to the beginning). Example 1.13 Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). a. A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group of boys aged 11 to 12, and three players from a group of boys aged 13 to 14 to form a recreational soccer team. b. A pollster interviews all human resource personnel in five different high tech companies. c. A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers. d. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital. e. A high school counselor uses a computer to |
generate 50 random numbers and then picks students whose names correspond to the numbers. f. A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on average. Solution 1.13 a. stratified b. cluster c. stratified d. systematic e. simple random f. convenience 1.13 Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). A high school principal polls 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors regarding policy changes for after school activities. If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural. Example 1.14 Suppose ABC high school has 10,000 upperclassman (junior and senior level) students (the population). We are interested in the average amount of money a upperclassmen spends on books in the fall term. Asking all 10,000 upperclassmen is an almost impossible task. Suppose we take two different samples. First, we use convenience sampling and survey ten upperclassman students from a first term organic chemistry class. Many of these students are taking first term calculus in addition to the organic chemistry class. The amount of money they spend on books is as follows: $128, $87, $173, $116, $130, $204, $147, $189, $93, $153. The second sample is taken using a list of seniors who take P.E. classes and taking every fifth seniors on the list, for a total of ten seniors. They spend the following: $50, $40, $36, $15, $50, $100, $40, $53, $22, $22. It is unlikely that any student is in both samples. 26 Chapter 1 | Sampling and Data a. Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population? Solution 1.14 a. No. The first sample probably consists of science-oriented students. Besides the chemistry course, some of them are also taking first-term calculus. Books for these classes tend to be expensive. Most of these students are, more than likely, paying more than the average part-time student for their books. The |
second sample is a group of senior citizens who are, more than likely, taking courses for health and interest. The amount of money they spend on books is probably much less than the average parttime student. Both samples are biased. Also, in both cases, not all students have a chance to be in either sample. b. Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population? Solution 1.14 b. No. For these samples, each member of the population did not have an equally likely chance of being chosen. Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. We assume that these are the only disciplines in which part-time students at ABC College are enrolled and that an equal number of part-time students are enrolled in each of the disciplines. Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he or she has a corresponding number. The students spend the following amounts: $180, $50, $150, $85, $260, $75, $180, $200, $200, $150. c. Is the sample biased? Solution 1.14 c. The sample is unbiased, but a larger sample would be recommended to increase the likelihood that the sample will be close to representative of the population. However, for a biased sampling technique, even a large sample runs the risk of not being representative of the population. Students often ask if it is good enough to take a sample, instead of surveying the entire population. If the survey is done well, the answer is yes. 1.14 A local radio station has a fan base of 20,000 listeners. The station wants to know if its audience would prefer more music or more talk shows. Asking all 20,000 listeners is an almost impossible task. The station uses convenience sampling and surveys the first 200 people they meet at one of the station’s music concert events. Twenty-four people said they’d prefer more talk shows, and 176 people said they’d prefer more music. Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population? Variation in Data Variation is present in any set of data. For example, 16-ounce |
cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage: 15.8, 16.1, 15.2, 14.8, 15.8, 15.9, 16.0, 15.5. Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range. Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 27 This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy. Variation in Samples It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their high school sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however. Think about what contributes to making Doreen’s and Jung’s samples different. If Doreen and Jung took larger samples, that is, the number of data values is increased, their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This is called sampling variability. In other words, it refers to how much a statistic varies from sample to sample within a population. The larger the sample size, the smaller the variability between samples will be. So, the large sample size makes for a better, more reliable statistic. Size of a Sample The size of a sample (often |
called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200–1,500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals. Be aware that many large samples are biased. For example, internet surveys are invariably biased, because people choose to respond or not. 28 Chapter 1 | Sampling and Data Divide into groups of two, three, or four. Your instructor will give each group one six-sided die. Try this experiment twice. Roll one fair die (six-sided) 20 times. Record the number of ones, twos, threes, fours, fives, and sixes you get in Table 1.10 and Table 1.11 (frequency is the number of times a particular face of the die occurs) Face on Die Frequency 1 2 3 4 5 6 Table 1.10 First Experiment (20 rolls) Face on Die Frequency 1 2 3 4 5 6 Table 1.11 Second Experiment (20 rolls) Did the two experiments have the same results? Probably not. If you did the experiment a third time, do you expect the results to be identical to the first or second experiment? Why or why not? Which experiment had the correct results? They both did. The job of the statistician is to see through the variability and draw appropriate conclusions. 1.3 | Frequency, Frequency Tables, and Levels of Measurement Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible. Answers and Rounding Off A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results, if possible. If it becomes necessary to round off intermediate results, carry them to at least twice as many decimal places as the final answer. Expect that some of your answers will vary from the text due to rounding errors. It is not necessary to reduce most fractions in this course. Especially in Probability Topics, the chapter on probability, it This OpenStax book is available for free at http://cnx.org/ |
content/col30309/1.8 Chapter 1 | Sampling and Data 29 is more helpful to leave an answer as an unreduced fraction. Levels of Measurement The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are as follows (from lowest to highest level): • Nominal scale level • Ordinal scale level • Interval scale level • Ratio scale level Data that is measured using a nominal scale is qualitative (categorical). Categories, colors, names, labels, and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful. Smartphone companies are another example of nominal scale data. The data are the names of the companies that make smartphones, but there is no agreed upon order of these brands, even though people may have personal preferences. Nominal scale data cannot be used in calculations. Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure differences between the data. Another example of using the ordinal scale is a cruise survey where the responses to questions about the cruise are excellent, good, satisfactory, and unsatisfactory. These responses are ordered from the most desired response to the least desired. But the differences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations. Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a difference between data. The differences between interval scale data can be measured though the data does not have a starting point. Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40° is equal to 100° minus 60°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like –10 °F and –15 |
°C exist and are colder than 0. Interval level data can be used in calculations, but one type of comparison cannot be done. 80 °C is not four times as hot as 20 °C (nor is 80 °F four times as hot as 20 °F). There is no meaning to the ratio of 80 to 20 (or four to one). Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data, but it has a 0 point and ratios can be calculated. For example, four multiple choice statistics final exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams are machine-graded. The data can be put in order from lowest to highest 20, 68, 80, 92. The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20. Frequency Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3. Table 1.12 lists the different data values in ascending order and their frequencies. DATA VALUE FREQUENCY 2 3 3 5 Table 1.12 Frequency Table of Student Work Hours 30 Chapter 1 | Sampling and Data DATA VALUE FREQUENCY 4 5 6 7 3 6 2 1 Table 1.12 Frequency Table of Student Work Hours A frequency is the number of times a value of the data occurs. According to Table 1.12, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample. A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample, in this case, 20. Relative frequencies can be written as fractions, percents, or decimals. DATA VALUE FREQUENCY RELATIVE FREQUENCY or.15 or.25 |
or.15 or.30 or.10 or.05 3 20 5 20 3 20 6 20 2 20 1 20 Table 1.13 Frequency Table of Student Work Hours with Relative Frequencies The sum of the values in the relative frequency column of Table 1.13 is 20 20, or 1. Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table 1.14. In the first row, the cumulative frequency is simply.15 because it is the only one. In the second row, the relative frequency was.25, so adding that to.15, we get a relative frequency of.40. Continue adding the relative frequencies in each row to get the rest of the column. DATA VALUE FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 2 3 or.15 3 20.15 Table 1.14 Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 31 DATA VALUE FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY or.25 or.15 or.30 or.10 or.05 5 20 3 20 6 20 2 20 1 20.15 +.25 =.40.40 +.15 =.55.55 +.30 =.85.85 +.10 =.95.95 +.05 = 1.00 Table 1.14 Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated. NOTE Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one. Table 1.15 represents the heights, in inches, of a sample of 100 male semiprofessional soccer players. HEIGHTS (INCHES) FREQUENCY RELATIVE FREQUENCY 59.95–61.95 61.95–63.95 63.95–65.95 65.95–67.95 67.95–69.95 69.95–71.95 71.95–73.95 73.95–75. |
95 5 3 15 40 17 12 7 1 5 100 3 100 15 100 40 100 17 100 12 100 7 100 1 100 =.05 =.03 =.15 =.40 =.17 =.12 =.07 =.01 CUMULATIVE RELATIVE FREQUENCY.05.05 +.03 =.08.08 +.15 =.23.23 +.40 =.63.63 +.17 =.80.80 +.12 =.92.92 +.07 =.99.99 +.01 = 1.00 Total = 100 Total = 1.00 Table 1.15 Frequency Table of Soccer Player Height 32 Chapter 1 | Sampling and Data The data in this table have been grouped into the following intervals: • 59.95–61.95 inches • 61.95–63.95 inches • 63.95–65.95 inches • 65.95–67.95 inches • 67.95–69.95 inches • 69.95–71.95 inches • 71.95–73.95 inches • 73.95–75.95 inches NOTE This example is used again in Descriptive Statistics, where the method used to compute the intervals will be explained. In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints. Example 1.15 From Table 1.15, find the percentage of heights that are less than 65.95 inches. Solution 1.15 If you look at the first, second, and third rows, the heights are all less than 65.95 inches. There are 5 + 3 + 15 = 23 players whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then 23 100 or 23 percent. This percentage is the cumulative relative frequency entry in the third row |
. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 33 1.15 Table 1.16 shows the amount, in inches, of annual rainfall in a sample of towns. Rainfall (Inches) Frequency Relative Frequency Cumulative Relative Frequency 2.95–4.97 4.97–6.99 6.99–9.01 9.01–11.03 11.03–13.05 13.05–15.07 6 7 15 8 9 5 =.12 =.14 =.30 =.16 =.18 =.10 6 50 7 50 15 50 8 50 9 50 5 50.12.12 +.14 =.26.26 +.30 =.56.56 +.16 =.72.72 +.18 =.90.90 +.10 = 1.00 Total = 50 Total = 1.00 Table 1.16 From Table 1.16, find the percentage of rainfall that is less than 9.01 inches. Example 1.16 From Table 1.15, find the percentage of heights that fall between 61.95 and 65.95 inches. Solution 1.16 Add the relative frequencies in the second and third rows:.03 +.15 =.18 or 18 percent. 1.16 From Table 1.16, find the percentage of rainfall that is between 6.99 and 13.05 inches. Example 1.17 Use the heights of the 100 male semiprofessional soccer players in Table 1.15. Fill in the blanks and check your answers. a. The percentage of heights that are from 67.95–71.95 inches is ________. b. The percentage of heights that are from 67.95–73.95 inches is ________. c. The percentage of heights that are more than 65.95 inches is ________. d. The number of players in the sample who are between 61.95 and 71.95 inches tall is ________. e. What kind of data are the heights? 34 Chapter 1 | Sampling and Data f. Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players. Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the |
relative frequency for the current row. Solution 1.17 a. 29 percent b. 36 percent c. 77 percent d. 87 e. quantitative continuous f. get rosters from each team and choose a simple random sample from each 1.17 From Table 1.16, find the number of towns that have rainfall between 2.95 and 9.01 inches. In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions: 1. What percentage of the students in your class have no siblings? 2. What percentage of the students have from one to three siblings? 3. What percentage of the students have fewer than three siblings? Example 1.18 Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. Table 1.17 was produced. DATA FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 3 4 3 1 3 19 1 19.1579.2105 Table 1.17 Frequency of Commuting Distances This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 35 DATA FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 5 7 10 12 13 15 18 20 3 2 3 2 1 1 1 1 3 19 2 19 4 19 2 19 1 19 1 19 1 19 1 19.1579.2632.4737.7895.8421.8948.9474 1.0000 Table 1.17 Frequency of Commuting Distances a. Is the table correct? If it is not correct, what is wrong? b. True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections. c. What fraction of the people surveyed commute five or seven miles? d. What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)? Solution 1.18 a. No. The frequency column sums to |
18, not 19. Not all cumulative relative frequencies are correct. b. False. The frequency for three miles should be one; for two miles (left out), two. The cumulative relative frequency column should read 1052, 01579, 02105, 03684, 04737, 06316, 07368, 07895, 08421, 09474, 1.0000. c. d. 5 19 7 19, 12 19, 7 19 1.18 Table 1.16 represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year? 36 Chapter 1 | Sampling and Data Example 1.19 Table 1.18 contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012. Year Total Number of Deaths 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Total Table 1.18 231 21,357 11,685 33,819 228,802 88,003 6,605 712 88,011 1,790 320,120 21,953 768 823,856 Answer the following questions: a. What is the frequency of deaths measured from 2006 through 2009? b. What percentage of deaths occurred after 2009? c. What is the relative frequency of deaths that occurred in 2003 or earlier? d. What is the percentage of deaths that occurred in 2004? e. What kind of data are the numbers of deaths? f. The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers? Solution 1.19 a. 97,118 (11.8 percent) b. 41.6 percent c. 67,092/823,356 or 0.081 or 8.1 percent d. 27.8 percent e. quantitative discrete f. quantitative continuous This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 37 1.19 Table 1.19 contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994–2011. Year Total Number of Crashes Year Total Number of Crashes 1994 36,254 1995 37,241 1996 37,494 1997 37,324 1998 37,107 1999 37,140 |
2000 37,526 2001 37,862 2002 38,491 2003 38,477 Table 1.19 2004 38,444 2005 39,252 2006 38,648 2007 37,435 2008 34,172 2009 30,862 2010 30,296 2011 29,757 Total 653,782 Answer the following questions: a. What is the frequency of deaths measured from 2000 through 2004? b. What percentage of deaths occurred after 2006? c. What is the relative frequency of deaths that occurred in 2000 or before? d. What is the percentage of deaths that occurred in 2011? e. What is the cumulative relative frequency for 2006? Explain what this number tells you about the data. 1.4 | Experimental Design and Ethics Does aspirin reduce the risk of heart attacks? Is one brand of fertilizer more effective at growing roses than another? Is fatigue as dangerous to a driver as speeding? Questions like these are answered using randomized experiments. In this module, you will learn important aspects of experimental design. Proper study design ensures the production of reliable, accurate data. The purpose of an experiment is to investigate the relationship between two variables. In an experiment, there is the explanatory variable which affects the response variable. In a randomized experiment, the researcher manipulates the explanatory variable and then observes the response variable. Each value of the explanatory variable used in an experiment is called a treatment. You want to investigate the effectiveness of vitamin E in preventing disease. You recruit a group of subjects and ask them if they regularly take vitamin E. You notice that the subjects who take vitamin E exhibit better health on average than those who do not. Does this prove that vitamin E is effective in disease prevention? It does not. There are many differences between the two groups compared in addition to vitamin E consumption. People who take vitamin E regularly often take other steps to improve their health: exercise, diet, other vitamin supplements. Any one of these factors could be influencing health. As described, this study does not prove that vitamin E is the key to disease prevention. Additional variables that can cloud a study are called lurking variables. In order to prove that the explanatory variable is causing a change in the response variable, it is necessary to isolate the explanatory variable. The researcher must design her experiment in such a way that there is only one difference between groups being compared: the planned treatments. This is accomplished by the random assignment of experimental units to treatment groups. When subjects are assigned treatments 38 Chapter 1 | Sampling and Data randomly, all of the potential lurking variables are spread equally |
among the groups. At this point the only difference between groups is the one imposed by the researcher. Different outcomes measured in the response variable, therefore, must be a direct result of the different treatments. In this way, an experiment can prove a cause-and-effect connection between the explanatory and response variables. Confounding occurs when the effects of multiple factors on a response cannot be separated, for instance, if a student guesses on the even-numbered questions on an exam and sits in a favorite spot on exam day. Why does the student get a high test scores on the exam? It could be the increased study time or sitting in the favorite spot or both. Confounding makes it difficult to draw valid conclusions about the effect of each factor on the outcome. The way around this is to test several outcomes with one method (treatment). This way, we know which treatment really works. The power of suggestion can have an important influence on the outcome of an experiment. Studies have shown that the expectation of the study participant can be as important as the actual medication. In one study of performance-enhancing substances, researchers noted the following: Results showed that believing one had taken the substance resulted in [performance] times almost as fast as those associated with consuming the substance itself. In contrast, taking the substance without knowledge yielded no significant performance increment.[1] When participation in a study prompts a physical response from a participant, it is difficult to isolate the effects of the explanatory variable. To counter the power of suggestion, researchers set aside one treatment group as a control group. This group is given a placebo treatment, a treatment that cannot influence the response variable. The control group helps researchers balance the effects of being in an experiment with the effects of the active treatments. Of course, if you are participating in a study and you know that you are receiving a pill that contains no actual medication, then the power of suggestion is no longer a factor. Blinding in a randomized experiment designed to reduce bias by hiding information. When a person involved in a research study is blinded, he does not know who is receiving the active treatment(s) and who is receiving the placebo treatment. A double-blind experiment is one in which both the subjects and the researchers involved with the subjects are blinded. Sometimes, it is neither possible nor ethical for researchers to conduct experimental studies. For example, if you want to investigate whether malnutrition affects elementary school performance in children, it would not be appropriate to assign an experimental group to be malnourished. In these cases, observational studies or |
surveys may be used. In an observational study, the researcher does not directly manipulate the independent variable. Instead, he or she takes recordings and measurements of naturally occurring phenomena. By sorting these data into control and experimental conditions, the relationship between the dependent and independent variables can be drawn. In a survey, a researcher’s measurements consist of questionnaires that are answered by the research participants. Example 1.20 Researchers want to investigate whether taking aspirin regularly reduces the risk of a heart attack. 400 men between the ages of 50 and 84 are recruited as participants. The men are divided randomly into two groups: one group will take aspirin, and the other group will take a placebo. Each man takes one pill each day for three years, but he does not know whether he is taking aspirin or the placebo. At the end of the study, researchers count the number of men in each group who have had heart attacks. Identify the following values for this study: population, sample, experimental units, explanatory variable, response variable, treatments. Solution 1.20 The population is men aged 50 to 84. The sample is the 400 men who participated. The experimental units are the individual men in the study. The explanatory variable is oral medication. The treatments are aspirin and a placebo. The response variable is whether a subject had a heart attack. 1. McClung, M. and Collins, D. (2007 June). "Because I know it will!" Placebo effects of an ergogenic aid on athletic performance. Journal of Sport & Exercise Psychology, 29(3), 382-94. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 39 Example 1.21 The Smell & Taste Treatment and Research Foundation conducted a study to investigate whether smell can affect learning. Subjects completed mazes multiple times while wearing masks. They completed the pencil and paper mazes three times wearing floral-scented masks, and three times with unscented masks. Participants were assigned at random to wear the floral mask during the first three trials or during the last three trials. For each trial, researchers recorded the time it took to complete the maze and the subject’s impression of the mask’s scent: positive, negative, or neutral. a. Describe the explanatory and response variables in this study. b. What are the treatments? c. d. Identify any lurking variables that could interfere with this study. Is it possible to |
use blinding in this study? Solution 1.21 a. The explanatory variable is scent, and the response variable is the time it takes to complete the maze. b. There are two treatments: a floral-scented mask and an unscented mask. c. All subjects experienced both treatments. The order of treatments was randomly assigned so there were no differences between the treatment groups. Random assignment eliminates the problem of lurking variables. d. Subjects will clearly know whether they can smell flowers or not, so subjects cannot be blinded in this study. Researchers timing the mazes can be blinded, though. The researcher who is observing a subject will not know which mask is being worn. Example 1.22 A researcher wants to study the effects of birth order on personality. Explain why this study could not be conducted as a randomized experiment. What is the main problem in a study that cannot be designed as a randomized experiment? Solution 1.22 The explanatory variable is birth order. You cannot randomly assign a person’s birth order. Random assignment eliminates the impact of lurking variables. When you cannot assign subjects to treatment groups at random, there will be differences between the groups other than the explanatory variable. 1.22 You are concerned about the effects of texting on driving performance. Design a study to test the response time of drivers while texting and while driving only. How many seconds does it take for a driver to respond when a leading car hits the brakes? a. Describe the explanatory and response variables in the study. b. What are the treatments? c. What should you consider when selecting participants? d. Your research partner wants to divide participants randomly into two groups: one to drive without distraction and one to text and drive simultaneously. Is this a good idea? Why or why not? e. Identify any lurking variables that could interfere with this study. f. How can blinding be used in this study? 40 Ethics Chapter 1 | Sampling and Data The widespread misuse and misrepresentation of statistical information often gives the field a bad name. Some say that “numbers don’t lie,” but the people who use numbers to support their claims often do. A recent investigation of famous social psychologist, Diederik Stapel, has led to the retraction of his articles from some of the world’s top journals including, Journal of Experimental Social Psychology, Social Psychology, Basic and Applied Social Psychology, British Journal of Social Psychology, and the magazine Science. Diederik Stapel is a former professor |
at Tilburg University in the Netherlands. Over the past two years, an extensive investigation involving three universities where Stapel has worked concluded that the psychologist is guilty of fraud on a colossal scale. Falsified data taints over 55 papers he authored and 10 Ph.D. dissertations that he supervised. Stapel did not deny that his deceit was driven by ambition. But it was more complicated than that, he told me. He insisted that he loved social psychology but had been frustrated by the messiness of experimental data, which rarely led to clear conclusions. His lifelong obsession with elegance and order, he said, led him to concoct results that journals found attractive. “It was a quest for aesthetics, for beauty—instead of the truth,” he said. He described his behavior as an addiction that drove him to carry out acts of increasingly daring fraud.[2] The committee investigating Stapel concluded that he is guilty of several practices including • creating datasets, which largely confirmed the prior expectations, • altering data in existing datasets, • changing measuring instruments without reporting the change, and • misrepresenting the number of experimental subjects. Clearly, it is never acceptable to falsify data the way this researcher did. Sometimes, however, violations of ethics are not as easy to spot. Researchers have a responsibility to verify that proper methods are being followed. The report describing the investigation of Stapel’s fraud states that, “statistical flaws frequently revealed a lack of familiarity with elementary statistics.”[3] Many of Stapel’s co-authors should have spotted irregularities in his data. Unfortunately, they did not know very much about statistical analysis, and they simply trusted that he was collecting and reporting data properly. Many types of statistical fraud are difficult to spot. Some researchers simply stop collecting data once they have just enough to prove what they had hoped to prove. They don’t want to take the chance that a more extensive study would complicate their lives by producing data contradicting their hypothesis. Professional organizations, like the American Statistical Association, clearly define expectations for researchers. There are even laws in the federal code about the use of research data. When a statistical study uses human participants, as in medical studies, both ethics and the law dictate that researchers should be mindful of the safety of their research subjects. The U.S. Department of Health and Human Services oversees federal regulations of research studies with the aim of protecting participants. When a university or other research institution engages in research, it must ensure |
the safety of all human subjects. For this reason, research institutions establish oversight committees known as Institutional Review Boards (IRB). All planned studies must be approved in advance by the IRB. Key protections that are mandated by law include the following: • Risks to participants must be minimized and reasonable with respect to projected benefits. • Participants must give informed consent. This means that the risks of participation must be clearly explained to the subjects of the study. Subjects must consent in writing, and researchers are required to keep documentation of their consent. • Data collected from individuals must be guarded carefully to protect their privacy. These ideas may seem fundamental, but they can be very difficult to verify in practice. Is removing a participant’s name from the data record sufficient to protect privacy? Perhaps the person’s identity could be discovered from the data that remains. What happens if the study does not proceed as planned and risks arise that were not anticipated? When is informed consent really necessary? Suppose your doctor wants a blood sample to check your cholesterol level. Once the sample has been tested, you expect the lab to dispose of the remaining blood. At that point the blood becomes biological waste. Does a 2. Bhattacharjee, Y. (2013, April 26). The mind of a con man. The New York Times. Retrieved from http://www.nytimes.com/2013/04/28/magazine/diederik-stapels-audacious-academic-fraud.html?_r=3&src=dayp&. 3. Tillburg University. (2012, Nov. 28). Flawed science: the fraudulent research practices of social psychologist Diederik Stapel. Retrieved from https://www.tilburguniversity.edu/upload/3ff904d7-547b-40ae-85febea38e05a34a_Final%20report%20Flawed%20Science.pdf. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 41 researcher have the right to take it for use in a study? It is important that students of statistics take time to consider the ethical questions that arise in statistical studies. is fraud in statistical studies? You might be surprised—and disappointed. There is a website How prevalent (http://openstaxcollege.org/l/40introone) dedicated to catalog |
ing retractions of study articles that have been proven fraudulent. A quick glance will show that the misuse of statistics is a bigger problem than most people realize. Vigilance against fraud requires knowledge. Learning the basic theory of statistics will empower you to analyze statistical studies critically. Example 1.23 Describe the unethical behavior in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected. A researcher is collecting data in a community. a. She selects a block where she is comfortable walking because she knows many of the people living on the street. b. No one seems to be home at four houses on her route. She does not record the addresses and does not return at a later time to try to find residents at home. c. She skips four houses on her route because she is running late for an appointment. When she gets home, she fills in the forms by selecting random answers from other residents in the neighborhood. Solution 1.23 a. By selecting a convenient sample, the researcher is intentionally selecting a sample that could be biased. Claiming that this sample represents the community is misleading. The researcher needs to select areas in the community at random. b. c. Intentionally omitting relevant data will create bias in the sample. Suppose the researcher is gathering information about jobs and child care. By ignoring people who are not home, she may be missing data from working families that are relevant to her study. She needs to make every effort to interview all members of the target sample. It is never acceptable to fake data. Even though the responses she uses are real responses provided by other participants, the duplication is fraudulent and can create bias in the data. She needs to work diligently to interview everyone on her route. 1.23 Describe the unethical behavior, if any, in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected. A study is commissioned to determine the favorite brand of fruit juice among teens in California. a. The survey is commissioned by the seller of a popular brand of apple juice. b. There are only two types of juice included in the study: apple juice and cranberry juice. c. Researchers allow participants to see the brand of juice as samples are poured for a taste test. d. Twenty-five percent of participants prefer Brand X, 33 percent prefer Brand Y and 42 percent have no preference between the two brands. Brand X references the study in a commercial saying “Most teens like Brand X |
as much as or more than Brand Y.” 1.5 | Data Collection Experiment 42 Chapter 1 | Sampling and Data 1.1 Data Collection Experiment Student Learning Outcomes • The student will demonstrate the systematic sampling technique. • The student will construct relative frequency tables. • The student will interpret results and their differences from different data groupings. Movie Survey Get a class roster/list. Randomly mark a person’s name, and then mark every fourth name on the list until you get 12 names. You may have to go back to the start of the list. For each name marked, record the number of movies they saw at the theater last month. Order the Data Complete the two relative frequency tables below using your class data. Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0 1 2 3 4 5 6 7+ Table 1.20 Frequency of Number of Movies Viewed Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0–1 2–3 4–5 6–7+ Table 1.21 Frequency of Number of Movies Viewed 1. Using the tables, find the percent of data that is at most two. Which table did you use and why? 2. Using the tables, find the percent of data that is at most three. Which table did you use and why? 3. Using the tables, find the percent of data that is more than two. Which table did you use and why? 4. Using the tables, find the percent of data that is more than three. Which table did you use and why? Discussion Questions This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 43 1. 2. Is one of the tables more correct than the other? Why or why not? In general, how could you group the data differently? Are there any advantages to either way of grouping the data? 3. Why did you switch between tables, if you did, when answering the question above? 1.6 | Sampling Experiment 44 Chapter 1 | Sampling and Data 1.2 Sampling Experiment Student Learning Outcomes • The student will demonstrate the simple random, systematic, stratified, and cluster sampling techniques. • The student will explain the details of each procedure used. In this lab, you will be asked to pick several random samples of restaurants. In each case, describe your procedure briefly, including how you might have used the random number generator, and then list the restaurants in the |
. __________ 2. __________ 7. __________ 12. __________ 3. __________ 8. __________ 13. __________ 4. __________ 9. __________ 14. __________ 5. __________ 10. __________ 15. __________ Table 1.23 A Systematic Sample Pick a systematic sample of 15 restaurants. 1. Describe your procedure. 2. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 2. __________ 7. __________ 12. __________ 3. __________ 8. __________ 13. __________ 4. __________ 9. __________ 14. __________ 5. __________ 10. __________ 15. __________ Table 1.24 A Stratified Sample Pick a stratified sample, by city, of 20 restaurants. Use 25 percent of the restaurants from each stratum. Round to the nearest whole number. 1. Describe your procedure. 2. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 16. __________ 2. __________ 7. __________ 12. __________ 17. __________ 3. __________ 8. __________ 13. __________ 18. __________ 4. __________ 9. __________ 14. __________ 19. __________ 5. __________ 10. __________ 15. __________ 20. __________ Table 1.25 A Stratified Sample Pick a stratified sample, by entree cost, of 21 restaurants. Use 25 percent of the restaurants from each stratum. Round 46 Chapter 1 | Sampling and Data to the nearest whole number. 1. Describe your procedure. 2. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 16. __________ 2. __________ 7. __________ 12. __________ 17. __________ 3. __________ 8. __________ 13. __________ 18. __________ 4. __________ 9. __________ 14. __________ 19. __________ 5. __________ 10. __________ 15. __________ 20. __________ 21. __________ Table 1.26 A Cluster Sample Pick a cluster sample of restaurants from two |
cities. The number of restaurants will vary. 1. Describe your procedure. 2. Complete the table with your sample. 1. ________ 6. ________ 11. ________ 16. ________ 21. ________ 2. ________ 7. ________ 12. ________ 17. ________ 22. ________ 3. ________ 8. ________ 13. ________ 18. ________ 23. ________ 4. ________ 9. ________ 14. ________ 19. ________ 24. ________ 5. ________ 10. ________ 15. ________ 20. ________ 25. ________ Table 1.27 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 47 KEY TERMS average also called mean; a number that describes the central tendency of the data blinding not telling participants which treatment a subject is receiving categorical variable variables that take on values that are names or labels cluster sampling a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters; every individual in the chosen clusters is included in the sample continuous random variable a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV control group a group in a randomized experiment that receives an inactive treatment but is otherwise managed exactly as the other groups convenience sampling a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data cumulative relative frequency the term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value data a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number) Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage) discrete random variable a random variable (RV) whose outcomes are counted double-blinding the act of blinding both the subjects of an experiment and the researchers who work with the subjects experimental |
unit any individual or object to be measured explanatory variable the independent variable in an experiment; the value controlled by researchers frequency the number of times a value of the data occurs informed consent any human subject in a research study must be cognizant of any risks or costs associated with the study; the subject has the right to know the nature of the treatments included in the study, their potential risks, and their potential benefits; consent must be given freely by an informed, fit participant institutional review board a committee tasked with oversight of research programs that involve human subjects lurking variable variable a variable that has an effect on a study even though it is neither an explanatory variable nor a response mathematical models a description of a phenomenon using mathematical concepts, such as equations, inequalities, distributions, etc. nonsampling error an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis numerical Variable variables that take on values that are indicated by numbers observational study a study in which the independent variable is not manipulated by the researcher parameter a number that is used to represent a population characteristic and that generally cannot be determined easily placebo an inactive treatment that has no real effect on the explanatory variable 48 Chapter 1 | Sampling and Data population all individuals, objects, or measurements whose properties are being studied probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur proportion the number of successes divided by the total number in the sample qualitative data see data quantitative data see data random assignment the act of organizing experimental units into treatment groups using random methods random sampling selected a method of selecting a sample that gives every member of the population an equal chance of being relative frequency the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes to the total number of outcomes reliability the consistency of a measure; a measure is reliable when the same results are produced given the same circumstances representative sample a subset of the population that has the same characteristics as the population response variable experiment the dependent variable in an experiment; the value that is measured for change at the end of an sample a subset of the population studied sampling bias not all members of the population are equally likely to be selected sampling error the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error sampling with replacement once a member of the population is |
selected for inclusion in a sample, that member is returned to the population for the selection of the next individual sampling without replacement a member of the population may be chosen for inclusion in a sample only once; if chosen, the member is not returned to the population before the next selection simple random sampling a straightforward method for selecting a random sample; give each member of the population a number Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample statistic a numerical characteristic of the sample; a statistic estimates the corresponding population parameter statistical models a description of a phenomenon using probability distributions that describe the expected behavior of the phenomenon and the variability in the expected observations stratified sampling a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum survey a study in which data is collected as reported by individuals. systematic sampling a method for selecting a random sample; list the members of the population Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with the one that was randomly selected. If necessary, return to the beginning of the population list to complete your sample treatments different values or components of the explanatory variable applied in an experiment validity refers to how much a measure or conclusion accurately reflects real world This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 49 variable a characteristic of interest for each person or object in a population CHAPTER REVIEW 1.1 Definitions of Statistics, Probability, and Key Terms The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. 1.2 Data, Sampling, and Variation in Data and Sampling Data are individual items of information that come from a population or sample. Data may be classified as qualitative (categorical), quantitative continuous, or quantitative discrete. Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling |
, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data. Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted. 1.3 Frequency, Frequency Tables, and Levels of Measurement Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth. Expect that some of your answers will vary from the text due to rounding errors. In addition to rounding your answers, you can measure your data using the following four levels of measurement: • Nominal scale level data that cannot be ordered nor can it be used in calculations • Ordinal scale level data that can be ordered; the differences cannot be measured • Interval scale level data with a definite ordering but no starting point; the differences can be measured, but there is no such thing as a ratio • Ratio scale level data with a starting point that can be ordered; the differences have meaning and ratios can be calculated When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets? Frequency, relative frequency, and cumulative relative frequency are measures that answer questions like these. 1.4 Experimental Design and Ethics A poorly designed study will not produce reliable data. There are certain key components that must be included in every experiment. To eliminate lurking variables, subjects must be assigned randomly to different treatment groups. One of the groups must act as a control group, demonstrating what happens when the active treatment is not applied. Participants in the control group receive a placebo treatment that looks exactly like the active treatments but cannot influence the response variable. To preserve the integrity of the placebo, both researchers and subjects may be blinded. When a study is designed properly, the only difference between treatment groups is the one imposed by the researcher. Therefore, when groups respond differently to different treatments, the difference must be due |
to the influence of the explanatory variable. “An ethics problem arises when you are considering an action that benefits you or some cause you support, hurts or reduces benefits to others, and violates some rule.”[4] Ethical violations in statistics are not always easy to spot. Professional 4. Gelman, A. (2013, May 1). Open data and open methods. Ethics and Statistics. Retrieved from http://www.stat.columbia.edu/~gelman/research/published/ChanceEthics1.pdf. 50 Chapter 1 | Sampling and Data associations and federal agencies post guidelines for proper conduct. It is important that you learn basic statistical procedures so that you can recognize proper data analysis. PRACTICE 1.1 Definitions of Statistics, Probability, and Key Terms 1. Below is a two-way table showing the types of college sports played by men and women. Soccer Basketball Lacrosse Total Women Men Total 8 4 12 Table 1.28 8 12 20 4 4 8 20 20 40 Given these data, calculate the marginal distributions of college sports for the people surveyed. 2. Below is a two-way table showing the types of college sports played by men and women. Soccer Basketball Lacrosse Total Women Men Total 8 4 12 Table 1.29 8 12 20 4 4 8 20 20 40 Given these data, calculate the conditional distributions for the subpopulation of women who play college sports. Use the following information to answer the next five exercises. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new viral antibody drug is currently under study. It is given to patients once the virus's symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once they start the treatment. Two researchers each follow a different set of 40 patients with the viral disease from the start of treatment until their deaths. The following data (in months) are collected. Researcher A 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34 Researcher B 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22 |
; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29 Determine what the key terms refer to in the example for Researcher A. 3. population 4. sample 5. parameter 6. statistic 7. variable This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 51 1.2 Data, Sampling, and Variation in Data and Sampling 8. Number of times per week is what type of data? a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous Use the following information to answer the next four exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Antonio, Texas. The first house in the neighborhood around the park was selected randomly, and then the resident of every eighth house in the neighborhood around the park was interviewed. 9. The sampling method was a. simple random; b. systematic; c. stratified; d. cluster 10. Duration (amount of time) is what type of data? a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous 11. The colors of the houses around the park are what kind of data? a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous 12. The population is ________. 52 Chapter 1 | Sampling and Data 13. Table 1.30 contains the total number of deaths worldwide as a result of earthquakes from 2000–2012. Year Total Number of Deaths 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Total 231 21,357 11,685 33,819 228,802 88,003 6,605 712 88,011 1,790 320,120 21,953 768 823,856 Table 1.30 Use Table 1.30 to answer the following questions. a. What is the proportion of deaths between 2007–2012? b. What percent of deaths occurred before 2001? c. What is the percent of deaths that occurred in 2003 or after 2010? d. What is the fraction of deaths that happened before 2012? e. What kind of data is the number of deaths? f. Earthquakes are quantified according to the amount of |
energy they produce (examples are 2.1, 5.0, 6.7). What type of data is that? g. What contributed to the large number of deaths in 2010? In 2004? Explain. h. If you were asked to present these data in an oral presentation, what type of graph would you choose to present and why? Explain what features you would point out on the graph during your presentation. For the following four exercises, determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). 14. A group of test subjects is divided into twelve groups; then four of the groups are chosen at random. 15. A market researcher polls every tenth person who walks into a store. 16. The first 50 people who walk into a sporting event are polled on their television preferences. 17. A computer generates 100 random numbers, and 100 people whose names correspond with the numbers on the list are chosen. Use the following information to answer the next seven exercises: Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new viral antibody drug is currently under study. It is given to patients once the virus's symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 patients with the viral disease from the start of treatment until their deaths. The following data (in months) are collected: Researcher A: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 53 Researcher B: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29 18. Complete the tables using the data provided. Survival Length ( |
in months) Frequency Relative Frequency Cumulative Relative Frequency.5–6.5 6.5–12.5 12.5–18.5 18.5–24.5 24.5–30.5 30.5–36.5 36.5–42.5 42.5–48.5 Table 1.31 Researcher A Survival Length (in months) Frequency Relative Frequency Cumulative Relative Frequency.5–6.5 6.5–12.5 12.5–18.5 18.5–24.5 24.5–30.5 30.5–36.5 36.5-45.5 Table 1.32 Researcher B 19. Determine what the key term data refers to in the above example for Researcher A. 20. List two reasons why the data may differ. 21. Can you tell if one researcher is correct and the other one is incorrect? Why? 22. Would you expect the data to be identical? Why or why not? 23. Suggest at least two methods the researchers might use to gather random data. 24. Suppose that the first researcher conducted his survey by randomly choosing one state in the nation and then randomly picking 40 patients from that state. What sampling method would that researcher have used? 25. Suppose that the second researcher conducted his survey by choosing 40 patients he knew. What sampling method would that researcher have used? What concerns would you have about this data set, based upon the data collection method? Use the following data to answer the next five exercises: Two researchers are gathering data on hours of video games played by school-aged children and young adults. They each randomly sample different groups of 150 students from the same school. They collect the following data: 54 Chapter 1 | Sampling and Data Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency 0–2 2–4 4–6 6–8 8–10 10–12 26 30 49 25 12 8.17.20.33.17.08.05 Table 1.33 Researcher A.17.37.70.87.95 1 Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency 0–2 2–4 4–6 6–8 8–10 10–12 48 51 24 12 11 4.32.34.16.08.07.03 Table 1.34 Researcher B 26. Give a reason why the data may differ..32.66.82.90.97 1 27. Would the sample size be large enough |
if the population is the students in the school? 28. Would the sample size be large enough if the population is school-aged children and young adults in the United States? 29. Researcher A concludes that most students play video games between four and six hours each week. Researcher B concludes that most students play video games between two and four hours each week. Who is correct? 30. Suppose you were asked to present the data from researchers A and B in an oral presentation. When would a pie graph be appropriate? When would a bar graph more desirable? Explain which features you would point out on each type of graph and what potential display problems you would try to avoid. 31. As part of a way to reward students for participating in the survey, the researchers gave each student a gift card to a video game store. Would this affect the data if students knew about the award before the study? Use the following data to answer the next five exercises: A pair of studies was performed to measure the effectiveness of a new software program designed to help stroke patients regain their problem-solving skills. Patients were asked to use the software program twice a day, once in the morning, and once in the evening. The studies observed 200 stroke patients recovering over a period of several weeks. The first study collected the data in Table 1.35. The second study collected the data in Table 1.36. Group Showed Improvement No Improvement Deterioration Used program Did not use program Table 1.35 142 72 43 110 15 18 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 55 Group Showed Improvement No Improvement Deterioration Used program Did not use program Table 1.36 105 89 74 99 19 12 32. Given what you know, which study is correct? 33. The first study was performed by the company that designed the software program. The second study was performed by the American Medical Association. Which study is more reliable? 34. Both groups that performed the study concluded that the software works. Is this accurate? 35. The company takes the two studies as proof that their software causes mental improvement in stroke patients. Is this a fair statement? 36. Patients who used the software were also a part of an exercise program whereas patients who did not use the software were not. Does this change the validity of the conclusions from Exercise 1.34? 37. Is a sample size of 1,000 a reliable measure for a population of |
5,000? 38. Is a sample of 500 volunteers a reliable measure for a population of 2,500? 39. A question on a survey reads: "Do you prefer the delicious taste of Brand X or the taste of Brand Y?" Is this a fair question? 40. Is a sample size of two representative of a population of five? 41. Is it possible for two experiments to be well run with similar sample sizes to get different data? 1.3 Frequency, Frequency Tables, and Levels of Measurement 42. What type of measure scale is being used? Nominal, ordinal, interval or ratio. Incomes measured in dollars a. High school soccer players classified by their athletic ability: superior, average, above average b. Baking temperatures for various main dishes: 350, 400, 325, 250, 300 c. The colors of crayons in a 24-crayon box d. Social security numbers e. f. A satisfaction survey of a social website by number: 1 = very satisfied, 2 = somewhat satisfied, 3 = not satisfied g. Preferred TV shows: comedy, drama, science fiction, sports, news h. Time of day on an analog watch i. The distance in miles to the closest grocery store j. The dates 1066, 1492, 1644, 1947, and 1944 k. The heights of 21–65-year-old women l. Common letter grades: A, B, C, D, and F 1.4 Experimental Design and Ethics 43. Design an experiment. Identify the explanatory and response variables. Describe the population being studied and the experimental units. Explain the treatments that will be used and how they will be assigned to the experimental units. Describe how blinding and placebos may be used to counter the power of suggestion. 44. Discuss potential violations of the rule requiring informed consent. Inmates in a correctional facility are offered good behavior credit in return for participation in a study. a. b. A research study is designed to investigate a new children’s allergy medication. c. Participants in a study are told that the new medication being tested is highly promising, but they are not told that only a small portion of participants will receive the new medication. Others will receive placebo treatments and traditional treatments. 56 Chapter 1 | Sampling and Data HOMEWORK 1.1 Definitions of Statistics, Probability, and Key Terms 45. For each of the following situations, indicate whether it would be best modeled with a mathematical model or a statistical model. |
Explain your answers. a. driving time from New York to Florida b. departure time of a commuter train at rush hour c. distance from your house to school d. e. weight of a bag of rice at the store temperature of a refrigerator at any given time For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate. 46. A fitness center is interested in the mean amount of time a client exercises in the center each week. 47. Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally. 48. A cardiologist is interested in the mean recovery period of her patients who have had heart attacks. 49. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance. 50. A politician is interested in the proportion of voters in his district who think he is doing a new good job. 51. A marriage counselor is interested in the proportion of clients she counsels who stay married. 52. Political pollsters may be interested in the proportion of people who will vote for a particular cause. 53. A marketing company is interested in the proportion of people who will buy a particular product. Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter. 54. What is the population she is interested in? a. all Lake Tahoe Community College students b. all Lake Tahoe Community College English students c. all Lake Tahoe Community College students in her classes d. all Lake Tahoe Community College math students 55. Consider the following X = number of days a Lake Tahoe Community College math student is absent. In this case, X is an example of which of the following? a. variable b. population c. statistic d. data 56. The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of which of the following? a. parameter b. data statistic c. d. variable 1.2 Data, Sampling, and Variation in Data and Sampling For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous |
, or qualitative), and give an example of the data. 57. number of tickets sold to a concert This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 57 58. percent of body fat 59. favorite baseball team 60. time in line to buy groceries 61. number of students enrolled at Evergreen Valley College 62. most-watched television show 63. brand of toothpaste 64. distance to the closest movie theatre 65. age of executives in Fortune 500 companies 66. number of competing computer spreadsheet software packages Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed. 67. Number of times per week is what type of data? a. qualitative b. quantitative discrete c. quantitative continuous 68. Duration (amount of time) is what type of data? a. qualitative b. quantitative discrete c. quantitative continuous 69. Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study. a. Using complete sentences, list three things wrong with the way the survey was conducted. b. Using complete sentences, list three ways that you would improve the survey if it were to be repeated. 70. Suppose you want to determine the mean number of students per statistics class in your state. Describe a possible sampling method in three to five complete sentences. Make the description detailed. 71. Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed. 72. List some practical difficulties involved in getting accurate results from a telephone survey. 73. List some practical difficulties involved in getting accurate results from a mailed survey. 74. With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey. 75. The instructor takes her sample by gathering data on |
five randomly selected students from each Lake Tahoe Community College math class. The type of sampling she used is which of the following? a. cluster sampling b. c. d. convenience sampling stratified sampling simple random sampling 76. A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was which of the following? simple random systematic stratified a. b. c. d. cluster 58 Chapter 1 | Sampling and Data 77. Name the sampling method used in each of the following situations: a. A woman in the airport is handing out questionnaires to travelers asking them to evaluate the airport’s service. She does not ask travelers who are hurrying through the airport with their hands full of luggage, but instead asks all travelers who are sitting near gates and not taking naps while they wait. b. A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class. c. The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, 100 randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest. d. The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books. e. A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party’s polling staff calls 1,200 randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates. 78. A random survey was conducted of 3,274 people of the microprocessor generation—people born since 1971, the year the microprocessor was invented. It was reported that 48 percent of those individuals surveyed stated that if they had $2,000 to spend, they |
would use it for computer equipment. Also, 66 percent of those surveyed considered themselves relatively savvy computer users. a. Do you consider the sample size large enough for a study of this type? Why or why not? b. Based on your gut feeling, do you believe the percents accurately reflect the U.S. population for those individuals born since 1971? If not, do you think the percents of the population are actually higher or lower than the sample statistics? Why? Additional information: The survey, reported by Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute's road show called “America’s Smithsonian.” c. With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not? d. With the additional information, comment on how accurately you think the sample statistics reflect the population parameters. 79. The Well-Being Index is a survey that follows trends of U.S. residents on a regular basis. There are six areas of health and wellness covered in the survey: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment, and Basic Access. Some of the questions used to measure the Index are listed below. Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous. a. Do you have any health problems that prevent you from doing any of the things people your age can normally do? b. During the past 30 days, for about how many days did poor health keep you from doing your usual activities? c. d. Do you have health insurance coverage? In the last seven days, on how many days did you exercise for 30 minutes or more? 80. In advance of the 1936 presidential election, a magazine released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards. a. Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time. b. What effect does the low response rate have on the reliability |
of the sample? c. Are these problems examples of sampling error or nonsampling error? d. During the same year, another pollster conducted a poll of 30,000 prospective voters. These researchers used a method they called quota sampling to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 59 81. Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI's Uniform Crime Report. One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates. Which of the potential problems with samples discussed in Data, Sampling, and Variation in Data and Sampling could explain this connection? 82. A website that allows anyone to create and respond to polls had a question posted on April 15 which asked: “Do you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?”[5] As of April 25, 11 people responded to this question. Each participant answered “NO!” Which of the potential problems with samples discussed in this module could explain this connection? 83. A scholarly article about response rates begins with the following quote: “Declining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about the validity of estimates drawn from such research.”[6] The Pew Research Center for People and the Press admits “The percentage of people we interview—out of all we try to interview—has been declining over the past decade or more.”[7] a. What are some reasons for the decline in response rate over the past decade? b. Explain why researchers are concerned with the impact of the declining response rate on public opinion polls. 1.3 Frequency, Frequency Tables, and Levels of Measurement 84. Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below. # of Courses Frequency Relative Frequency Cumulative Relative Frequency 1 2 3 30 15.6 Table 1.37 Part-time Student Course Loads a. Fill in the blanks in Table 1.37. b. What percent of students take exactly two courses? c. What percent of students |
take one or two courses? lastbaldeagle. Retrieved from http://www.youpolls.com/details.aspx?id=12328. 5. 6. Keeter, S., et al. (2006). Gauging the impact of growing nonresponse on estimates from a national RDD telephone survey. Public Opinion Quarterly, 70(5). Retrieved from http://hbanaszak.mjr.uw.edu.pl/TempTxt/Links/ GAUGING%20THE%20IMPACT%20OF%20GROWING.pdf. 7. Pew Research Center. (n.d.). Frequently asked questions. Retrieved from http://www.pewresearch.org/methodology/us-survey-research/frequently-asked-questions/#dont-you-have-trouble-getting-people-to-answer-your-polls. 60 Chapter 1 | Sampling and Data 85. Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown in Table 1.38. # Flossing per Week Frequency Relative Frequency Cumulative Relative Frequency 0 1 3 6 7 27 18 3 1.4500.0500.0167 Table 1.38 Flossing Frequency for Adults with Gum Disease a. Fill in the blanks in Table 1.38. b. What percent of adults flossed six times per week? c. What percent flossed at most three times per week?.9333 86. Nineteen immigrants to the United States were asked how many years, to the nearest year, they have lived in the United States The data are as follows: 2, 5, 7, 2, 2, 10, 20, 15, 0, 7, 0, 20, 5, 12, 15, 12, 4, 5, 10. Table 1.39 was produced. Data Frequency Relative Frequency Cumulative Relative Frequency 0 2 4 5 7 10 12 15 20 19 3 19 1 19 3 19 2 19 2 19 2 19 1 19 1 19.1053.2632.3158.4737.5789.6842.7895.8421 1.0000 Table 1.39 Frequency of Immigrant Survey Responses a. Fix the errors in Table 1.39. Also, explain how someone might have arrived at the incorrect number(s). b. Explain what is wrong with |
this statement: “47 percent of the people surveyed have lived in the United States for 5 years.” c. Fix the statement in b to make it correct. d. What fraction of the people surveyed have lived in the United States five or seven years? e. What fraction of the people surveyed have lived in the United States at most 12 years? f. What fraction of the people surveyed have lived in the United States fewer than 12 years? g. What fraction of the people surveyed have lived in the United States from five to 20 years, inclusive? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 61 87. How much time does it take to travel to work? Table 1.40 shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly. 24.0 24.3 25.9 18.9 27.5 17.9 21.8 20.9 16.7 27.3 18.2 24.7 20.0 22.6 23.9 18.0 31.4 22.3 24.0 25.5 24.7 24.6 28.1 24.9 22.6 23.6 23.4 25.7 24.8 25.5 21.2 25.7 23.1 23.0 23.9 26.0 16.3 23.1 21.4 21.5 27.0 27.0 18.6 31.7 23.3 30.1 22.9 23.3 21.7 18.6 Table 1.40 88. A business magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. Table 1.41 shows the ages of the chief executive officers for the first 60 ranked firms. Age Frequency Relative Frequency Cumulative Relative Frequency 40–44 45–49 50–54 55–59 60–64 65–69 70–74 Table 1.41 3 11 13 16 10 6 1 a. What is the frequency for CEO ages between 54 and 65? b. What percentage of CEOs are 65 years or older? c. What is the relative frequency of ages under 50? d. What is the cumulative |
relative frequency for CEOs younger than 55? e. Which graph shows the relative frequency and which shows the cumulative relative frequency? (a) Figure 1.13 (b) 62 Chapter 1 | Sampling and Data Use the following information to answer the next two exercises: Table 1.42 contains data on hurricanes that have made direct hits on the United States. Between 1851-2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm. Category Number of Direct Hits Relative Frequency Cumulative Frequency 1 2 3 4 5 109 72 71 18 3.3993.2637.2601.0110.3993.6630.9890 1.0000 Total = 273 Table 1.42 Frequency of Hurricane Direct Hits 89. What is the relative frequency of direct hits that were category 4 hurricanes?.0768.0659.2601 a. b. c. d. not enough information to calculate 90. What is the relative frequency of direct hits that were AT MOST a category 3 storm? a. b. c. d..3480.9231.2601.3370 1.4 Experimental Design and Ethics 91. How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation. Use key terms from this module to describe the design of this experiment. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 63 92. An advertisement for Acme Investments displays the two graphs in Figure 1.14 to show the value of Acme’s product in comparison with the Other Guy’s product. Describe the potentially misleading visual effect of these comparison graphs. How can this be corrected? (a) (b) Figure 1.14 As the graphs show, Acme consistently outperforms the Other Guys! 93. The graph in Figure 1.15 shows the number of complaints for six different airlines as reported to the U.S. Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the |
most complaints? Figure 1.15 94. An epidemiologist is studying the spread of the common cold among college students. He is interested in how the temperature of the dorm room correlates with the incidence of new infections. How can he design an observational study to answer this question? If he chooses to use surveys in his measurements, what type of questions should he include in the survey? BRINGING IT TOGETHER: HOMEWORK 64 Chapter 1 | Sampling and Data 95. Seven hundred and seventy-one distance learning students at Long Beach City College responded to surveys in the 2010–11 academic year. Highlights of the summary report are listed in Table 1.43. Have computer at home Unable to come to campus for classes Age 41 or over 96% 65% 24% Would like LBCC to offer more DL courses 95% Took DL classes due to a disability Live at least 16 miles from campus 17% 13% Took DL courses to fulfill transfer requirements 71% Table 1.43 LBCC Distance Learning Survey Results a. What percent of the students surveyed do not have a computer at home? b. About how many students in the survey live at least 16 miles from campus? c. If the same survey were done at Great Basin College in Elko, Nevada, do you think the percentages would be the same? Why? 96. Several online textbook retailers advertise that they have lower prices than on-campus bookstores. However, an important factor is whether the Internet retailers actually have the textbooks that students need in stock. Students need to be able to get textbooks promptly at the beginning of the college term. If the book is not available, then a student would not be able to get the textbook at all, or might get a delayed delivery if the book is back ordered. A college newspaper reporter is investigating textbook availability at online retailers. He decides to investigate one textbook for each of the following seven subjects: calculus, biology, chemistry, physics, statistics, geology, and general engineering. He consults textbook industry sales data and selects the most popular nationally used textbook in each of these subjects. He visits websites for a random sample of major online textbook sellers and looks up each of these seven textbooks to see if they are available in stock for quick delivery through these retailers. Based on his investigation, he writes an article in which he draws conclusions about the overall availability of all college textbooks through online textbook retailers. Write an analysis of his study that addresses the following issues: Is his sample representative of the population of all college textbooks |
? Explain why or why not. Describe some possible sources of bias in this study, and how it might affect the results of the study. Give some suggestions about what could be done to improve the study. REFERENCES 1.1 Definitions of Statistics, Probability, and Key Terms The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html. 1.2 Data, Sampling, and Variation in Data and Sampling Gallup. Retrieved from http://www.well-beingindex.com/. Gallup. Retrieved from http://www.gallup.com/poll/110548/gallup-presidential-election-trialheat-trends-19362004.aspx#4. Gallup. Retrieved from http://www.gallup.com/175196/gallup-healthways-index-methodology.aspx. Data from http://www.bookofodds.com/Relationships-Society/Articles/A0374-How-George-Gallup-Picked-the-President. LBCC Distance Learning (DL) Program. Retrieved from http://de.lbcc.edu/reports/2010-11/future/highlights.html#focus. Lusinchi, D. (2012). “President” Landon and the 1936 Literary Digest poll: Were automobile and telephone owners to blame? Social Science History 36(1), 23-54. Retrieved from https://muse.jhu.edu/article/471582/pdf. The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Datafiles/USCrime.html. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 65 The Mercury News. Retrieved from http://www.mercurynews.com/. Virtual Laboratories in Probability and Statistics. Retrieved from http://www.math.uah.edu/stat/data/LiteraryDigest.html. The Mercury News. Retrieved from http://www.mercurynews.com/. 1.3 Frequency, Frequency Tables, and Levels of Measurement Levels of Measurement. Retrieved from http://cnx.org/content/m10809/latest/. National Hurricane Center. Retrieved from http://www.nh |
c.noaa.gov/gifs/table5.gif. ThoughtCo. Retrieved from https://www.thoughtco.com/levels-of-measurement-in-statistics-3126349. U.S. Census Bureau. Retrieved from https://www.census.gov/quickfacts/table/PST045216/00. Levels of measurement. Retrieved from https://www.cos.edu/Faculty/georgew/Tutorial/Data_Levels.htm. 1.4 Experimental Design and Ethics Econoclass.com. Retrieved from http://www.econoclass.com/misleadingstats.html. Bloomberg Businessweek. Retrieved from www.businessweek.com. Ethics in statistics. Retrieved from http://cnx.org/content/m15555/latest/. Forbes. Retrieved from www.forbes.com. Forbes. http://www.forbes.com/best-small-companies/list/. Harvard School of Public Health. Retrieved from https://www.hsph.harvard.edu/nutritionsource/vitamin-e/. Jacskon, M.L., et al. (2013). Cognitive components of simulated driving performance: Sleep loss effect and predictors. Accident Analysis and Prevention Journal 50, 438-44. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/22721550. International Business Times. Retrieved from http://www.ibtimes.com/daily-dose-aspirin-helps-reduce-heart-attacksstudy-300443. National Highway Traffic Safety Administration. Retrieved from http://www-fars.nhtsa.dot.gov/Main/index.aspx. Athleteinme.com. Retrieved from http://www.athleteinme.com/ArticleView.aspx?id=1053. The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Stories/ScentsandLearning.html. U.S. Department of Health and Human Services. Retrieved from https://www.hhs.gov/ohrp/regulations-and-policy/ regulations/45-cfr-46/index.html. U.S. Department of Transportation. Retrieved from http://www.dot.gov/airconsumer/april-2013-air-travel-consumer-report. U. |
S. Geological Survey. Retrieved from http://earthquake.usgs.gov/earthquakes/eqarchives/year/. SOLUTIONS 1 soccer = 12/40 = ; basketball = 20/40 = ; lacrosse = 8/40 = 0.2 2 women who play soccer = 8/20 = ; women who play basketball = 8/20 = ; women who play lacrosse = 4/20 = ; 3 patients with the virus 5 The average length of time (in months) patients live after treatment. 7 X = the length of time (in months) patients live after treatment 9 b 11 a 13 a..5242 b..03 percent 66 c. 6.86 percent d. 823,088 823,856 e. quantitative discrete f. quantitative continuous Chapter 1 | Sampling and Data g. In both years, underwater earthquakes produced massive tsunamis. h. Answers may vary. Sample answer: A bar graph with one bar for each year, in order, would be best since it would show the change in the number of deaths from year to year. In my presentation, I would point out that the scale of the graph is in thousands, and I would discuss which specific earthquakes were responsible for the greatest numbers of deaths in those years. 15 systematic 17 simple random 19 values for X, such as 3, 4, 11, and so on 21 No, we do not have enough information to make such a claim. 23 Take a simple random sample from each group. One way is by assigning a number to each patient and using a random number generator to randomly select patients. 25 This would be convenience sampling and is not random. 27 Yes, the sample size of 150 would be large enough to reflect a population of one school. 29 Even though the specific data support each researcher’s conclusions, the different results suggest that more data need to be collected before the researchers can reach a conclusion. 30 Answers may vary. Sample answer: A pie graph would be best for showing the percentage of students that fall into each Hours Played category. A bar graph would be more desirable if knowing the total numbers of students in each category is important. I would be sure that the colors used on the two pie graphs are the same for each category and are clearly distinguishable when displayed. The percentages should be legible, and the pie graph should be large enough to show the smaller sections clearly. For the bar graph, I would display the bars in chronological order and make sure that the colors used for |
each researcher’s data are clearly distinguishable. The numbers and the scale should be legible and clear when the bar graph is displayed. 32 There is not enough information given to judge if either one is correct or incorrect. 34 The software program seems to work because the second study shows that more patients improve while using the software than not. Even though the difference is not as large as that in the first study, the results from the second study are likely more reliable and still show improvement. 36 Yes, because we cannot tell if the improvement was due to the software or the exercise; the data is confounded, and a reliable conclusion cannot be drawn. New studies should be performed. 38 No, even though the sample is large enough, the fact that the sample consists of volunteers makes it a self-selected sample, which is not reliable. 40 No, even though the sample is a large portion of the population, two responses are not enough to justify any conclusions. Because the population is so small, it would be better to include everyone in the population to get the most accurate data. 42 a. ordinal b. interval c. nominal d. nominal e. ratio f. ordinal g. nominal This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 67 h. interval i. j. ratio interval k. ratio l. ordinal 44 a. Inmates may not feel comfortable refusing participation, or may feel obligated to take advantage of the promised benefits. They may not feel truly free to refuse participation. b. Parents can provide consent on behalf of their children, but children are not competent to provide consent for themselves. c. All risks and benefits must be clearly outlined. Study participants must be informed of relevant aspects of the study in order to give appropriate consent. 45 a. statistical model: The time any journey takes from New York to Florida is variable and depends on traffic and other driving conditions. b. statistical model: Although trains try to leave on time, the exact time of departure differs slightly from day to day. c. mathematical model: The distance from your house to school is the same every day and can be precisely determined. statistical model: The temperature of a refrigerator fluctuates as the compressor turns on and off. statistical model: The fill weight of a bag of rice is different for each bag. Manufacturers spend considerable effort to minimize the variance from bag to bag. d. e. 47 a. all |
children who take ski or snowboard lessons b. a group of these children c. d. the population mean age of children who take their first snowboard lesson the sample mean age of children who take their first snowboard lesson e. X = the age of one child who takes his or her first ski or snowboard lesson f. values for X, such as 3, 7, and so on 49 a. the clients of the insurance companies b. a group of the clients c. d. the mean health costs of the clients the mean health costs of the sample e. X = the health costs of one client f. values for X, such as 34, 9, 82, and so on 51 a. all the clients of this counselor b. a group of clients of this marriage counselor c. d. the proportion of all her clients who stay married the proportion of the sample of the counselor’s clients who stay married e. X = the number of couples who stay married f. yes, no 68 53 a. all people (maybe in a certain geographic area, such as the United States) b. a group of the people c. d. the proportion of all people who will buy the product the proportion of the sample who will buy the product e. X = the number of people who will buy it Chapter 1 | Sampling and Data f. buy, not buy 55 a 57 quantitative discrete, 150 59 qualitative, Oakland A’s 61 quantitative discrete, 11,234 students 63 qualitative, Crest 65 quantitative continuous, 47.3 years 67 b 69 a. The survey was conducted using six similar flights. The survey would not be a true representation of the entire population of air travelers. Conducting the survey on a holiday weekend will not produce representative results. b. Conduct the survey during different times of the year. Conduct the survey using flights to and from various locations. Conduct the survey on different days of the week. 71 Answers will vary. Sample Answer: You could use a systematic sampling method. Stop the tenth person as they leave one of the buildings on campus at 9:50 in the morning. Then stop the tenth person as they leave a different building on campus at 1:50 in the afternoon. 73 Answers will vary. Sample Answer: Many people will not respond to mail surveys. If they do respond to the surveys, you can’t be sure who is responding. In addition, mailing lists can be incomplete. 75 b 77 convenience; cluster; stratified ; systematic; simple random 79 a |
. qualitative b. quantitative discrete c. quantitative discrete d. qualitative 81 Causality: The fact that two variables are related does not guarantee that one variable is influencing the other. We cannot assume that crime rate impacts education level or that education level impacts crime rate. Confounding: There are many factors that define a community other than education level and crime rate. Communities with high crime rates and high education levels may have other lurking variables that distinguish them from communities with lower crime rates and lower education levels. Because we cannot isolate these variables of interest, we cannot draw valid conclusions about the connection between education and crime. Possible lurking variables include police expenditures, unemployment levels, region, average age, and size. 83 a. Possible reasons: increased use of caller id, decreased use of landlines, increased use of private numbers, voice mail, privacy managers, hectic nature of personal schedules, decreased willingness to be interviewed b. When a large number of people refuse to participate, then the sample may not have the same characteristics of the population. Perhaps the majority of people willing to participate are doing so because they feel strongly about the This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 | Sampling and Data 69 subject of the survey. 85 a. # Flossing per Week Frequency Relative Frequency Cumulative Relative Frequency 0 1 3 6 7 27 18 11 3 1.4500.3000.1833.0500.0167.4500.7500.9333.9833 1 Table 1.44 b. 5.00 percent c. 93.33 percent 87 The sum of the travel times is 1,173.1. Divide the sum by 50 to calculate the mean value: 23.462. Because each state’s travel time was measured to the nearest tenth, round this calculation to the nearest hundredth: 23.46. 89 b 91 Explanatory variable: amount of sleep Response variable: performance measured in assigned tasks Treatments: normal sleep and 27 hours of total sleep deprivation Experimental Units: 19 professional drivers Lurking variables: none – all drivers participated in both treatments Random assignment: treatments were assigned in random order; this eliminated the effect of any learning that may take place during the first experimental session Control/Placebo: completing the experimental session under normal sleep conditions Blinding: researchers evaluating subjects’ performance must not know which treatment is being applied at the time 93 You cannot assume that the numbers of complaints reflect the quality of |
the airlines. The airlines shown with the greatest number of complaints are the ones with the most passengers. You must consider the appropriateness of methods for presenting data; in this case displaying totals is misleading. 94 He can observe a population of 100 college students on campus. He can collect data about the temperature of their dorm rooms and track how many of them catch a cold. If he uses a survey, the temperature of the dorm rooms can be determined from the survey. He can also ask them to self-report when they catch a cold. 96 Answers will vary. Sample answer: The sample is not representative of the population of all college textbooks. Two reasons why it is not representative are that he only sampled seven subjects and he only investigated one textbook in each subject. There are several possible sources of bias in the study. The seven subjects that he investigated are all in mathematics and the sciences; there are many subjects in the humanities, social sciences, and other subject areas, for example: literature, art, history, psychology, sociology, business, that he did not investigate at all. It may be that different subject areas exhibit different patterns of textbook availability, but his sample would not detect such results. He also looked only at the most popular textbook in each of the subjects he investigated. The availability of the most popular textbooks may differ from the availability of other textbooks in one of two ways: • The most popular textbooks may be more readily available online, because more new copies are printed, and more students nationwide are selling back their used copies • The most popular textbooks may be harder to find available online, because more student demand exhausts the supply more quickly. In reality, many college students do not use the most popular textbook in their subject, and this study gives no useful information about the situation for those less popular textbooks. He could improve this study by • expanding the selection of subjects he investigates so that it is more representative of all subjects studied by college 70 students, and Chapter 1 | Sampling and Data • expanding the selection of textbooks he investigates within each subject to include a mixed representation of both the most popular and less popular textbooks. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 71 2 | DESCRIPTIVE STATISTICS Figure 2.1 When you have a large amount of data, you will need to organize it in a way that makes sense. These ballots from an election are rolled together with similar ballots to keep them |
organized. (credit: William Greeson) Introduction By the end of this chapter, the student should be able to do the following: Chapter Objectives • Display data graphically and interpret the following graphs: stem-and-leaf plots, line graphs, bar graphs, frequency polygons, time series graphs, histograms, box plots, and dot plots • Recognize, describe, and calculate the measures of location of data with quartiles and percentiles • Recognize, describe, and calculate the measures of the center of data with mean, median, and mode • Recognize, describe, and calculate the measures of the spread of data with variance, standard deviation, and range Once you have a data collection, what will you do with it? Data can be described and presented in many different formats. For example, suppose you are interested in buying a house in a particular area. You may have no clue about the house prices, so you might ask your real estate agent to give you a sample data set of prices. Looking at all the prices in the sample often is overwhelming. A better way might be to look at the median price and the variation of prices. The median and variation 72 Chapter 2 | Descriptive Statistics are just two ways that you will learn to describe data. Your agent might also provide you with a graph of the data. In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called descriptive statistics. You will learn how to calculate and, even more important, how to interpret these measurements and graphs. A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data values cluster and where there are only a few data values. Newspapers and the internet use graphs to show trends and to enable readers to compare facts and figures quickly. Statisticians often graph data first to get a picture of the data. Then more formal tools may be applied. Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon—a type of broken line graph—the pie chart, and the box plot. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs as well as frequency polygons, time series graphs, and dot plots |
. Our emphasis will be on histograms and box plots. NOTE This book contains instructions for constructing a histogram and a box plot for the TI-83+ and TI-84 calculators. The Texas Instruments (TI) website (http://education.ti.com/educationportal/sites/US/sectionHome/ support.html) provides additional instructions for using these calculators. 2.1 | Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The stem consists of the leading digit(s), while the leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem. Make sure the leaves show a space between values, so that the exact data values may be easily determined. The frequency of data values for each stem provides information about the shape of the distribution. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 73 Example 2.1 For Susan Dean's spring precalculus class, scores for the first exam were as follows (smallest to largest): 33, 42, 49, 49, 53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100 Stem Leaf 10 0 Table 2.1 Stem-andLeaf Graph The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26 percent ⎛ ⎝ were in the 90s or 100, a fairly high number of As. ⎞ ⎠ 8 31 2.1 For the Park City basketball team |
, scores for the last 30 games were as follows (smallest to largest): 32, 32, 33, 34, 38, 40, 42, 42, 43, 44, 46, 47, 47, 48, 48, 48, 49, 50, 50, 51, 52, 52, 52, 53, 54, 56, 57, 57, 60, 61 Construct a stemplot for the data. The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes, for example, writing 50 instead of 500, while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later. Example 2.2 The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data. 1.1, 1.5, 2.3, 2.5, 2.7, 3.2, 3.3, 3.3, 3.5, 3.8, 4.0, 4.2, 4.5, 4.5, 4.7, 4.8, 5.5, 5.6, 6.5, 6.7, 12.3 Do the data seem to have any concentration of values? The leaves are to the right of the decimal. 74 Chapter 2 | Descriptive Statistics Solution 2.2 The value 12.3 may be an outlier. Values appear to concentrate at 3 and 4 kilometers. Stem Leaf 1 2 3 4 5 6 7 8 9 10 11 12 Table 2.2 2.2 The data below show the distances (in miles) from the homes of high school students to the school. Create a stemplot using the following data and identify any outliers. 0.5, 0.7, 1.1, 1.2, 1.2, 1.3, 1.3, 1.5, 1.5, 1.7, 1.7, 1.8, 1.9, 2.0, 2.2, 2.5, 2.6, 2.8, 2.8, 2.8, 3 |
.5, 3.8, 4.4, 4.8, 4.9, 5.2, 5.5, 5.7, 5.8, 8.0 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 75 Example 2.3 A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. Table 2.3 and Table 2.4 show the ages of presidents at their inauguration and at their death. Construct a sideby-side stem-and-leaf plot using these data. President Age President Age President Age Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren 57 61 57 57 58 57 61 54 Tyler Polk Taylor Fillmore Pierce Buchanan 51 49 64 50 48 65 Lincoln A. Johnson Grant Hayes Garfield Arthur Cleveland 52 56 46 54 49 51 47 Hoover 54 F. Roosevelt 51 Truman Eisenhower Kennedy L. Johnson Nixon 60 62 43 55 56 61 52 69 B. Harrison 55 Ford McKinley 55 54 Carter Reagan T. Roosevelt 42 G.H.W. Bush 64 Taft Wilson Harding Coolidge 51 56 55 51 Clinton G. W. Bush Obama 47 54 47 W. H. Harrison 68 Cleveland Table 2.3 Presidential Ages at Inauguration President Age President Age President Age Hoover 90 F. Roosevelt 63 Truman 88 Eisenhower 78 Kennedy L. Johnson Nixon 46 64 81 93 93 Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren 67 90 83 85 73 80 78 79 Lincoln A. Johnson Grant Hayes Garfield Arthur Cleveland 56 66 63 70 49 56 71 B. Harrison 67 Ford W. H. Harrison 68 Cleveland Tyler Polk Taylor 71 53 65 McKinley T. Roosevelt 60 Taft 72 Reagan 71 58 Table 2.4 Presidential Age at Death 76 Chapter 2 | Descriptive Statistics President Age President Age President Age Fillmore Pierce Buchanan 74 64 77 Wilson Harding Coolidge 67 57 60 Table 2.4 Presidential Age at Death Solution 2.3 Ages at Inauguration Ages at Death Table 2.5 Notice that the leaf values increase in order, from right to left, for leaves shown to the left of the stem, while the leaf values increase in order from left to right, for leaves shown |
to the right of the stem. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 77 2.3 The table shows the number of wins and losses a sports team has had in 42 seasons. Create a side-by-side stemand-leaf plot of these wins and losses. Losses Wins Year Losses Wins Year 48 48 36 36 46 35 31 29 31 41 46 50 31 42 43 40 34 50 57 50 52 1968–1969 41 1969–1970 39 1970–1971 44 1971–1972 39 1972–1973 25 1973–1974 40 1974–1975 36 1975–1976 26 1976–1977 32 1977–1978 19 1978–1979 54 1979–1980 57 1980–1981 49 1981–1982 47 1982–1983 54 1983–1984 69 1984–1985 56 1985–1986 52 1986–1987 45 1987–1988 35 1988–1989 29 34 34 46 46 36 47 51 53 51 41 36 32 51 40 39 42 48 32 25 32 30 Table 2.6 41 43 38 43 57 42 46 56 50 31 28 25 33 35 28 13 26 30 37 47 53 1989–1990 1990–1991 1991–1992 1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 1997–1998 1998–1999 1999–2000 2000–2001 2001–2002 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008 2008–2009 2009–2010 Another type of graph that is useful for specific data values is a line graph. In the particular line graph shown in Example 2.4, the x-axis (horizontal axis) consists of data values and the y-axis (vertical axis) consists of frequency points. The frequency points are connected using line segments. Example 2.4 In a survey, 40 mothers were asked how many times per week a teenager must be reminded to do his or her chores. The results are shown in Table 2.7 and in Figure 2.2. 78 Chapter 2 | Descriptive Statistics Number of Times Teenager Is Reminded Frequency 0 1 2 3 4 5 Table 2.7 2 5 8 14 7 4 Figure 2.2 2.4 In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in Table 2.8. Construct a line graph. Number of Times in Shop Frequency 0 1 2 3 Table 2.8 7 10 14 9 Bar graphs |
consist of bars that are separated from each other. The bars can be rectangles, or they can be rectangular boxes, used in three-dimensional plots, and they can be vertical or horizontal. The bar graph shown in Example 2.5 has age- This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 79 groups represented on the x-axis and proportions on the y-axis. Example 2.5 By the end of 2011, a social media site had more than 146 million users in the United States. Table 2.9 shows three age-groups, the number of users in each age-group, and the proportion (percentage) of users in each agegroup. Construct a bar graph using this data. Age-Groups Number of Site Users Proportion (%) of Site Users 13–25 26–44 45–64 Table 2.9 65,082,280 53,300,200 27,885,100 45% 36% 19% Solution 2.5 Figure 2.3 80 Chapter 2 | Descriptive Statistics 2.5 The population in Park City is made up of children, working-age adults, and retirees. Table 2.10 shows the three age-groups, the number of people in the town from each age-group, and the proportion (%) of people in each agegroup. Construct a bar graph showing the proportions. Age-Groups Number of People Proportion of Population Children 67,059 Working-age adults 152,198 Retirees 131,662 Table 2.10 19% 43% 38% Example 2.6 The columns in Table 2.11 contain the race or ethnicity of students in U.S. public schools for the class of 2011, percentages for the Advanced Placement (AP) examinee population for that class, and percentages for the overall student population. Create a bar graph with the student race or ethnicity (qualitative data) on the x-axis and the AP examinee population percentages on the y-axis. Race/Ethnicity 1 = Asian, Asian American, or Pacific Islander 2 = Black or African American 3 = Hispanic or Latino 4 = American Indian or Alaska Native 5 = White 6 = Not reported/other Table 2.11 AP Examinee Population Overall Student Population 10.3% 9.0% 17.0% 0.6% 57.1% 6.0% 5.7% 14.7% 17 |
.6% 1.1% 59.2% 1.7% This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 81 Solution 2.6 Figure 2.4 2.6 Park City is broken down into six voting districts. The table shows the percentage of the total registered voter population that lives in each district as well as the percentage of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district. District Registered Voter Population Overall City Population 1 2 3 4 5 6 15.5% 12.2% 9.8% 17.4% 22.8% 22.3% Table 2.12 19.4% 15.6% 9.0% 18.5% 20.7% 16.8% Example 2.7 Table 2.13 is a two-way table showing the types of pets owned by men and women. 82 Chapter 2 | Descriptive Statistics Dogs Cats Fish Total Men 4 Women 4 Total 8 Table 2.13 2 6 8 2 2 4 8 12 20 Given these data, calculate the marginal distributions of pets for the people surveyed. Solution 2.7 Dogs = 8/20 = 0.4 Cats = 8/20 = 0.4 Fish = 4/20 = 0.2 Note—The sum of all the marginal distributions must equal one. In this case, 0.4 + 0.4 + 0.2 = 1; therefore, the solution checks. Example 2.8 Table 2.14 is a two-way table showing the types of pets owned by men and women. Dogs Cats Fish Total Men 4 Women 4 Total 8 Table 2.14 2 6 8 2 2 4 8 12 20 Given these data, calculate the conditional distributions for the subpopulation of men who own each pet type. Solution 2.8 Men who own dogs = 4/8 = 0.5 Men who own cats = 2/8 = 0.25 Men who own fish = 2/8 = 0.25 Note—The sum of all the conditional distributions must equal one. In this case, 0.5 + 0.25 + 0.25 = 1; therefore, the solution checks. 2.2 | Histograms, Frequency Polygons, and Time Series Graphs For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it |
can readily display large data sets. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 83 A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is more or less a number line, labeled with what the data represents, for example, distance from your home to school. The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. The shape of the data refers to the shape of the distribution, whether normal, approximately normal, or skewed in some direction, whereas the center is thought of as the middle of a data set, and the spread indicates how far the values are dispersed about the center. In a skewed distribution, the mean is pulled toward the tail of the distribution. The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. Remember, frequency is defined as the number of times an answer occurs. If • f = frequency, • n = total number of data values (or the sum of the individual frequencies), and • RF = relative frequency, then RF = f n. For example, if three students in Mr. Ahab's English class of 40 students received from ninety to 100 percent, then f = 3, n = 40, and RF = f n = 0.075. Thus, 7.5 percent of the students received 90 to 100 percent. Ninety to 100 percent is a = 3 40 quantitative measures. To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The width of each bar is also referred to as the bin size, which may be calculated by dividing the range of the data values by the desired number of bins (or bars). There is not a set procedure for determining the number of bars or bar width/bin size; however, consistency is key when determining which data values to place inside each interval. Example 2.9 The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data since height is measured |
. 60, 60.5, 61, 61, 61.5, 63.5, 63.5, 63.5, 64, 64, 64, 64, 64, 64, 64, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69.5, 69.5, 69.5, 69.5, 69.5, 70, 70, 70, 70, 70, 70, 70.5, 70.5, 70.5, 71, 71, 71, 72, 72, 72, 72.5, 72.5, 73, 73.5, 74 The smallest data value is 60, and the largest data value is 74. To make sure each is included in an interval, we can use 59.95 as the smallest value and 74.05 as the largest value, subtracting and adding.05 to these values, respectively. We have a small range here of 14.1 (74.05 – 59.95), so we will want a fewer number of bins; let’'s say eight. So, 14.1 divided by eight bins gives a bin size (or interval size) of approximately 1.76. NOTE We will round up to two and make each bar or class interval two units wide. Rounding up to two is a way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary |
. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals. 84 Chapter 2 | Descriptive Statistics The boundaries are as follows: • 59.95 • 59.95 + 2 = 61.95 • 61.95 + 2 = 63.95 • 63.95 + 2 = 65.95 • 65.95 + 2 = 67.95 • 67.95 + 2 = 69.95 • 69.95 + 2 = 71.95 • 71.95 + 2 = 73.95 • 73.95 + 2 = 75.95 The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95. The following histogram displays the heights on the x-axis and relative frequency on the y-axis. Figure 2.5 Interval Frequency Relative Frequency 59.95–61.95 61.95–63.95 63.95–65.95 65.95–67.95 67.95–69.95 69.95–71.95 71.95–73.95 Table 2.15 5 3 15 40 17 12 7 5/100 = 0.05 3/100 = 0.03 15/100 = 0.15 40/100 = 0.40 17/100 = 0.17 12/100 = 0.12 7/100 = 0.07 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 85 Interval Frequency Relative Frequency 73.95–75.95 1 1/100 = 0.01 Table 2.15 2.9 The following data are the shoe sizes of 50 male students. The sizes are continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. |
Use six bars on the histogram. 9, 9, 9.5, 9.5, 10, 10, 10, 10, 10, 10, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5, 12, 12, 12, 12, 12, 12, 12, 12.5, 12.5, 12.5, 12.5, 14 Example 2.10 The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data since books are counted. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6 Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books. Calculate the width of each bar/bin size/interval size. Solution 2.10 The smallest data value is 1, and the largest data value is 6. To make sure each is included in an interval, we can use 0.5 as the smallest value and 6.5 as the largest value by subtracting and adding 0.5 to these values. We have a small range here of 6 (6.5 –– 0.5), so we will want a fewer number of bins; let’'s say six this time. So, six divided by six bins gives a bin size (or interval size) of one. Notice that we may choose different rational numbers to add to, or subtract from, our maximum and minimum values when calculating bin size. In the previous example, we added and subtracted.05, while this time, we added and subtracted.5. Given a data set, you will be |
able to determine what is appropriate and reasonable. The following histogram displays the number of books on the x-axis and the frequency on the y-axis. 86 Chapter 2 | Descriptive Statistics Figure 2.6 Go to Appendix G. There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for Example 2.10. • Press Y=. Press CLEAR to delete any equations. • Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2. • • Into L1, enter 1, 2, 3, 4, 5, 6. Note that these values represent the numbers of books. Into L2, enter 11, 10, 16, 6, 5, 2. Note that these numbers represent the frequencies for the numbers of books. • Press WINDOW. Set Xmin =.5, Xscl = (6.5 –.5)/6, Ymin = –1, Ymax = 20, Yscl = 1, Xres = 1. The window settings are chosen to accurately and completely show the data value range and the frequency range. • Press second Y=. Start by pressing 4:Plotsoff ENTER. • Press second Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the third picture (histogram). Press ENTER. • Arrow down to Xlist: Enter L1 (2nd 1). Arrow down to Freq. Enter L2 (second 2). • Press GRAPH. • Use the TRACE key and the arrow keys to examine the histogram. 2.10 The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 Twenty student athletes play one sport. Twenty-two student athletes play two sports. Eight student athletes play three sports. Calculate a desired bin size for the data. |
Create a histogram and clearly label the endpoints of the intervals. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 87 Example 2.11 Using this data set, construct a histogram. Number of Hours My Classmates Spent Playing Video Games on Weekends 9.95 19.5 5.5 23 20 Table 2.16 10 22.5 11 21.9 15 2.25 7.5 10 24 22.9 16.75 15 20.75 23.75 18.8 0 12.75 17.5 18 20.5 Solution 2.11 Figure 2.7 Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram. 2.11 The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram. 22, 35, 15, 26, 40, 28, 18, 20, 25, 34, 39, 42, 24, 22, 19, 27, 22, 34, 40, 20, 38, 28 88 Chapter 2 | Descriptive Statistics Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think would be appropriate. You may want to experiment with the number of intervals. Frequency Polygons Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals and resulting interval size, for both the x-axis and y-axis. The x-axis will show the lower and upper bound for each interval, containing the data values, whereas the y-axis will represent the frequencies of the values. Each data point represents the frequency for each interval. For example, if an interval has three data values in it, the frequency polygon will show a 3 at the upper endpoint of that interval. After choosing the appropriate intervals, begin plotting the data points. After all the points are plotted, draw line segments to connect them. Example |
2.12 A frequency polygon was constructed from the frequency table below. Frequency Distribution for Calculus Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 69.5 79.5 89.5 Table 2.17 59.5 69.5 79.5 89.5 99.5 5 10 30 40 15 5 15 45 85 100 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 89 Figure 2.8 Notice that each point represents frequency for a particular interval. These points are located halfway between the lower bound and upper bound. In fact, the horizontal axis, or x-axis, shows only these midpoint values. For the interval 49.5–59.5 the value 54.5 is represented by a point, showing the correct frequency of 5. For the interval occurring before 49.5–59.5, (as well as 39.5–49.5), the value of the midpoint, or 44.5, is represented by a point, showing a frequency of 0, since we do not have any values in that range. The same idea applies to the last interval of 99.5–109.5, which has a midpoint of 104.5 and correctly shows a point representing a frequency of 0. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side. 2.12 Construct a frequency polygon of U.S. presidents’ ages at inauguration shown in Table 2.18. Age at Inauguration Frequency 41.5–46.5 46.5–51.5 51.5–56.5 56.5–61.5 61.5–66.5 66.5–71.5 Table 2.18 4 11 14 9 4 2 Frequency polygons are useful for comparing distributions. This comparison is achieved by overlaying the frequency polygons drawn for different data sets. 90 Chapter 2 | Descriptive Statistics Example 2.13 We will construct an overlay frequency polygon comparing the scores from Example 2.12 with the students’ final numeric grades. Frequency Distribution for Calculus Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 69.5 79.5 89.5 Table 2.19 59.5 69.5 79.5 89.5 99.5 5 10 30 40 15 5 |
15 45 85 100 Frequency Distribution for Calculus Final Grades Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 69.5 79.5 89.5 Table 2.20 59.5 69.5 79.5 89.5 99.5 10 10 30 45 5 10 20 50 95 100 Figure 2.9 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 91 Suppose that we want to study the temperature range of a region for an entire month. Every day at noon, we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected. One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don't have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph. Constructing a Time Series Graph To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By using the axes in that way, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur. 92 Chapter 2 | Descriptive Statistics Example 2.14 The following data show the Annual Consumer Price Index each month for 10 years. Construct a time series graph for the Annual Consumer Price Index data only. Year Jan Feb Mar Apr May Jun Jul 2003 181.7 183.1 184.2 183.8 183.5 183.7 183.9 2004 185.2 186.2 187.4 188.0 189.1 189.7 189.4 2005 190.7 191.8 193.3 194.6 194.4 194.5 195.4 2006 198.3 198.7 199. |
8 201.5 202.5 202.9 203.5 2007 202.416 203.499 205.352 206.686 207.949 208.352 208.299 2008 211.080 211.693 213.528 214.823 216.632 218.815 219.964 2009 211.143 212.193 212.709 213.240 213.856 215.693 215.351 2010 216.687 216.741 217.631 218.009 218.178 217.965 218.011 2011 220.223 221.309 223.467 224.906 225.964 225.722 225.922 2012 226.665 227.663 229.392 230.085 229.815 229.478 229.104 Table 2.21 Year Aug Sep Oct Nov Dec Annual 2003 184.6 185.2 185.0 184.5 184.3 184.0 2004 189.5 189.9 190.9 191.0 190.3 188.9 2005 196.4 198.8 199.2 197.6 196.8 195.3 2006 203.9 202.9 201.8 201.5 201.8 201.6 2007 207.917 208.490 208.936 210.177 210.036 207.342 2008 219.086 218.783 216.573 212.425 210.228 215.303 2009 215.834 215.969 216.177 216.330 215.949 214.537 2010 218.312 218.439 218.711 218.803 219.179 218.056 2011 226.545 226.889 226.421 226.230 225.672 224.939 2012 230.379 231.407 231.317 230.221 229.601 229.594 Table 2.22 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 93 Solution 2.14 Figure 2.10 The annual amounts are plotted for each year. Then, consecutive points are connected with a line. 2.14 The following table is a portion of a data set from a banking website. Use the table to construct a time series graph for CO2 emissions for the United States. CO2 Emissions Ukraine United Kingdom United States 2003 352,259 2004 343,121 2005 339,029 2006 327,797 2007 328,357 2008 323,657 2009 272,176 Table 2.23 540,640 540 |
,409 541,990 542,045 528,631 522,247 474,579 5,681,664 5,790,761 5,826,394 5,737,615 5,828,697 5,656,839 5,299,563 Uses of a Time Series Graph Time series graphs are important tools in various applications of statistics. When a researcher records values of the same variable over an extended period of time, it is sometimes difficult for him or her to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot. 2.3 | Measures of the Location of the Data The common measures of location are quartiles and percentiles. Quartiles are special percentiles. The first quartile, Q1, is the same as the 25th percentile, and the third quartile, Q3, is the same as the 75th percentile. The median, M, is called both the second quartile and the 50th percentile. To calculate quartiles and percentiles, you must order the data from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. Recall that a percent means one-hundredth. So, percentiles mean the data is divided into 100 sections. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90 percent on a test. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the 94 Chapter 2 | Descriptive Statistics test scores are the same as or greater than your test score. Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75th percentile. That translates into a score of at least 1220. Percentiles are mostly used with very large populations. Therefore, if you were to say that 90 percent of the test scores are less, and not the same or less, than your score, it would be acceptable because removing one particular data value is not significant. The median is a number that measures the center of the data. You can think of the median as the middle value, but it does not actually have to |
be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data: 1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1 Ordered from smallest to largest: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 When a data set has an even number of data values, the median is equal to the average of the two middle values when the data are arranged in ascending order (least to greatest). When a data set has an odd number of data values, the median is equal to the middle value when the data are arranged in ascending order. Since there are 14 observations (an even number of data values), the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two. 6.8 + 7.2 2 = 7 The median is seven. Half of the values are smaller than seven and half of the values are larger than seven. Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second, quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q3, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 The data set has an even number of values (14 data values), so the median will be the average of the two middle values (the average of 6.8 and 7.2), which is calculated as 6.8 + 7.2 and equals 7. 2 So, the median, or second quartile ( Q2 ), is 7. The first quartile is the median of the lower half of the data, so if we divide the data into seven values in the lower half and seven values in the upper |
half, we can see that we have an odd number of values in the lower half. Thus, the median of the lower half, or the first quartile ( Q1 ) will be the middle value, or 2. Using the same procedure, we can see that the median of the upper half, or the third quartile ( Q3 ) will be the middle value of the upper half, or 9. The quartiles are illustrated below: Figure 2.11 The interquartile range is a number that indicates the spread of the middle half, or the middle 50 percent of the data. It is the difference between the third quartile (Q3) and the first quartile (Q1) IQR = Q3 – Q1. The IQR for this data set is calculated as 9 minus 2, or 7. The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than 1.5 × IQR below the first quartile or more than 1.5 × IQR above the third quartile. Potential outliers always require further This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 95 investigation. NOTE A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality, or they may be a key to understanding the data. Example 2.15 For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. Prices are in dollars. 389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000 Solution 2.15 Order the following data from smallest to largest: 114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000 M = 488,800 Q1 = 230,500 + 387,000 2 = 308,750 Q3 = 639,000 + 659,000 2 = 649,000 |
IQR = 649,000 – 308,750 = 340,250 (1.5)(IQR) = (1.5)(340,250) = 510,375 Q1 – (1.5)(IQR) = 308,750 – 510,375 = –201,625 Q3 + (1.5)(IQR) = 649,000 + 510,375 = 1,159,375 No house price is less than –201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier. 2.15 For the 11 salaries, calculate the IQR and determine if any salaries are outliers. The following salaries are in dollars. $33,000; $64,500; $28,000; $54,000; $72,000; $68,500; $69,000; $42,000; $54,000; $120,000; $40,500 In the example above, you just saw the calculation of the median, first quartile, and third quartile. These three values are part of the five number summary. The other two values are the minimum value (or min) and the maximum value (or max). The five number summary is used to create a box plot. 2.15 Find the interquartile range for the following two data sets and compare them. Test Scores for Class A: 69, 96, 81, 79, 65, 76, 83, 99, 89, 67, 90, 77, 85, 98, 66, 91, 77, 69, 80, 94 Test Scores for Class B: 90, 72, 80, 92, 90, 97, 92, 75, 79, 68, 70, 80, 99, 95, 78, 73, 71, 68, 95, 100 96 Chapter 2 | Descriptive Statistics Example 2.16 Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were as follows: Amount of Sleep per School Night (Hours) Frequency Relative Frequency Cumulative Relative Frequency 4 5 6 7 8 9 10 Table 2.24 2 5 7 12 14 7 3.04.10.14.24.28.14.06.04.14.28.52.80.94 1.00 Find the 28th percentile. Notice the.28 in the Cumulative Relative Frequency column. Twenty-eight percent of 50 data values |
is 14 values. There are 14 values less than the 28th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28th percentile is between the last six and the first seven. The 28th percentile is 6.5. Find the median. Look again at the Cumulative Relative Frequency column and find.52. The median is the 50th percentile or the second quartile. Fifty percent of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and 11 of the 7s. The median or 50th percentile is between the 25th, or seven, and 26th, or seven, values. The median is seven. Find the third quartile. The third quartile is the same as the 75th percentile. You can eyeball this answer. If you look at the Cumulative Relative Frequency column, you find.52 and.80. When you have all the fours, fives, sixes, and sevens, you have 52 percent of the data. When you include all the 8s, you have 80 percent of the data. The 75th percentile, then, must be an eight. Another way to look at the problem is to find 75 percent of 50, which is 37.5, and round up to 38. The third quartile, Q3, is the 38th value, which is an eight. You can check this answer by counting the values. There are 37 values below the third quartile and 12 values above. 2.16 Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65th percentile. Amount of Time Spent on Route (Hours) Frequency Relative Frequency Cumulative Relative Frequency 2 3 4 5 Table 2.25 12 14 10 4.30.35.25.10.30.65.90 1.00 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 97 Example 2.17 Using Table 2.24: a. Find the 80th percentile. b. Find the 90th percentile. c. Find the first quartile. What is another name for the first quartile? Solution 2.17 Using the data from the frequency table, we have the following: a. The 80th percentile is between the last eight and the first |
nine in the table (between the 40th and 41st values). Therefore, we need to take the mean of the 40th an 41st values. The 80th percentile = 8 + 9 2 = 8.5. b. The 90th percentile will be the 45th data value (location is 0.90(50) = 45), and the 45th data value is nine. c. Q1 is also the 25th percentile. The 25th percentile location calculation: P25 =.25(50) = 12.5 ≈ 13, the 13th data value. Thus, the 25th percentile is six. 2.17 Refer to Table 2.25. Find the third quartile. What is another name for the third quartile? Your instructor or a member of the class will ask everyone in class how many sweaters he or she owns. Answer the following questions: 1. How many students were surveyed? 2. What kind of sampling did you do? 3. Construct two different histograms. For each, starting value = ________ and ending value = ________. 4. Find the median, first quartile, and third quartile. 5. Construct a table of the data to find the following: a. The 10th percentile b. The 70th percentile c. The percentage of students who own fewer than four sweaters A Formula for Finding the kth Percentile If you were to do a little research, you would find several formulas for calculating the kth percentile. Here is one of them. k = the kth percentile. It may or may not be part of the data. i = the index (ranking or position of a data value) n = the total number of data • Order the data from smallest to largest. 98 Chapter 2 | Descriptive Statistics • Calculate i = k 100 (n + 1). • • If i is an integer, then the kth percentile is the data value in the ith position in the ordered set of data. If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set. The formula and calculation are easier to understand in an example. Example 2.18 Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, |
57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 a. Find the 70th percentile. b. Find the 83rd percentile. Solution 2.18 a. b. k = 70 i = the index n = 29 i = k 100 (n + 1) = ( 70 100 k = 83rd percentile i = the index n = 29 i = k 100 (n + 1) = ( 83 100 )(29 + 1) = 21. This equation tells us that i, or the position of the data value in the data set, is 21. So, we will count over to the 21st position, which shows a data value of 64. )(29 + 1) = 24.9, which is not an integer. Round it down to 24 and up to 25. The age in the 24th position is 71, and the age in the 25th position is 72. Average 71 and 72. The 83rd percentile is 71.5 years. 2.18 Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 Calculate the 20th percentile and the 55th percentile. NOTE You can calculate percentiles using calculators and computers. There are a variety of online calculators. A Formula for Finding the Percentile of a Value in a Data Set • Order the data from smallest to largest. • x = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile. • y = the number of data values equal to the data value for which you want to find the percentile. • n = the total number of data. • Calculate x +.5y n (100). Then round to the nearest integer. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 | Descriptive Statistics 99 Example 2.19 Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42 |
, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 a. Find the percentile for 58. b. Find the percentile for 25. Solution 2.19 a. Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58. x = 18 and y = 1. x +.5y n (100) = 18 +.5(1) 29 (100) = 63.80. Fifty-eight is the 64th percentile. b. Counting from the bottom of the list, there are three data values less than 25. There is one value of 25. x = 3 and y = 1. x +.5y n (100) = 3 +.5(1) 29 (100) = 12.07. Twenty-five is the 12th percentile. 2.19 Listed are 30 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 Find the percentiles for 47 and 31. Interpreting Percentiles, Quartiles, and Median A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15 percent of data values are less than or equal to the 15th percentile. • Low percentiles always correspond to lower data values. • High percentiles always correspond to higher data values. A percentile may or may not correspond to a value judgment about whether it is good or bad. The interpretation of whether a certain percentile is good or bad depends on the context of the situation to which the data apply. In some situations, a low percentile would be considered good; in other contexts a high percentile might be considered good. In many situations, there is no value judgment that applies. A high percentile on a standardized test is considered good, while a lower percentile on body mass index might be considered good. A percentile associated with a person's height doesn't carry any value judgment. Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of |
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