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closer to campus. 33. 25 0 0 5 10 15 20 25 30 35 1134 Answers to Selected Exercises 35. Sample answer: the histogram is not as symmetric as the stem plot. The histogram more accurately shows the distribution of the data due to the smaller class interval of 5. Section 13.2, page 862 1. approximately 43.429 3. approximately 7.583 5. 42 9. mean: 12.2 median: 12 mode: 13 7. 5.15 11. 53 13. Grains 15. The median is larger than the mean. 17. The mean and the median are the same. 19. approximately 2.828 23. 21. approximately 7.071 s 5.249; s 5.447 s 13.176; s 13.518 25. 27. 19 31. 7 29. 44 33. 21 35. five-number summary: 8, 17, 18.5, 24, 27 37. five-number summary: 50, 62.5, 73.5, 83.5, 94 50 94 39. mean: 43.167 median: 42.5 41. standard deviation: 20.3 range: 80 interquartile range: 35 43. The sample standard deviation is a good measure of dispersion because the data set is relatively small. 3 represents 23 6 represents 56 8 27 5910aans_1114-1147 9/21/05 2:33 PM Page 1135 45. 90.4 47. median mean skewed right 49. The median more accurately describes the βtypicalβ salary, since the outlier of 105,000 has a large effect on the mean. 51. Answers may vary. Sample: data 26,28,30,32,34 20,25,30,35,40 mean 30 30 standard deviation 3.162 7.906 53. The mean will increase by the value of k. 55. The mean will be multiplied by the constant k. 57. All data values must be the same. Section 13.3, page 872 1. {A, B, C, D} 5. Outcome Probability 7. 1 216 3. 0.8 black 1 2 red 1 3 white 1 6 Section 13.4, page 882 1. Outcome Probability red 0.50 blue 0.31 green 0.15 yellow 0.04 3. 0.0016 5. Outcome Probability 7. 0.0001 nun 0.1 |
gimel 0.45 hay 0.24 shin 0.21 Outcome 9. Answers may vary. Sample: 6 2 5 1 50 50 Probability 3 4 50 5 6 50 4 6 50 7 7 50 8 6 50 9 10 11 12 5 1 50 50 5 50 4 50 15. 0.0016 19. 2048 23. 362,880 27. 12,870 11. Answers may vary. Sample: 6.89 13. approximately 0.0027 17. 0.56; approximately 0.176 21. 3,268,760 25. approximately 0.04 29. approximately 3.108 10 4 31. 48,228,180; 1.03669 1051; 365! 365 n 1! 2 33. probability 1 365 Pn 365n n 3: approximately 0.008 n 20: approximately 0.411 n 35: approximately 0.814 35. 366 people 9. approximately 0.05 or 5% Section 13.4A, page 889 11. {0, 1, 2} 13. approximately 39% 15. Number of blue Probability 0 25 64 1 15 32 30 64 2 9 64 17. 12.52 19. 17.2 21. 0.41 23. 0.026 25. 0.5 0.4 0.3 0.2 0.1 1 2 3 4 5 27. 1 to 5 31. approximately 0.55 29. 1 2 33. The median appears to occur at 15 inches of rain. 35. 5 10 or 50% 3. 0.015 0.111 0.311 0.384 0.179. 0.35 5.316 0.422 0.211 0.047 0.004 7. approximately 0.738. approximately 0.0796 11. approximately 0.000023 13. expected value: 50; standard deviation: 5 15. approximately 0.160; approximately 0.185 Section 13.5, page 896 1. 20 3. approximately 0.68 Answers to Selected Exercises 1135 5910aans_1114-1147 9/21/05 2:34 PM Page 1136 5. 0.1 5. quantitative β40 60 β0.01 7. 0.1 0 β0.01 1000 9. the mean is 70. The standard deviation is 5.228. 11. 0.5 13. 0.68 15. 0.8385 17. 19. 560 S 0.6 z-value 450 S 0.5 |
z-value 640 S 1.4 z-value 530 S 0.3 z-value $0.95 S 4.29 z-value $1.00 S 3.57 z-value $1.35 S 1.43 z-value 23. 0.48 25. 0.16 21. 0.19 27. 0.815 31. 63.25; Q3 Q1 fall between 63.25 and 76.75. 29. 0.48 76.75; Fifty percent of the scores Chapter 13 Review, page 900 1. qualitative 3. 25 20 15 10 5 sparrow purple finch chickadee cardinal bluejay 1136 Answers to Selected Exercises 7. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Key 18 0 2 represents 18.2 9. The distribution is skewed left. 11. 31.2 13. 4.72 15. Q1 27.7; Q3 33 17. 18.2 39 19. Outcome Probability 1 0.1 21. 0.7 2 0.2 3 0.4 23. 0 to 3 4 0.2 5 0.1 25. Number of red Probability 0 0.512 1 0.384 2 0.096 3 0.008 27. 1.0 0.5 1 2 3 4 29. 0.4; 0.53; 0.07 31. 0.5 33. 0.86 35. approximately 0.0048 37. 3.08; 1.3 39. 0.1 40 β0.01 110 for h 4; area 1.856 41. 0.68 43. 99.85% 45. 0.55 Chapter 13 can do calculus, page 907 In Exercises 1β6, estimates may vary. 1. 1 3 0.14 2. 2 3. 4.5 4. 5. 0.51 5.67 7. Your answer should be close to 0.7301. A sample is shown of the program using the normal curve 1 1812p with equation a 100, x 138 648 y 1 e 6. ; 2 2 b 150, and h 0.025. h 5; area 2.02 3.7 x 5.7, 1 y 5 for (Scales The area estimates get larger for larger values of h. h 5 The value for is close to 2, which is the exact area under the curve from 0 to 1. |
2 9. A sample is shown of the program using the equation of the standard normal curve with a 3, h 0.5. close to the expected area of 0.997, or 99.7% of the area under the curve. The estimate is very b 3, and (Scales 84 X 192, 0.01 Y 0.05 ) 8. Samples of the program are shown for h 3; area 1.74 (Scales 3 x 3, 0.5 y 1 2 Chapter 14 Section 14.1, page 916 1. x f(x) 2.9 2.99 2.999 0.25641 0.25063 0.25006 Answers to Selected Exercises 1137 x 3.001 3.01 3.1 lim xS3 3. f(x) 0.25 x f(x) 0.24994 0.24938 0.2439 0.1 0.01 0.001 0.35809 0.354 0.3536 x 0.001 0.01 0.1 0.35351 0.35311 0.34924 f(x) 0.3535 x f(x) 7.1 7.01 7.001 0.1662 0.1666 0.1667 lim xS0 5. x 6.999 6.99 6.9 0.1667 f(x) 0.1667 0.1667 0.1671 x f(x) 0.9 0.99 0.999 0.1149 0.1115 0.1111 lim xS7 7. x 1.001 1.01 1.1 0.1111 0.1107 0.1075 f(x) 0.1111 lim xS1 0.1 0.01 0.001 0.04996 0.005 0.0005 29. 31. lim xS3 f x 1 2 2; lim xS0 f x 1 2 does not exist; lim xS2 f x 1 2 0 lim xS3 f x 1 2 2; lim xS0 f x 1 2 1 ; lim xS2 f x 1 2 1 33. a. y 5 4 3 2 1 β2 β1 1 2 x b. c. lim xS2 lim xS1 f f 2 x 1 x 2 1 does not exist. 3 d. lim xS2 f x 1 2 does not exist. 35. No matter how close x gets |
to c, there are still an infinite number of both rational and irrational numbers between x and c, so t(x) will take the values 0 and 1 an infinite number of times, but never get close to a single number for all values of x that are very close to c. Section 14.2, page 923 1. 5 5. 0 13. 214 17. 1 2 3. Limit does not exist. 7. 3 5 9. 47 11. 7 15. 1 25 19. 0 23. The limit does not exist. Values of x less than are mapped by the function to more than 3 are mapped by the function to 1. 1 1 21. 222 3 and values of x 0.001 0.01 0.0005 0.005 0.1 0.05 25. 12 27. 1 222 29. As the angle t in standard position gets closer to 13. 1.25 21. 0.2887 1; lim xS0 15 1.5 23. 0 17. 0.333 25. 1 1; f x 1 2 lim xS2 f x 2 1 does not, the x-coordinate of the point at which the p 2 terminal side of t crosses the unit circle gets closer to 0. Since this x-coordinate is the cosine of the angle t, we have cos t 0. lim tS p 2 x f(x) x f(x) 0 9. lim xS0 11. 1.5 2 19. 27. f x 1 2 lim xS3 exist 1138 Answers to Selected Exercises 31. Using the function, when x is When x is close to 2 but slightly less 2. x 2 close to 2 but slightly more than 2, we have 3. x 3 x than 2 we have In this case, the difference is either 5 or 4 depending on whether x is more than 2 or less than 2 and so the limit doesnβt exist. 1 x 2, 3 4 33. 35. Many correct answers, including f 0 x 0x,, and c 0. In this case, lim xS0 f x 1 2 does not exist by Example 10; a similar argument shows that f g 2 by Exercise 22. does not exist. But g 1 0 lim xS0 x x lim xS0 1 21 2 Section 14.2A, page 927 1. 0 3. 0 5. 6 7. a. 0 b. 1 c. 1 d. 1 |
9. a. 1 b. 0 d. Limit does not exist. c. Limit does not exist. 11. a. 5 b. 0 c. 4 d. 2 13. 3 19. 2.5 15. 1 8 21. 2 17. Limit does not exist. 23. lim xS2 x 3 4 2 and lim xS2 3 x 4 1 Section 14.3, page 935 The symbol means βimplies.β 1 1. Given e 7 0, let Then 0 6. d e 3 x 3 0 3x 2 0 d e 6x 20 Then 6 e 0 6 0 x 5 0 Then 0 6 6 e 1 15 3x 9 f x 0 7 2 1 3. Given x 5 0 5. Given, 6 e 1 let f 0 1 0 e 7 0, let 0 x 2 0 6x 12 f x 0 15. Given 0 e 7 0, 0 0 0 0 0 let be any positive number. Then d for every number x (including those satisfying let d e. x 6 Then. Then let 2x 5 2 0 1 22. Given x 4 0 x f 1 2 11. Given e 7 0 2x, 2 1 13. Given 0 6 x2 0 0 15. Let x 0 0 6 e 1 e 7 0. such that if let 6 d 1 x2 0 d 2e Then 6 2e Then there exist positive numbers 6 d1, x c then L 0 6 x f 0 d1, d2 6 e 2 0 x c 0 6 d2, and if 0 6 Choose d then g x 0 0 0 to be the smaller of 2 1 d1, d2. 0 Then if 2 1 0 M 0 6 e 2. we have both and 0 0 f x x c 6 d Then. rewritten as 0 Now using the triangle inequality we have x f 0 1 lim xSc 2 L M 2 0 L M. 0 6 e. Therefore. 2 0 x 1 1 1 22 1 2 2 1 1 e. This can be Section 14.4, page 946 x 3, x 6 3. Continuous at 1. x 0 x 2 and x 3; discontinuous at 5. Continuous at x 2 x 0 and x 3; discontinuous at 7. f lim 1 xS3 lim 5 1 xS3 9 5 x 2 x 2 1 3 2 lim xS3 7 x2 5 1 lim xS3 2 2 7 14 f 1 2 x 2 7 1 |
x2 2 2 lim xS3 1 5 2 lim xS3 lim xS3 1 x 2 x 2 1 22 7 3 1 2 x2 9 9. lim xS2 lim xS2 1 f 2 1 x lim xS2 1 lim xS2 1 x2 x 6 2 22 9 x2 x 6 x2 9 lim xS2 1 x2 6x 9 21 2 x2 6x 9 2 5 2 1 4 25 2 21 2 22 6 2 9 21 22 2 6 1 1 f 20 2 1 2 Answers to Selected Exercises 1139 f has a removable discontinuity at Chapter 14 Review, page 961 11. lim xS36 f x 1 2 lim xS36 x 6 lim xS36 1 36 6 30 30 2 B 2 2x x 6 x2x x 6 lim xS36 A lim xS36 1 x lim xS36 lim xS36 1 6 1 25 x 3. x 1. 1;, but f 0 f 2 2 216 900 0 36 1 2 1 2 13. f is not defined at 15. f is not defined at 17. lim xS0 f x 1 2 36136 36 6 36 6 21 2 hence lim xS0 f x 2 1 f 0 1 2 19. Continuous 23. Continuous 21. Continuous 25. Continuous at every real number except 27. Continuous at every real number except x 3 x 0 and. 29. 31. 33. 1 2 2x f has a removable discontinuity at g x 2 1 x 4. 35. If and 1 for all 1 x x 0, then x g 1 x 6 0. 2 1 Thus for all for all g lim xS0 x 2 2 does not exist. Hence the definition of continuity cannot be satisfied, no matter what g(0) is. 1 9. No vertical asymptotes; 0 x xSq f lim 1 2 xSq f lim x 1 2 q; 11. No vertical asymptotes; xSq f lim x 1 2 1 xSq f lim x 1 2 2; 13. Vertical asymptote x 10; q x 1 2 xSq f lim y 3 4 15. xSq f lim x 1 2 q; 17. y 2 19. No horizontal asymptote 21. 1 2 23. q 25. 5 27. 2 29. 3 22 31. 22 3 33. 23 35. 1 37. 1 39 |
. 0 41. 0 1140 Answers to Selected Exercises 43. With a parachute: 20 ft/sec Without a parachute: 177.78 ft/sec 45. 1 47. 1 49. The first part of the informal definition is included in the second part, which says βthe values of f(x) can be made arbitrarily close to L by taking large enough values of x.β This means that whenever you specify how close f(x) should be to L, we can tell you how large x must be to guarantee this. In other words, you specify how close you want f(x) to be to L by giving a positive number and we tell you how large x must be to guarantee that f(x) is within of L, that is, to guarantee that f 0 0 number k such that x 7 k. e, positive number e (depending on ) with this property: 6 e. 0 This can be reworded as follows: For every We do this by giving a positive there is a positive number k e 6 e. whenever x 7 k, then L L L 6 e If. 2 9. 1 2 3. 2 5. 5 7. 2 11. 4 13. 1 15. Given e 7 0, let Then 2x 6 0 7 f x 17. Continuous at 2x ; discontinuous at x 2 x lim xS2 2 1 x2 x 6 x2 9 2 lim 1 xS2 lim 1 xS2 2 22 2 6 22 9 4 5 f 2 4 5 2 b. f is not defined at 1 x 3 and hence is discontinuous there. 21. Vertical asymptotes at x 1 and graph moves upward as x approaches 1 the left and downward as x approaches the right. The graph moves downward as x approaches 2 from the left and upward as x approaches 2 from the right. x 2. 1 The from from 23. 1 2 25. 10 3 27. y 1 2 Chapter 14 can do calculus, page 967 1. a. upper estimate: 259 ft. lower estimate: 157 ft. Section 14.5, page 957 1. 1 3. 0 5. 0 7. 0 19. a. lim xS2 f x2 x 6 x2 9 b. 110 0 β5 77. Many possible examples, including but 6 3 22 4 2 26 23 64 8 8, 79. False for all nonzero a; for instance, 3 32 9 9, but 3 3 2 |
1 2 1 21 2 5 Section A.2, page 976 c. less than 5 ft: less than 1 ft: Β’t 6 0.04902 sec. Β’t 6 0.0098 sec. 2. Lower estimate: 21 Upper estimate: 25 For the lower estimate, count all of the complete squares beneath the curve. For the upper estimate, count all of the complete squares below the curve and estimate the number of partial squares below the curve. 3. a. 1.798 b. The degree of accuracy can be Β’t. increased by lowering the 5. 1.0 4. between 15 and 17 ft. 6. between 0.67 and 0.72 ft. Algebra Review Section A.1, page 972 1. 36 7. 125 64 13. 19. 25. 31. 37. 43. 81 16 x10 9x4y2 ab3 3xy x7 49. a12b8 55. a7c b6 3. 73 9. 1 3 15. 21. 27. 33. 39. 45. 51. 57. 211 216 0.03y9 21a6 1y3 8x 212 ce9 1 c10d6 1 c3d6 2 5. 5 11. 112 17. 23. 29. 35. 41. 47. 53. 59. 129 8 24x7 384w6 3 a8x 12 2 b2c2d6 1 108x 1 a 1 a 2 61. Negative 63. Negative 67. 3s 69. a6tb4t 65. Negative brsst c2rt 71. 73. Many possible examples, including 32 42 9 16 25, but 2 75. Many possible examples, including 1 3 4 2 72 49 32 23 9 8 72; 3 2 23 65 7776 but 1 2 2a2b 5u3 u 4 5xy x 15. 15y3 5y 12a2b 18ab2 6a3b2 2x2 2x 12 6x2 x 35 x2 16 y2 22y 121 16x6 8x3y4 y8 2y3 9y2 7y 3 x3 6x2 11x 6 61. 1 9 61y y 3ax2 3b 2a 1 63. 5 x 2b 2 ab ac bc x abc xmn 2xn 3xm 6 2 2 2 32; 2; 2 1 3 4 1 2 3y 6 2 2 3 then |
2 x2 2xy y2 7 2 1 7y 21 2 2 2 23; 7 3 2 21 49xy then 7x 1 2 2 correct 1. 5. 9. 11. 17. 19. 23. 27. 31. 35. 39. 43. 47. 49. 53. 55. 65. 69. 73. 75. 77. 81. 3. 25. 29. 37. 33. 45. 41. 21. 8x x3 4x2 2x 3 7. 4z 12z2w 6z3w2 zw3 8 3x3 15x 8 13. 12a2x2 6a3xy 6a2xy 12z4 30z3 x2 x 2 y2 7y 12 3y3 9y2 4y 12 16a2 25b2 25x2 10bx b2 9x4 12x2y4 4y8 15w3 2w2 9w 18 24x3 4x2 4x 51. 3x3 5x2y 26xy2 8y3 6 59. 6 x ab x2 a2 b2 2 1 a b c 3 x 25 13x2 4x 13 abx2 x3 1 34rt 2x4n 5x3n 8x2n 18xn 5 y 4, 4 2 correct statement: x 2, correct statement: x 2, 3 then 1 y 2 3 1 y 3, x y 2 1 y 3, 67. 79. 57. 71. 1 2 83. Example: if 85. Example: if 87. Example: if 7 2 3; 89. Example: if statement: 91. Example: if correct statement: then y 2, y y y 3y x 4, then 42 5 4 6; x2 5x 6 4 3 1 correct statement: 21 4 2 2 x 3 1 x 2 2 21 93. If x is the chosen number, then adding 1 and squaring the result gives 1 from the original number x and squaring the result gives Subtracting the second of these Subtracting 1 x 1 2. 2 x 1 2. 1 2 Answers to Selected Exercises 1141 squares from the first yields: 1 x2 2x 1 x2 2x 1 1 1 the original number x now gives 2 2 2 4x. 4x x x 1 2 1 2 2 x 1 Dividing by 4. So the answer is always 4, no matter what number x is |
chosen. 95. Many correct answers Section A.3, page 981 21 21 9x 2 7 2z x 3 21 y 9 x 5 21 3x 1 21 1 x 8 9x 4u 3 x 5 x 2 2 x 5 2x y 21 x3 23 21 2u 3 21 x2 5x 25 2 3. 7. 11. 15. 19. 23. 27. 31. 35. 41. 45. 2 B x y 2 1 1 A 21 3y 5 2 15 x BA x y 3y 5 21 1 15 x x2 y2 21 z 1 z 3 21 x 3 2 x 9 21 2z 3 21 1 x 1 2 21 1 2x 5y x 2 2 z 4 2 5x 21 21 4 2x x2 x2 x 1 2 2 x2 2x 4 53. 2 21 4x2 2xy y2 x3 23 2 x 2 21 1 y2 2 21 z2 z 1 x2 y 21 a2 b 21 x 4 21 y2 5 z 1 21 x2 3y a 2b x2 8 21 x2 1 x 2 x2 2x 4 21 3 y 3 y 2 21 2 1 21 9 y2 21 z2 z 1 x z 21 1 21 59. 2 x y 65. If 1 2 x c x 18 A x d cd 1. and hence that 21 and c d 0 c d then that or equivalently, that number with this property, factor in this way. 2 d2 1. x 18 x 4 1 2 x cd, BA x2 But 1 cd B 1 c d 2 c d 0 implies d d2, d Since there is no real cannot possibly x2 1 2 1 Section A.4, page 985 3. 9. 195 8 1 x 15. 21. ce 3cd de x 3 x 4 2 2 1 5. x 2 x 1 11. 17. 23. 29 35 b2 c2 bc 2x 4 3x 4 x 1 9 7 a b a2 ab b2 121 42 1 x 1 2 x 1 x2 xy y2 x y x3 y3 1. 5. 9. 13. 17. 21. 25. 29. 33. 37. 39. 43. 47. 49. 51. 55. 57. 61. 63. 1. 7. 13. 19. 25. 1142 Answers to Selected Exercises 27. 6x5 38x4 84x3 71x2 14x |
1 3 3 4x x 1 x 2 1 2 1 2 29. 2 31. 37. 43. 49. 55. 2 5y2 y 5 2 3 1 35 24 x2y2 x 2y x y 21 1 3y 3 y 2 2 3c 1 2 39. u 1 u 45. u2 v w 33. 3y x2 35. 41. 47. u v 21 1 2u v 21 1 x 3 2x 12x x 3 4u 3v 2 2u 3v 2 53. 59. y x xy xy x y 1 1 2 ; cd 51. 57 then 1 b a ab then 1 1 a b b 9, ; correct statement: 61. Example: if a 1, b 2, correct statement: a 4, 2 1 4 9 63. Example: if 1 14 19b 2 1 1a 1bb a a 1 a 21ab b 65. Example: if statement: 67. Example: if 24 29 A u 1, v 2, u2 v2 vu y 9, 1 24 29 u v u v x 4, B then 1 2 2 1 1; correct then 4 9; correct 2 statement: 1x 1y A B 1 1x 1y 1 Section A.5, page 991 1. 13; 1 a 2, 1 b 3. 217; 3 2 a, 3 b, 3 2b 22 23 a 2 a b 2 b, (6, 36) 5. 26 216 1.05. 22 9. a. 40 M 30 M 20 M 10 M (0, 12) 2 0 (1992) 1 2 3 4 5 6 7 b. 40 M 30 M 20 M 10 M (3, 24) 0 (1992) 1 2 3 4 5 6 7 c. In 1995, about 24 million personal computers were sold. We must assume that sales increased steadily. 2 x 3 2 1 1 x2 y2 2 y 4 2 4 2 35. x2 y2 4x 2y 0 3, 4 and 2 1 k 7 d. 39. Assume 2, 1 37. 2 1 The other two vertices of one 1 possible square are those of another square are, d c k d those of a third square are 41. (0, 0), (6, 0) 43. 45. 3, 5 111 A x 6, A B 3, 5 111 B 47. M has coordinates r 2b formula. Hence the distance from M to ( |
0, 0) is 2 by the midpoint and the distance from M to (0, r) is the same: B a b s2 4 r2 as is the distance from M to 1 s 2b a 2 s 2b r2 4 s2 4 2 2 s 2b r2 4 s2 49. Place one vertex of the rectangle at the origin, 11. 13. 15. 17. 4 1 β2 β4 1169.25 2, 2 has length 2 18. and Since 1 12 this is a right triangle. 110; 1 other sides have lengths 2 2 1 2 145 2, to 3, 2 15 has length 145. 2 and Since this is a right triangle. 19. Center 21. Center 23. Center 1 1 1 radius radius 2, 4, 3 3, 2, 2 12.5, 5 2110 217, radius 2 25. Hypotenuse from (1, 1) to other sides have lengths 110 12 2, B 2, 3 A 27. Hypotenuse from 18 2 2 B A B A A 150; 15 29. 31. 33 150 A 2 8 2 8 2 16 1 1 2 the be the coordinates 0, b with one side on the positive x-axis and another on the positive y-axis. Let a, 0 2 of the vertex on the x-axis and coordinates of the vertex on the y-axis. Then the fourth vertex has coordinates picture!). One diagonal has endpoints and 2 2a2 b2. b 0 2 diagonal has endpoints (0, 0) and has the same length: 2 2 so that its length is, 2 The other a, b a, 0 2 1 0 a 2 (draw a 0, b and hence a 2a2 b2. 0 a 2 1 51. The circle k, 0 to (0, 0)). So the radius family consists of every circle that is tangent to the y-axis and has center on the x-axis. 1 x k 2 y2 k2 1 (the distance from has center and k 53. The points are on opposite sides of the origin because one first coordinate is positive and one is negative. They are equidistant from the origin because the midpoint on the line segment joining them is, 0. 2 1 Answers to Selected Exercises 1143 35 8 5 u 33. 4032 35. 160 1 Advanced Topics Section B.1, page 1000 1. 720 3. 220 5. 0 7. |
64 9. 3,921,225 x5 5x4y 10x3y2 10x2y3 5xy4 y5 a5 5a4b 10a3b2 10a2b3 5ab4 b5 32x5 80x4y2 80x3y4 40x2y6 10xy8 y10 x3 6x21x 15x2 20x1x 15x 61x 1 1 10c 45c2 120c3 210c6 120c7 12 4x 8 6x x 8i 45c8 10c9 4 4 x4 10x3y2 210c4 252c5 23. 56 29. 27. c10 35c3d4 11. 13. 15. 17. 19. 21. 25. 31. 37. a. b. 9 1b a n 1b a 9; 9! 1!8! n n 1b a 39. 2n 1 1 1 2 n 1n 9 8b a 9! 8!1! 9 n! n 1! 2 1n1 1 1! 1 n 1b 2b a 1n2 12 n 3b a 1n3 13 p 11 1n1 1n n 0b a n 1b a n nb n n 1b a 6 cos2 u sin2 u 41. 43. a. n n 1b a n 2b n 3b p a a a cos4 u 4i cos3 u sin u 4i cos u sin3 u x h f f 2 1 x5 5 1b 1 2 1 5 x4h 2b a xh4 h5 x5 sin4 u x a a 5 4b 5 2b x h 2 h 1 a f b. 5 3b a f x 1 2 x h 2 x3h2 5 x5 5 3b a x2h3 x4h 5 1b a 5 4b a 5 1b a x4 5 2b a x3h 5 3b a x2h2 5 4b a xh3 h4 c. When h is very close to 0, so are the last four f f x terms in part b, so 1 2 1 x h 2 h x4 5x4. 5 1b a x h 2 h f 1 45. f x 1 2 1 x h 12 2 b a x10h 12 3 b a x9h2 12 4 b a 12 x12 2 h |
12 1 b a x8h3 p x11 1144 Answers to Selected Exercises n r 3 1 21 47. a. x11 x2h9 a 12 11 b a xh10 h11 12 1 b when h is very close to 0 21 r 1 r 1 2 21 n r n r 12 10 b a 12x11, n r 1 n r 1 b. Since, n r! 2 n 1b n! n r n! 1 r 1 1 3 1 a r rb n. For example, rows 2 and 3 of Pascalβs triangle 1b are 1 1 2 1 3 3 1 that is, 2 0b a 2 1b a 2 2b a a 3 1b a 3 2b a 3 3 0b 3b a The circled 3 is the sum of the two closest entries in the row above: 2, 1b 2 3 0b a 1b a r 0. Similarly, in the general case, and n 2 verify that the two closest entries in the row But this just says that 1 2. which is part c with a n 1 r 1b a are n rb a and n r 1b a and use above part c. 1. Step 1: For n 1 1 21 1, to 2k Add n k: the statement is which is true. Step 2: Assume that the statement is true for that is, 1 2 22 23 p 2k1 2k 1. both sides, and rearrange terms: 1 2 22 23 p 2k1 2k 2k 1 2k 1 2 22 23 p 2k1 21 1 2 1 2k 1 1 2 22 23 p 2k1 21 1 2k1 1 But this last line says that the statement is true for n k 1. Therefore, by the Principle of Mathematical Induction the statement is true for every positive integer n. k1 k1 2 2 2 x3h2 x2h3 xh4 h5 Section B.2, page 1009 Note: Hereafter, in these answers, step 1 will be omitted if it is trivial (as in Exercise 1), and only the essential parts of step 2 will be given. n k: 3. Assume that the statement is true for 2 2 2 2k 1 k2. 1 to both sides: 2 1 3 5 p 1 k 1 Add 1 1 3 5 p 2k 1 1 k2 2k 1 k2 2 k 1 The first and |
last parts of this equation say that the statement is true for n k 1. 1 2. Assume that the statement is true for k 1 12 22 32 p k2 k k 1 Add to both sides: 2 12 22 32 p k2 k 1 1 1 2 n k: 2k 1 21 2k 1 2 21 6 k 1 k k 1 1 21 2 2k 1 6 2k 1 6 2k2 7k 6 k 1 2 2 2k 3 2 21 2 3 21 21 The first and last parts of this equation say that the statement is true for n k 1. 7. Assume that the statement is true for n k 21 k 1 1 k 2. 2 to 1 k 1 1 4 2 Adding k 1 2 3 1 1 both sides yields 21 21 21 21 k2 2k 21 2 k 1 k 1 1 2 1 The first and last parts of this equation show that the statement is true for n k 1. n k: 9. Assume the statement is true for k 2 1 7 Adding 1 to both sides, we have: k 1 k 1,. 1 Therefore, the statement is true for n k 1. or equivalently. 11. Assume the statement is true for and hence that 3k1 3 3k. Multiplying both sides by 3 yields: or equivalently, 3k 3 know that 2 3k 3k 3 3k, Therefore, 3 3k 3k 3. with the fact that 3k1 3k 3, Therefore, the statement is true for 3k1 3 3k, or equivalently, n k: 3k 3k. 3 3k 3 3k, we k 1, Now since 2 3k 3. or equivalently, we see that 3k1 3 k 1 2 1 n k 1.. Combining this last inequality 13. Assume the statement is true for n k: Adding 3 to both sides yields: k 1 3k 7 k 1. 3k 3 7 k 1 3, 3. 7 3 2 1 2 certainly greater than k 1 k 1 that statement is true for k 1 7 3 1 2 1 1 3 or equivalently, k 1 1 1, 2 Therefore, the Since k 1 1 1. 2 n k 1. 2 is we conclude then 3 is a n k ; 15. Assume the statement is true for 22k1 1; that is, 22k1 1 3M for Now 22k1 3M 1. 1 k1 2 |
1 3M 1 4M 1 factor of some integer M. Thus, 22k21 22 4 1. 3 1 1 this equation we see that k1 1 1. 22 3 2 1 n k 1. Therefore, the statement is true for 2 From the first and last terms of 2 22 22k1 3 1 222k1 4M 3 Hence, 3 is a factor of 12M 4 4M 1 k1 22 1 1 2 2. 1 2 1 1 17. Assume the statement is true for n k: 64 is a 32k2 8k 9 32k2 9 32 2k2 k1 2 Then 32k2 8k 9. factor of 8k 64N for some integer N so that k1 32k22 2 32 64N. Now 2 1 1 32 32k2 8k 9 64N Consequently, 9. 2 1 2 8 k1 9 32 k 1 32 2 8k 17 k1 2 1 32 9 72K 81 9 64N 8k 17 64k 64 9 64N 64 8k 9 64N k 1 9N 8k 17 8k 8 9 From the first and last parts of this equation we 2 8 9. see that 64 is a factor of Therefore, the statement is true for k1 32 2 2 1 19. Assuming that the statement is true for c 2d 1 d p c k 1 2c c d c 2 Adding 3 c kd 2. 4 3 1 2 to both sides, we. n k: d k 1 have c 1 c d 1 2 c 2d kd 2 d 2 4 c kd 2c 2c 2ck k 1 4 1 2 c kd d 2 2 k 1 d 2c 2kd 2 2 2 Answers to Selected Exercises 1145 2 1 k 1 2ck 2c kd 1 2 2c kd 2 2c kd 2 2c k 1 k 1 1 2 1 21 k 1 2 2kd 2c kd 21. 21. a. Therefore, the statement is true for x y 21 x y 21 x y 21 xn yn x2yn3 x y ; 2 x2 xy y2 2 x3 x2y xy2 y3 xn1 xyn2 yn1 x2 y2 1 x3 y3 1 x4 y4 1 b. Conjecture: x y 21 ; 1 2 xn2y. 2 n 2, 3, 4, xn3y2 |
p Proof: The statement is true for by part a. Assume that the statement is true for n k: xk yk x y xk1 xk2y p xy k2 y k1 yxk yxk 0 to write 21 1 Now use the fact that xk1 yk1 as follows: xk1 yk1 xk1 yxk yxk yk1 xk1 yxk x y x y 2 xk y 1 xk y 1 1 xk yk 2 xk1 xk2 y x y 21 xk3y2 p xyk2 yk1 yxk yk1 xk2y2 xk3y3 p xyk1 yk xk1y xk 21 1 2 1 x y xk xk1y xk2y2 2 3 xk3y3 p xyk1 yk The first and last parts of this equation show that the conjecture is true for Therefore, by mathematical induction, the n 2. conjecture is true for every integer n k 1 12 1, 25. True: Proof: Since n 1. 23. False; counterexample: 1 1 the statement 2 1 Assume the statement is true for 2 7 k2 1. 2 k 1 1 k2 2k 2 2 7 is true for n k: k 1 2 1 1 k2 2k 1 1 The first and last terms of this inequality say that the statement is true for statement is true for every positive integer n. k 1 3 1 1 2 1 k2 2k 2 Therefore, by induction the Then 1 7 k2 1 2 k 1 n k 1. 2 2 1. k 1 4 2 2 2 1 27. False; counterexample: n 2 the statement is true for n k 2 5 4 7 5, 29. Since n 5. Assume the statement is true for k 5 (with shows that k 1 2 2 1 k 1 that 2 n k 1. for 4 7 k 1. Therefore, by the Extended : 2 2k 4 2 7 k 2, Adding 2 to both sides or equivalently, k 2 7 k 1, we see So the statement is true 4 7 k 2. 2k 4 7 k. Since 2 1 1146 Answers to Selected Exercises 31. Since 33. Since n 2. n 5. the statement is true for k 2 Principle of Mathematical Induction, the statement is true |
for all 22 7 2, Assume that n k: for k2 1 7 k 1. inequality show that the statement is true for n k 1. is true for all 34 81 and that the statement is true k2 2k 1 7 Therefore, by induction, the statement 24 10 4 16 40 56, The first and last terms of this k2 7 k. n 2. k 1 Then 2 2 1 we So the statement is true n 4. and 34 7 24 10 4. Assume that see that for statement is true for Multiplying both sides by 3 yields: 2k 10k 3 3k 7 3, 2 1 3k1 7 3 2k 30k. or equivalently, But n k: k 4 and that the 3k 7 2k 10k. 3 2k 30k 7 2 2k 30k 2k1 30k. Now we shall show 3k1 7 2k1 30k. Since. 20k 7 80 7 10. k 4, Therefore, 30k 7 10 that 20k 20 4, both sides of equivalently, k 1 2 1 so that 20k 7 10 30k 7 10 yields: k 1. 1 3k1 7 2k1 30k 7 2k1 10 2 we have Adding 30k 7 10k 10, Consequently, k 1. 1 2 to 10k or The first and last terms of this inequality show that the statement is true for the statement is true for all n k 1. n 4 Therefore, 35. a. 3 (that is, n 3; for 22 1 for 2 15 (that is, n 2; 24 1 b. Conjecture: The smallest possible number of by induction. 23 1 7 (that is, n 4. for 2 2 2n 1. n 1 or and that we Assume it is true for k 1 n k rings to move. In order to move the moves for n rings is Proof: This conjecture is easily seen to be true for n 2. have bottom ring from the first peg to another peg (say, the second one), it is first necessary to move the top k rings off the first peg and leave the second peg vacant at the end (the second peg will have to be used during this moving process). If this is to be done according to the rules, we will end up with the top k rings on the third peg in the same order they were on the first peg. According to the induction assumption, the least possible number of |
moves needed to do this is one move to transfer the bottom ring [the k 1 st] from the first to the second peg. 1 Finally, the top k rings now on the third peg must be moved to the second peg. Once again by the induction hypothesis, the least number of moves for doing this is smallest total number of moves needed to k 1 transfer all rings from the first to the second peg is 1 1 2k 2k 1 2k 1 2 2 2k 1 2k 1 2 1 2k1 1. It now takes 2k 1. 2k 1. Hence, Therefore, the 2 1 2 the conjecture is true for by induction it is true for all positive integers n. Therefore, n k 1. 37. De Moivreβs Theorem: For any complex number and any positive integer n,. Proof: The theorem is Assume that the zk that is, 3 2 1 cos cos u i sin u nu z r 2 1 i sin zn rn nu 2 4 1 n 1. obviously true when n k, theorem is true for rk ku ku. 1 zk1 z zk r cos u i sin u i sin cos rk 2 4 1 2 3 Then cos 2 4 1 3 3 1 According to the multiplication rule for complex numbers in polar form (multiply the moduli and add the arguments) we have: cos 3 cos u ku k 1 zk1 r rk rk1 u ku 2 4 k 1 This statement says the theorem is true for n k 1. true for every positive integer n. i sin 1 i sin Therefore, by induction, the theorem is u 2. 4 6 ku 1 2 i sin ku 1 2 4 2. Answers to Selected Exercises 1147 Index angles (continued) applications (continued) direction, 662β664 double-angle identities, 593β595, 602β603, 611 of elevation and depression, 425β429 with the horizontal, 620β621 identity, 620 inclination, 589β592 intersecting lines, 590β592 inverse trigonometric functions as, 530 negative angle identities, 459β460, 464 parts of, 413, 433β434 radian measure, 436β439, 444β446 reference, 448β451 rotation, 730β731, 771 solving triangles, 421, 422 special, 418β419, 437, 462 standard position, 4 |
34 between vectors, 671β673, 683 angle-side-angle (ASA) information, 631β632 Angle Theorem, 672 angular speed, 439β440 applications box construction, 103β104, 323 break-even point, 824 composition of functions, 195β196 compound interest, 345β349, 382, 402 distance, 101β102 exponential equations, 345β352, 379β384 food webs, 810β811 gravity, 677β678 guidelines, 97 height and elevation, 425β429 interest, 100β101 ladder safety, 426 linear programming, 831β832 LORAN, 724β725 lottery probabilities, 870, 881β882, 886β887 matrices, 795β801 mixtures, 104 multiple choice exams, 887β888 optimization, 322β324, 468β471 parabolas, 711β714 population growth, 340β342, 349β350, 382β384, 389β392, 395 radiocarbon dating, 352, 381β382 response times, 895β896 rotating wheels, 550β551 sequences, 13β19, 26β29 solutions in context, 98β100 sound waves, 558β562 spring motion, 551β553 trigonometry, 421β429 vector, 661β667 width of walkway, 102β103 approaching infinity, 201, 303β305, 914, 948β957 arccosine function, 532β534, 538, 539, 541 Archimedian spiral, 742 arc length, 434β435, 437β439, 776β777 arcsine function, 529β534, 539 arctangent function, 534β536, 540 area circular puddle, 195 under a curve, 904β906 maximum, 138β139, 169β170, 468β471 quarter-circle, 905β906 Riemann sums, 964β967 triangle, 632β633, 682 z-values, 895β896 A absolute value complex numbers, 638β639 definitions, 107β108, 134 deviations, 858 distance and, 108, 128 equations, 109β111 extraneous solutions, 110β111 functions, 156, 173β174 inequalities, 127β132 properties, 108β109 square roots, 109 acoustics, 696 addition identities, 582β |
585, 593β600, 604, 610 matrix, 804β805 vector, 657β659, 662, 683 adjacency matrix, 809β811 adjacent sides, 415 algebraic expressions, 973β977 amplitude, 494β497, 502β505, 516, 563 amplitude modulation (AM), 472, 499 analytic geometry, definition, 691 angle-angle-side (AAS) information, 626 Angle of Inclination Theorem, 589β590 angles arc length, 434β435, 438β439, 776β777 argument, 639 central, 434 coterminal, 434, 436, 450 degree measure, 437β438 1148 Index arguments, 639 arithmetic progression. see arithmetic sequences arithmetic sequences definition, 21, 66 explicit forms, 22β24, 34, 66 finding terms, 24 graphs, 22, 33β34 lines and, 34 partial sums, 26β29 recursive forms, 22, 24, 66 summation notation, 25β26 arithmetic series, 25β29 asymptotes horizontal, 284β285, 288, 951β953 hyperbolas, 701β706, 720 oblique, 285β286 parabolic, 286 slant, 285β286, 289 vertical, 280β281, 288β289, 479, 486, 950 augmented matrices, 795β798 auxiliary rectangles, 703 average rates of change, 214β220, 226 averages, 853β854 axes (singular: axis) definition, 5 ellipses, 692, 694β695 hyperbolas, 701β702 imaginary, 638 parabolas, 709, 711 polar, 734 rotation of, 728β732 three-dimensional, 790β791 axis of symmetry, 163, 711 B bacteria growth, 350, 382β383 bar graphs, 845 beats, sound, 561 Bernoulli experiments, 884β889 Big-Little Concept, 281, 954 bimodal data, 855 binomial distributions, 887β888, 898 binomial expansion, 997β1000 binomial experiments, 842, 884β888 Binomial Theorem, 994β1000 boiling points, 335 bounds, 254β256 bounds test, 256 box plots, 861β862 box volume, 99β100, 323 break-even point, 824 C Calculator Expl |
orations, 8, 16, 27, 201, 299, 409, 411, 457, 857, 861, 862, 877, 911, 994, 995, 1025 calculators absolute value, 110, 638 area under the normal curve, 895 complex numbers, 297, 299, 302β303, 638 composite functions, 193 conic sections, 721 continuity, 945 discontinuity, 115, 910, 937 dot mode, 157 ellipses, 695 factorials, 520 function graphers, 34β35 function notation, 143 geometric sequences, 60 graphical root finder, 84β85, 122 graphic solutions, 84 greatest integer function, 147 histograms, 849β850 holes in graphs, 115, 910, 937 horizontal shifts, 175β176 hyperbolas, 703β704 inequality symbols, 156 infinite geometric series, 76β78 instantaneous rates of change, 237 intersection finders, 525 inverse sine function, 531 iterations of functions, 200 limits of functions, 913 linear regression, 47β51 logarithms, 357, 375β376, 393 matrix operations, 299, 798β801, 808β809 numerical derivatives, 237 orbits, 302β303 parabolas, 712 parametric mode, 159, 756, 760β761 periodic graphs, 493 piecewise-functions, 157 points of inflection, 267 polar form, 641 polynomial regression, 274β275 probabilities, 875β876, 887 radian mode, 445, 480 radical equations, 113 rational exponents, 330 calculators (continued) rational functions, 283β285 rational zeros, 252 reflections, 177 sequences and sums, 16, 26β27 shading on graphs, 827β828 sine functions, 526 sound wave data collection, 559β561 statistics, 857 stretches and compressions, 177 sum and difference functions, 191 systems of equations, 780 tables of values, 910 trigonometric ratios, 416β417, 421β423, 426, 454, 473β479 tuning programs, 559 vertical shifts, 174 viewing windows, 512 calculus, 76, 138. see also Can Do Calculus Can Do Calculus approximating functions with infinite series, 520β521 arc length of a polar graph, 776β777 area under a curve, 904β906 Eulerβs formula, 688β689 infinite geometric series, |
76β79 instantaneous rates of change, 234β237, 614β615 limits of trigonometric functions, 566β568 maximum area of a triangle, 138β139 optimization applications, 322β324, 468β471 partial fractions, 838β841 Riemann sums, 964β967 tangents to exponential functions, 408β411 carbon-14 dating, 352, 381β382 cardioid graphs, 742 Cartesian coordinate systems, 5, 739 Cassegrain telescopes, 705 center, statistical, 857 chances, 866. see also probability change in x or y, 31 change-of-base formula, 374β375, 402 chord frequency, 561β562 chords, 18 circle equation, 989 circle graphs, 742 circles, parameterization of, 766β767 Index 1149 circles, unit, 445β446, 448β449, 474, conic sections (continued) cosines (continued) 475, 478, 650β651 circumference, ellipses, 699 closed intervals, 118 coefficient of determination, 47 coefficients, 239 cofunction identities, 585β587, 611 combinations, in counting, 880β882, 899 common differences, 22β24, 34 common logarithmic functions, 356β357, 364 common ratios, 58β61, 66, 77 complements, event, 866β867 completing the square, 90β92, 169 complex numbers absolute values, 638β639 arithmetic of, 295β296 definition, 294 equal, 294 Eulerβs formula, 688β689 factorization, 308, 310β313 imaginary powers of, 688β689 Mandelbrot set, 304β306 nth roots, 645β651 orbits, 301β304 polar form, 639β640 polar multiplication and division, 640β642 polynomial coefficients, 307 powers of, 644β645, 688β689 properties, 293β294 quotients of, 296 real and imaginary parts, 300 roots of unity, 648β651 square roots, 297 zeros, 308β310, 317 complex number system, 293β294 complex plane, 301, 637β642, 650 components, vector, 655, 676β677 composite functions, 193β195, 211β212 compound inequalities, 118β120 compound interest, 345β349, 360β361, 382, 402 compressions, 177β180 concavity, 154 |
conic sections definitions, 691, 747 degenerate, 691 discriminants, 723β724 eccentricity, 745β748 ellipses, 692β698, 716β722, 745β747, 767, 771β772 horizontal and vertical shifts, 716β717 hyperbolas, 700β706, 721β725, 745β750, 771β772 1150 Index identifying, 717β719, 722β723 nonstandard equations, 721β722 parabolas, 163, 709β714, 719, 756β760, 771β772 parameterizations, 766β769 polar equations, 745β752, 772 rotations, 722β724, 728β732 standard equations, 720 conjugate pairs, 296 conjugates, complex, 296, 309 conjugate solutions, 299 conjugate zeros, 309β313 Conjugate Zero Theorem, 309 consistent systems, 781 constant functions, 152, 173, 192, 953 constant polynomials, 240 constants, 239 constraints, 829 continuity analytic description, 937β939 calculators and, 937 composite functions, 944 definition, 939 at endpoints, 941β942 informal definition, 936β937 on intervals, 940β942 from the left and right, 941 at a point, 938β940 polynomial equations, 261β262 properties of continuous functions, 942β943 removable discontinuities, 947 of special functions, 940 continuous compounding, 347β349 convergence, 77, 200, 203, 520β521 coordinate planes, 5, 790 coordinates, 5 coordinate systems comparison of, 792 conversions, 737β738 polar, 734β743, 776β777 rectangular, 5, 736β739 three-dimensional, 790β793 corner points, 829, 831β832 correlation coefficients, 47, 52, 66 cosecants, 416β419, 444β446, 485β487, 490 cosines addition and subtraction identities, 610 amplitude, 493β498, 516 basic equations, 538β541 coterminal angles, 451 damping, 512β514 definition, 416, 444β445 domain and range, 447, 477β478, 483 double-angle identities, |
593β595, 602β603, 611 exact values, 448β451, 536 graphs of, 475β478, 497β498 half-angle identities, 596β597, 611 inverse function, 532β534, 539β541, 563 law of, 617β622, 682 oscillating behavior, 568 periodicity, 456β458, 493β497, 516 phase shifts, 501β505, 516, 549 power-reducing identities, 595β596 product-to-sum identities, 599 property summary, 483 restricted, 532β533 roots of unity, 648β651, 682 special angles, 418β419, 462 sum-to-product identities, 599β600 transformations, 481β482, 503β505 trigonometric identities, 454β460, 463 unit circle, 445β446 cotangents, 416β419, 444β446, 489β490 coterminal angles, 434, 436β437, 450β451 counting numbers, 3 counting techniques, 879β882, 899 Critical Thinking, 20, 21, 64, 132, 172, 214, 250, 259, 273, 300, 344, 363, 388, 420β421, 433, 453, 492, 529, 538, 547, 624, 637, 643β644, 652, 700, 708, 727, 733, 744, 765β766, 794, 820, 852, 864, 884, 935, 947, 959 crystal lattices, 802 cube root of one, 299 cubic functions, 173, 240 cubic models, 396 cubic regression, 274β276 curve fitting, 818. see also models; regression cycles, 492, 559 cycloids, 761β763 cylinder surface area, 149, 324 D damping, 512β514 data. see also statistics comparing, 893β894 definition, 843 displays of, 844β850 data (continued) division (continued) end behavior, 262β264, 284β287, distribution shapes, 846β847 outliers, 847, 862 qualitative, 843, 845β846 quantitative, 843, 846β850 standardized, 893 types of, 843 variability, 857 z-values, 894β |
896 decomposition, partial fraction, 838β841 definite integrals, 967 degenerate conic sections, 691 degree measure, 94, 436β437, 462, 528 degree of a polynomial, 240β242, 260β261, 263, 313 DeMoivreβs Theorem, 644β645, 682 denominators, partial fractions, polynomial, 240β245 remainders and factors, 243β245 synthetic, 241β242 domains convention, 145β146 exponential and logarithmic functions, 359β360, 375β376 functions, 142, 145 inverse functions, 533 rational functions, 279 of relations, 6β7 restricting, 210β211, 529β530, 532, 534β535 sequences, 14β15 sum, difference, product, quotient functions, 192 trigonometric functions, 447, 477, 480, 483, 486β488, 490 838β841 dot products density functions, 871β872, 891, angles between vectors, 898 depression, angles of, 425β429 derivatives, numerical, 237 deviations, data, 857β859 difference functions, 191, 943 difference of cubes, 298 difference quotients, 143β144, 671β673 gravity, 677β678 projections and components, 674β677 properties, 670β671 double-angle identities, 593β595, 602β603, 611 219β220, 234β235, 408, 584, 922β923 dreidels, 882 dynamical systems, 199 differential calculus, 76 directed networks, 809β811 direction angles, 662β664 directrix, 709β711, 720 discriminants, 93β94, 172, 723β724 distance absolute value and, 108, 128 applications, 101β102, 113β114 average rates of change, 214β220 between two moving objects, 619 formula, 987β988 from velocities, 964β967 distance difference, 700 distributions binomial, 887β888, 898 definition, 846 mean, median, and mode, 856β857 normal, 889β896 probability, 865β866 shapes, 846β847, 857 divergence, 77 division algorithm, 243 checking, 242β243 polar, of complex numbers, 640β642 E e, 341, 347β |
349. see also exponential functions; logarithmic functions eccentricity, 745β748 effective rate of interest, 354 elementary row operations, 795β796 elevation, angles of, 425β429 eliminating the parameter, 757β759 elimination method, 783β786, 797β798, 821 ellipses applications, 696β698 characteristics, 694 circumference, 699 definitions, 692, 745β747 eccentricity, 745β747 equations, 692β694, 696, 720β722, 771β772 graphing, 695 parameterization, 767 polar equations, 749 translations, 716β718 empirical rule, 892β893, 899 289, 316, 954β955 endpoints, 118 Engelsohnβs equations, 685 equations. see also quadratic equations; systems of equations; trigonometric equations absolute value, 109β111 applications of, 97β104 basic, 524β528, 538β542 conditional, 523 conic sections, 720 degrees, 94 ellipses, 692β694, 696, 720β722, 770β771 Engelsohnβs, 685 exponential, 379β384 fractional, 114β115 functions, 144β145 graphical solutions, 81β86, 524β528 hyperbolas, 701β702, 704, 720, 771β772 linear, 33β37, 39 logarithmic, 379, 384β386 matrix, 814, 817β818 normal curve, 891, 899 number relations, 97β98 parabolas, 709β710, 712, 720 parametric, 157β159, 755β757, 767β769 polar, 745β752, 772 polynomial, 94β95, 260β262 radical, 111β113 rotation, 728β730 second-degree, 722β724 solutions in context, 98β100 tangent lines, 236β237 translated conics, 718 equivalent inequalities, 119 equivalent statements, 81, 134 equivalent systems, 795 equivalent vectors, 653β655 Eulerβs formula, 688β689 even degree, 261, 263, 313 even functions, 188, 482β483, 489β490 events, definition, 865. see also probability eventually fixed points, 202β203 eventually periodic points, 202 expected values, 869β870 |
experiments binomial, 842, 884β888 definition, 864β865 probability estimates from, 874β877 Index 1151 exponential decay, 350β352, 402 exponential equations, 379β384 exponential functions. see also logarithmic functions applications, 345β352 bases, 336β337, 371, 380, 402 bases other than e, 410β411 common logarithms and, 356β357 compound interest, 345β347, 382 graphs, 59, 336β342, 375 growth and decay, 339β342, 349β352, 402 horizontal stretches, 338β339 natural, 341 natural logarithms and, 358β359 Power Law, 366β367 Product Law, 365 solving, 379β381 tangent lines to, 408β410 translations, 338 trigonometric functions, 454 vertical stretches, 339 exponential growth, 349β350, 402 exponential models, 389β390, 392β394 exponents complex numbers, 295β296 fractional, 294 irrational, 333 laws of, 330β331, 402, 969β973 rational, 329β331, 333, 402 extraneous solutions, 110β111 extrema, 153, 170, 266, 468β469, 607β608, 829β830 F factorials, 520β521, 880 factoring common factors, 978 complex numbers, 308β313 cubic factoring patterns, 979 partial fractions, 839β841 polynomials, 243β245, 253β254, 310β313 quadratic equations, 88β89, 978 trigonometric equations, 542β544 Fibonacci sequence, 20β21 filtering, 350β351 finance, exponential change, 339β340 finance, interest applications, 100β101, 339β340, 345β349, 354, 360β361, 382 finite differences, 43β44, 47β48 first octant, 790 five-number summary, 861β862 fixed points, 202β203 flagpole height, 427 focal axis, 701 foci (singular: focus) ellipses, 692, 694, 696β697, 720 hyperbolas, 700β702, 720 parabolas, 709, 711, 720 food webs, 810β811 force, gravitational, 677β678 force, resultant, 664β667 fractional expressions, 114β115 fractions, partial, 838β841. see also rational functions |
free-fall, 958β959 frequency, definition, 844 frequency, wave, 558β562 frequency tables, 844β845 functions. see also logarithmic functions; polynomial functions; trigonometric functions absolute-value, 156, 173 composite, 193β195, 211β212, 944 concavity and inflection points, 154, 266 constant, 152, 173, 192, 953 continuity, 261β262, 936β945 cube root, 173 cubic, 173, 240 defined by graphs, 150β152 definition, 142β143 density, 871β872, 891, 900 difference quotients, 143β144, 219β220, 234β235, 408β410, 584, 922β923 domain, 142, 145 equations, 144 evaluating, 10, 143 even and odd, 188β189, 482β483, 489β490 exponential, 336β342 fractional, 114β115 graphs of, 8β9 greatest integer, 147, 157, 173 horizontal line test, 208β209, Factor Theorem, 245β246, 226 functions (continued) as infinite series, 520β521 input and output, 141β142, 204 instantaneous rates of change, 234β237 inverse, 204β212, 529β536, 539β540, 563 iterations of, 199β200 limits of, 566β568, 909β915 linear, 34β36 local maxima and minima, 153, 266, 468β471, 607β608, 829β831 notation, 9β10, 143 numerical representation, 7β8 objective, 829β831 odd, 188β189, 482β483, 489β490 one-to-one, 208β211 orbits, 200β203, 301β303 parametric, 157β159 parent, 172β173 piecewise-defined, 146β147, 155 polynomial, 145, 165β168, 188β189, 239β248 probability density, 871β872, 891, 898 product and quotient, 192 profit, 146 quartic, 240 range, 142 rates of change, 214β220 rational, 278β289, 954β957 reciprocal, 173 sinusoidal, 510β511, 521, 547β555 square root, 173 step, 157 sums and differences, 191 symmetry, 186β189 transformations, 172β181 trigon |
ometric ratios, 443β444 vertical line test, 151β152, 225 zeros, 240, 245β248, 250β257, 265, 308β313, 316β317 Fundamental Counting Principle, 879β882, 899 Fundamental Theorem of Algebra, 307β313 Fundamental Theorem of Linear Programming, 829β831 G Gateway Arch (St. Louis, MO), 342 Gauss-Jordan elimination, 797β798, 252β253 feasible regions, 829β831 Ferris wheels, 522, 555 1152 Index identity, 173, 918 increasing and decreasing, 152β154 817 geometric sequences applications, 62β63 geometric sequences (continued) graphs (continued) definition, 58, 66 examples of, 58 explicit form, 59β61, 66 infinite, 76β79 partial sums, 61β63 recursive form, 59, 66 geometric series, 76β79, 520β521 graphical root finder, 84β85 graphical testing, 572β573 graphing calculators. see calculators Graphing Explorations, 83, 122, 156, 157, 158, 174, 175, 176, 177, 181, 185, 186, 187, 207, 260, 261, 262, 266, 274, 337, 342, 365, 366, 446, 475, 482, 486, 487, 494, 496, 510, 513, 514, 562, 572, 581, 582, 651, 741, 751, 756, 761, 784, 891, 910, 913, 950, 1023, 1024 graphs absolute-value functions, 156 amplitude, 493β498, 516 arithmetic sequences, 22 bar, 845 binomial distributions, 888 box plots, 861β862 concavity, 154 continuity, 261β262 cosecant function, 486β487 cosine function, 475β478, 493β494, 501 cotangent function, 488β489 damped and compressed, 512β514 defining functions, 150β152 definition, 30 ellipses, 695β696 end behavior, 262β264, 289 of equations, 30, 173 equation solutions, 81β86, 524β528 exponential functions, 336β341 finding slopes, 32 function values from, 8β9 geometric sequences, 59 greatest integer functions, 157 histograms, 849β850 holes, 282β |
283, 289 horizontal asymptotes, 284, 951β953 horizontal shifts, 175β176 horizontal stretches, 338β339 hyperbolas, 702β704 identifying, 505 identities, 506β507 increasing and decreasing functions, 152β154 inequalities, 121β123 inflection points, 154, 266 inverse cosine function, 533 inverse relations, 205β207 inverse sine function, 530 inverse tangent function, 535 limits of functions, 566β568, 950β953 of lines, 34β37 local maxima and minima, 153β154, 266 logarithmic functions, 359β361, 375β376 maximum area of a triangle, 139 multiplicity, 265, 283 nth roots, 328 one-to-one, 208β209 open circles, 150 oscillating behavior, 514β515, 568 parabolas, 286, 711β712 parametric, 157β159, 206β207, 755β757 parent functions, 172β173 periodicity, 487, 489, 493β497 phase shifts, 501β505 piecewise-defined functions, 155 plane curves, 754β755 polar, 739β743 polynomial functions, 260β268 quadratic functions, 163β167 rational functions, 279β289 reflections, 176β177, 206, 226 roots of unity, 651, 681 rotated conics, 731β732 scatter plots, 5 secant function, 487β488 sequences, 14β15 sine function, 473β475, 494β495 sinusoidal, 511β512 slant asymptotes, 285β286 stem plots, 847β848 stretches and compressions, 177β181 symmetry, 184β189, 482β483 systems of equations, 780β788, 822β824 tangent function, 478β481 in three dimensions, 790β794 translations, 338, 716β717 in two dimensions, 792 vertical asymptotes, 282β283, 288β289, 950 vertical shifts, 174 vertical stretches, 339 viewing windows, 512 gravitational acceleration, 293 greatest integer functions, 147, 157, 173 H half-angle identities, 596β598, 604, 611 half-life, 351β352 half-open intervals, 118 Halleyβs Comet, 700, 754 headings, 432 height, from trigonometry, 425β429 Heronβ |
s formula, 633, 682 Hertz, 559 histograms, 849β850 holes, 282β283, 289, 910, 937β939 horizontal asymptotes, 284, 288, 951β953 horizontal lines, 37, 152, 792 horizontal line test, 209β210, 226, 530 horizontal shifts, 175β176 horizontal stretches and compressions, 178β180, 338β339 Hubble Space Telescope, 705 Huygens, Christiaan, 761 hyperbolas applications, 705β706, 724β725 characteristics, 702 definitions, 700β701, 747 eccentricity, 745β748 equations, 701β702, 704, 720, 771β772 graphing, 702β704, 721β724 parametric equations, 768 polar equations, 749 hypotenuses, 415 I identities. see trigonometric identities identity functions, 173, 918 identity matrices, 815 imaginary axis, 638 imaginary numbers. see complex numbers inconsistent systems, 781, 784β785 independent events, 867β868 index of refraction, 545 indirect measurement, 427β429 inequalities absolute value, 127β131 algebraic methods, 128 applications, 123β124 Index 1153 inequalities (continued) compound, 118β120 equivalent, 119 interval notation, 118β119 linear, 119β120, 827 multiplying by negative numbers, 119 nonlinear, 121β122 quadratic and factorable, 122β123, 129β130 solving, 119β123 systems of, 826β832 test-point method, 826 infinite geometric series, 76β79, 520β521 infinite sequences, 13 infinite series, 520β521 infinity concept, 119, 201, 948 horizontal asymptotes, 951β953 limits approaching, 201, 303β305, 914, 948β953 negative, 948, 951β953, 956 properties of limits, 953β957 vertical asymptotes and, 950 inflection points, 154, 266, 317 initial point (vectors), 653 initial sides, of rays, 433 input, 7β9 instantaneous rates of change, 234β237 integers, 3β4 integral calculus, 76 integrals, definite, 967 intensity, 562 interest applications, 100β101, 339β340, 345β349, 354, 360β361, 382, 404 Intermediate Value Theorem, 944 |
β945 interquartile range, 860β861 intersection method, 86, 127, 134, 524β525 interval notation, 118β119 interval of convergence, 520β521 inverse functions composition of, 532 cosine, 532β534, 538β539, 541, 563 definition, 210 horizontal line test, 530 restricting the domain, 210β211 sine, 529β532, 539, 563 tangent, 534β536, 540, 563 inverse matrices, 815β817 inverse relations algebraic representations, 207β208, 534 1154 Index inverse relations (continued) composite, 211β212 definition, 205 graphs, 205β207 horizontal line test, 209β210, 226 one-to-one functions, 208β211 restricting the domain, 210β211, 529, 532, 534 irrational exponents, 333 irrational numbers, 4, 341, 688 irreducible polynomials, 253, 315 iterations, 199β200 L ladder safety, 426 latitude, 442 law of cosines, 617β622, 682 law of sines AAS information, 626 ambiguous case, 627β628 area of a triangle, 632β633 ASA information, 631β632 definition, 625, 682 SSA information, 628β633 supplementary angle identity, 628 laws of exponents, 330β331, 402 laws of logarithms, 373β374, 402 least squares regression lines, 47β52 lemniscate graphs, 743 length, maximum, 469 light, reflection, 705β706, 712β714 limaΓ§on graphs, 743 limits of functions approaching infinity, 201, 303β305, 914, 948β957 approaching two values, 914β915 of constants, 918, 953 definition, 909β913, 929β935 difference quotients, 922β923 function values, 912β913 identity function, 918 nonexistence of, 913β915 notation, 910β911, 925, 931, 951 one-sided, 924β927, 935 oscillating functions, 915 polynomial functions, 919β920 properties of, 919, 933β934, 953β957 proving, 931β934 rational functions, 920β923, |
954β957 trigonometric functions, 566β568 two-sided, 926β927 limits of sequences, 76β77 Limit Theorem, 922, 954 linear combinations of vectors, 662 linear depreciation, 35β36 linear equations, 33β37, 39 linear functions, 34β36, 240 linear inequalities, 119β120, 827 linear models corresponding function, 396 finite differences, 43β44 least squares regression lines, 47β52 modeling terminology, 44β46 prediction from, 51β52 residuals, 44β46, 49β51, 66 linear programming, 829β832 linear regression, 47β52 linear speed, 439β440 linear systems, 779, 781β782, 796β797 lines. see also slope angles between intersecting, 590β592 arithmetic sequences and, 34 least squares regression, 47β52 parallel and perpendicular, 38β39, 66 parameterizations of, 755 point-slope form, 36β37, 39, 66, 792 secant, 218β219, 408, 409 slope-intercept form, 33β36, 39, 66, 792β793 standard form, 39, 66 tangent, 235β237, 408β411 vertical and horizontal, 37β38, 66 local maxima, 153, 266 local minima, 153, 266 logarithmic equations, 379, 384β386 logarithmic functions change-of-base formula, 374β375, 402 common, 356β357, 364 graphs, 359β361, 375β376 laws of, 373β374, 402 natural, 358β359, 364 other bases, 370β376 Power Law, 367 Product Law, 365 properties, 363β364, 372β373 Quotient Law, 366 solving, 384β386 transformations, 359β360, 375β376 logarithmic models, 389, 394 logistic models, 342, 389, 391β392, 395, 401 LORAN, 724β725 lottery probabilities, 870, 881β882, 886β887 lower bounds, 255β256 M Mach numbers, 529 magnitude, vector, 653, 655 Mandelbrot set, 304β306 mathematical induction, 1002β1010 mathematical models, definition, 43. see also models mathematical patterns, 13β19 matrices (singular: matrix) addition and subtraction, 804β805 adjacency, 809β810 applications, |
800β801, 808β811 augmented, 795β798 dimensions, 804 directed networks, 809β811 elementary row operations, 795β796 equivalent, 795β796 Gauss-Jordan elimination, 797β798, 817 identity, 815 inverse, 815β817 matrix equations, 814, 817β818 multiplication, 805β809 notation, 299, 795, 797, 804 reduced row echelon form, 797β801 square systems, 814β818 maxima, 153, 170, 266, 468, 607β608, 829β830 mean, 853β854, 856β857, 869, 898 measures of center, 853β857 measures of spread, 857β862 median, 854β856, 861β862 midpoint formula, 998 minima, 153, 266, 829β830 mixtures, 104, 787β788 mode, 855β856, 898 modeling terminology, 44β46 models cubic, 396 definition, 43 exponential, 389β390, 392β394, 396 linear, 43β52, 396 logarithmic, 389, 394β395, 396 logistic, 341β342, 389, 391β392, 396, 402 polynomial, 273β276 models (continued) power, 389, 392β394, 396 quadratic, 396 quartic, 274β276, 392 simple harmonic motion, 549β553 terminology, 44β46 modulus. see absolute value motion parameterization, 759β763 pendulum, 335 planetary, 393β394 projectile, 546, 602, 761β762 multimodal data, 855 multiple choice exams, 887β888 multiplication matrix, 805β809 by negative numbers, 119 polar, of complex numbers, 640β642 scalar, 655β656, 659, 661, 683, 805β806 multiplicity, 265, 283, 308β309, 316 music, 558β562 mutually exclusive events, 866β868 N natural logarithmic functions, 358β359, 364 natural numbers, 3β4 navigation systems, 724β725 negative angle identities, 459β460, 463, 574 negative correlation, 52 negative infinity, 948, 951β953, 956 negative numbers, 119, 297, |
330 n factorial, 520β521, 880 no correlation, 52 nonlinear systems, 779, 821β824 nonnegative integers, 3 nonrepeated linear factor denominators, 838β839 nonrepeated quadratic factor denominators, 838β839 nonsingular matrices, 815 normal curve area under, 906 definition, 889β890 empirical rule, 892β893, 899 equation of, 891, 900 properties, 890β892 quartiles, 897 normal curve (continued) standard, 890, 893β896 z-values, 894β896, 900 normal distributions, 889β896 notation angles, 413β415 complex numbers, 294, 296, 301 ellipses, 694 functions, 9β10, 143, 191β192 interval, 118β119 inverse functions, 210, 532, 534 iterated functions, 199β200 limits of functions, 910β911, 925, 929β930, 951 matrix, 299, 795, 797, 804 sequences, 14, 16β17 summation, 25β26, 61 triangles, 617 trigonometric functions, 483, 523, 581 vectors, 653, 655, 662 nth roots, 327β329, 645β651 nuclear wastes, 340 number e, 341, 347β349. see also exponential functions; logarithmic functions number lines, 4, 107β108, 128 number relations, 6β7, 97β98 numerical derivatives, 237 O objective functions, 829β830 oblique asymptotes, 285β286 oblique triangles, 617β622, 625β633 octants, 790β791 odd degree, 260β261, 313 odd functions, 188β189, 482, 489β490 one-stage paths, 809 one-to-one functions, 208β211 open intervals, 118 opposite sides, 415 optimization applications, 322β324, 468β469 orbits, 200β203, 301β304, 697β698, 705, 754 ordered pairs, 5, 118 order importance, in counting, 879β880 orientation, 757 origin, 5, 734 origin symmetry, 186β189 orthogonal vectors, 673, 683 Index 1155 oscillating behavior, 514β515, 568, polar coordinates Power Law of Logarith |
ms, 367, 915 outliers, 847, 862 output, 7, 9 P parabolas applications, 712β714 asymptotes, 286β287 characteristics, 711 curve fitting, 818 definitions, 163, 709, 747 equations, 712, 720, 770β771 graphs, 711β712 parameterization, 756β760, 769 polar equations, 749 translations, 719 parallel lines, 38β39, 66 parallel planes, 793 parallel vectors, 671 parameterization, 755, 757β763, 766β769 parameters, definition, 157, 785 parametric graphing, 157β159, 206β207, 755β757 parent functions, 172β173 parentheses, 9 partial fractions, 838β841 partial sums, 26β29 pendulum motion, 335 perfect square trinomials, 91 periodicity identities, 456β458, 460, 463, 574 periodic orbits, 202 periodic points, 202 periods determining, 495β497, 501β502 pendulum, 335 trigonometric functions, 483, 487, 489, 493β495, 516, 563 permutations, 880β881, 899 perpendicular lines, 38β39, 66 phase shifts, 501β503, 516, 549, 563 pi, 4 piecewise-defined functions, 146β147 pie charts, 845β846 plane curves, 754β755 planetary motion, 393β394, 697β698 plutonium, 340 point-slope form, 36β37, 39, 66, 792 points of ellipsis, 13 points of inflection, 154, 266, 317 1156 Index arc lengths, 776β777 graphs, 739β743 polar coordinate system, 734β 736 rectangular coordinates and, 736β738 polar equations, 745β752, 772 polar form of complex numbers, 639β640 373β374 power models, 389, 392β394, 396 Power Principle, 112β113 power-reducing identities, 595β596 powers of i, 295β296 principal, 100 probability area under a curve, 904β906 binomial experiments, 885β888, polar multiplication and division, 900 640β642 polynomial functions. see also polynomials complete graphs, 267β268 continuity, 261β262 definition, 240 end behavior, 262β264, |
267, 316 graphs, 260β268 intercepts, 264β265, 267 limits of, 919β920 local extrema, 266β268 multiplicity, 265 points of inflection, 266, 317 polynomial inequalities, 127 polynomial models, 273β276 polynomials. see also polynomial functions bounds, 254β256 complex coefficients, 307β310 complex zeros, 309β313, 317 conjugate solutions, 299 constant, 240 definition, 239β240 degree of, 240, 242, 248, 260β 261, 263, 308 division, 240β245 equations, 94β95, 240 factoring, 246β247, 253β254, 310β313, 317 complements, 866β867 counting techniques, 879β882, 899 definitions, 864β865 density functions, 871β872, 891, 900 distributions, 865β866 expected values, 869β870 experimental estimates, 874β 877 independent events, 867β868 mutually exclusive events, 866β868 random variables, 869β870 simulations, 875β876, 905β906 theoretical estimates, 877β878 probability density functions, 871β872, 891 product functions, 192, 943 Product Law of Exponents, 365 Product Law of Logarithms, 365, 373β374 product-to-sum identities, 599 profit functions, 146, 170, 196, 216β217 projectile motion, 546, 602, 761β762 projections, 674β677, 681 proofs, identities, 573β579 Pythagorean identities, 456, 460, Factor Theorem, 245β246, 463, 574β577 252β253 rational zero test, 251β253 regression, 273β276 remainders, 244β245 solutions, 246β247 x-intercepts, 246, 316 zeros, 240, 245β248, 250β257, 265, 308β313, 316β317 population growth, 340β342, 349β350, 382β384, 389β392, 395 populations, statistical, 843, 858β859, 899 positive correlation, 52 positive integers, 3 Power Law of Exponents, 366β367 Pythagorean Theorem, 421, 1012β1015 Q quadrants, 5 quadratic equations algebraic solutions, 88β95 applications, 169β |
170 changing forms, 167β169 completing the square, 90β92 complex solutions, 298 definition, 88 discriminant, 93β94 quadratic equations (continued) rates of change factoring, 89 graphs of, 173 irreducible, 253 number of solutions, 93β94 parabolas, 163 polynomial form, 94β95, 164β167, 169, 225, 240 regression, 274β276 summary of forms, 169 taking the square root of both sides, 90 transformation form, 164β165, 168β169, 225 vertex, 163β166, 169 x-intercept form, 164, 166β169, 225 quadratic formula, 92β94, 134, 165, 544β545 quadratic inequalities, 122, 129β 130 quadratic models, 396 quadratic regression, 274β276 quartic functions, 240 quartic regression, 274β276 quartiles, 860β861, 897 quotient functions, 192, 943 quotient identities, 455, 460, 463, 574 Quotient Law of Exponents, 365 Quotient Law of Logarithms, 366, 373β374 R radian/degree conversion, 436β437, 463 average, 214β219, 226 difference quotient, 219β220, 234 instantaneous, 234β237, 614β615 logistic models, 391 slope of tangent lines, 235β237 rational exponents, 329β331, 402 rational functions complete graphs, 287β289 definition, 278 domains, 279 end behavior, 284β285, 287, 289 holes, 282β283, 289 horizontal asymptotes, 284 intercepts, 279β280, 288β289, 317 limits, 920β923, 954β957 maximum of, 322 parabolic asymptotes, 286 partial fractions, 838β841 slant asymptotes, 285β286, 289 trigonometric identities, 578β579 vertical asymptotes, 281β282, 288β289, 317, 950 rational inequalities, 127β128 rationalizing denominators and numerators, 332β333 rational numbers, 4, 78β79 rational zeros, 250β254, 316 rays, rotation, 433 real axis, 638 real numbers, 3β4 real solutions, 89, 94 real zeros, 245, 248, 250β257 reciprocal identities, 455, 460, 463, 574 radian measure, 435β438 |
, 444β445 radicals, 111β113, 327β329, 332β rectangles, 98β99, 703 rectangular box volume, 99β100, 333. see also roots radioactive decay, 340, 351β352, 402 radiocarbon dating, 352, 381β382 radio signals, 472, 500, 724β725 radio telescopes, 713β714 Ramanujan, 699 random samples, definition, 843 random variables, 869β870 ranges definition, 142 exponential and logarithmic functions, 359, 375 relations, 6β7 statistical, 859β860 trigonometric functions, 446, 476, 479, 482, 486β489 323 rectangular coordinate systems, 5, 736β738 recursively defined sequences, 15, 66 reduced row echelon form, 797β801 reference angles, 449β451 reflection light, 705β706, 712β714 radio signals, 712β714 sound, 696β697 reflections, 177, 206, 226, 481, 487, 501, 696β697, 705β706, 712β714 refraction, 545 regression. see also models cubic, 274β276 regression (continued) exponential, 390 least squares, 47β52 linear, 47β52 polynomial, 273β276 power, 394 quadratic, 274β276 quartic, 274β276, 392 sinusoidal functions, 553β554 relations, 6β7, 97β98 relative frequency, 844β845 remainders, 244β245 Remainder Theorem, 244 removable discontinuities, 947 repeating decimals, 78β79 replacement, in counting, 879β880 residuals, 44β46, 48, 50β51, 66 response times, 895β896 restricted domains, 210β211, 529, 532, 534 resultant force, 664β667 Richter magnitudes, 368β369 Riemann sums, 964β967 right triangles, 415β418, 421β426 roots absolute value, 109 of complex numbers, 646β648 cube root of one, 299 extraneous, 110β111 graphical root finder, 84β85, 122 limits at infinity, 955β957 nth, 327β329, 645β651 square, 90, 109, 297, 955β956 of unity, 648β651, 682 roots of |
unity, 648β651, 682 rose graphs, 742 rotation angles, 730β731, 771 rotations, 722β723, 728β732, 771 rounding, 426, 525 rule of a relation, 7 rule of the function, 7 S samples, definition, 843 sample space, definition, 864β865 sample standard deviation, 858, 899 scalar multiplication, 655β656, 659, 661, 683, 805β806 scatter plots, 5β7, 48, 50β51 Schwarz inequality, 673, 683 secant lines, 218β219, 408 secants, 408, 416β418, 444β445, 486β487, 489 Index 1157 second-degree equations, 722β724. sines (continued) see also quadratic equations self-similar under magnification, 305 sequences applications, 17β19, 28β29 arithmetic, 21β29, 34, 66 definition, 13, 66 explicit forms, 34, 66 geometric, 58β63, 76β79 graphs, 14β16 limits, 76β77 notation, 14, 16β17 partial sums, 26β29 recursive forms, 15β16, 22, 66 summation notation, 25 series arithmetic, 25β29 infinite geometric, 76β79, 520β521 shading on graphs, 827β828 side-angle-side (SAS) information, 618β621 side-side-angle (SSA) information, 627β631 side-side-side (SSS) information, 619, 633 sides of angles, 413, 422, 424 simple harmonic motion. see also sinusoidal functions bouncing springs, 551β553 characteristics, 547 definition, 549 examples of, 522, 550β553 rotating wheel, 550β551 simple interest, 100 sines addition and subtraction identities, 582β583, 604, 610 amplitude, 497β498, 516 basic equations, 539β540, 542 calculators, 423 coterminal angles, 451 damping, 512β513 definition, 416, 444 domain and range, 446, 476, 482 double-angle identities, 593β594, 602β603, 611 exact values, 448β450 finding values, 449β450 graphical transformations, 481β482, 503β504 graph of, 473β475, |
497β498, 510β511 half-angle identities, 596β597, 604, 611 instantaneous rates of change, 614β615 inverse function, 529β532, 539, 563 1158 Index law of, 625β633, 682 oscillating behavior, 514β515 periodicity, 456β457, 493β494, 496β497, 516 phase shifts, 501β503, 516, 549 polar graphs, 740β741 power-reducing identities, 595β596 product-to-sum identities, 599 restricted, 529, 531 roots of unity, 648β651, 682 sinusoidal graphs, 510β511 special angles, 418, 462 summary of properties, 483 sum-to-product identities, 599β600 trigonometric identities, 454β460, 463 unit circle, 445 1 SIN key, 423 sinusoidal functions amplitude, 547β549 constructing, 548β549 examples, 522 graphs, 510β511, 548 modeling, 553β555 rotating wheels, 550β551 sound waves, 560β561 spring motion, 551β553 skewed distributions, 846 skid mark length, 335 slant asymptotes, 285β286, 289 slope correlation and, 52, 66 definition, 31β32, 66 from a graph, 32 horizontal and vertical lines, 37β38 parallel and perpendicular lines, 38β39 point-slope form, 36β37, 39, 66, 792 properties, 33 secant lines, 408, 410 slope-intercept form, 33β36, 39, 66, 792β793 tangent lines, 235β237, 408β409 Snellβs Law of Refraction, 545 solutions, definition, 81 solving a triangle, 421 sound, speed of, 705 sound waves, 558β562, 696β697 special angles, 418β419, 437, 462 speed angular, 439β440 average, 215β216, 219β220 instantaneous, 234β236 speed (continued) linear, 439β440 skid mark length, 335 sound, 705 spring motion, 551β553 square roots, 90, 109, 173, 297, 955β956 square systems, 814β818 standard deviation, 857β859, 888, 894, 899 standard form, of a line, |
39, 66 standard normal curve, 890, 893β896 standard position, of angles, 434 standard viewing window, 84 statistics. see also data box plots, 861β862 data displays, 844β850 five-number summary, 861β862 interquartile range, 860β861 mean, 853β854, 856β857, 869, 899 median, 854β856, 861 mode, 855β856, 898 range, 859β860 standard deviation, 857β859, 888, 894, 899 variance, 858 z-values, 894β896, 900 stem plots, 847β848 step functions, 157 substitution method, 782, 821β822 subtraction identities, 582β585, 593β600 matrix, 804β805 vector, 658β659 sum functions, 191, 943 summation notation, 25β26, 61 sum of the convergent series, 77 sum-to-product identities, 599β600 surface area of a cylinder, 149, 324 symmetric distributions, 846 symmetry, 184β189, 482β483 synthetic division, 241β242 systems of equations. see also matrices algebraic solutions, 782β786 applications, 786β787 augmented matrices and, 795β798 definition, 779 elimination method, 783β786, 797β798, 821 equivalent, 795 graphs of, 780β781, 783β785, 822β824 inconsistent and consistent, 781, 784β785, 799 systems of equations (continued) linear, 779, 781β782, 796β797 nonlinear, 779, 821β824 number of solutions, 781β782, 821 square, 814β818 substitution method, 782, 821β822 systems of inequalities, 826, 828β832 T tangent lines, 235β237, 408β411 tangents addition and subtraction identities, 584β585, 610 basic equations, 540 coterminal angles, 451 definition, 416, 444 domain and range, 447, 483 exact values, 448β450, 536 graphical transformations, 481β482 graph of, 478β479 half-angle identities, 596, 598, 611 inverse function, 534β536, 540 |
, 563 periodicity, 456β457, 495, 516 restricted, 534 special angles, 418, 462 summary of properties, 483 trigonometric identities, 454β460, 463 two intersecting lines, 590β593 unit circle, 446 technology tips. see calculators telescopes, 705, 713β714 temperature, rate of change, 217β218 terminal points, of vectors, 653 terminal sides, of rays, 433 terminal velocity, 908, 959 terms, of a sequence, 14 test-point method for inequalities, 826β827 theorems Angle of Inclination Theorem, 589β590 Angle Theorem, 672β673 Conjugate Zero Theorem, 309 DeMoivreβs Theorem, 644β645, 682 Factor Theorem, 245β246, 252β253 Fundamental Theorem of Algebra, 307β313 Fundamental Theorem of Linear Programming, 829β830 Intermediate Value Theorem, 944β945 theorems (continued) Limit Theorem, 922, 954 Pythagorean Theorem, 421 Remainder Theorem, 244 Triangle Sum Theorem, 421 three-dimensional coordinates, 790β791 TRACE feature, 176, 185 transformation form, 164β165, 168β169, 225 transformations amplitude, 497β498, 503β505, 516, 563 combined, 180, 503β505 conic sections, 716β725 horizontal shifts, 175β176 logarithmic functions, 359β360, 375β376 parameterization of, 757β759 parent functions, 172β173 phase shifts, 501β504, 516, 549, 563 reflections, 176β177, 206, 226, 481, 487, 501 rotations, 722β723, 728β732, 771 stretches and compressions, 177β181, 338β339, 481, 487β489, 497 transforming identities, 574β576 trigonometric functions, 481β482, 487β489, 501β507 vertical shifts, 174, 481β482, 501, 504, 516, 563 translations, 338 trapezoids, area of, 468β469 triangles angle of inclination, 589β592 angles, 413β414 area, 632β633, 682 Heronβs formula, 633, |
682 hypotenuses, 415 maximum area, 138β139 oblique, 617β622, 625β633 right, 414β419, 421β426 similar, 415β417 solving, 421β426 special, 418β419, 437, 462 standard notation, 617 trigonometric ratios, 415β417 Triangle Sum Theorem, 421 trigonometric equations algebraic solutions, 538β545 basic, 524β526, 538β542 complex numbers, 644β648 conditional, 523 factoring, 542β544 fractional, 577β579 graphical solutions, 524β528 trigonometric equations (continued) inverse trigonometric functions, 529β536, 539β540, 563 quadratic formula and, 544β545 roots of unity, 648β651, 682 rotation of conic sections, 728β730 solution algorithm, 540β541 special values, 541 substitutions, 542 trigonometric form of complex numbers, 639β640 trigonometric functions. see also trigonometric identities for all angles, 460 angle notation, 413 applications, 425β426 coordinate plane, 443β444 cosecants, 416, 418, 444β445, 486β487, 490 coterminal angles, 451 damped and compressed graphs, 512β514 definitions, 462 domain, 447, 477, 480, 483, 490 even and odd, 482β483, 489β490 exact values, 448β449, 536 finding values, 450β451 graphs of, 472β482, 510β514 identities, 454β460, 463 instantaneous rates of change, 614β615 inverse cosine, 532β534 inverse sine, 529β532 inverse tangent, 534β536 limits of, 566β569 maximum and minimum, 607β608 optimization with, 468β469 oscillating behavior, 514β515, 568 phase shifts, 501β504, 516, 549 polar coordinates, 736β738 powers of, 454 ranges, 447, 477, 480, 483, 487β488, 490 ratios, 415β419 restricted, 529, 531β532, 534 secants, 408, 419, 444β445, 487β488, 490 signs, 447β448 special angles, 418β419, 462 summary of |
properties, 483, 490 trigonometric functions. see also transformations, 481β482, 487β489, 501β507, 446β449 unit circle, 446β449 trigonometric identities addition and subtraction, 581β 587, 593β600, 604, 610 Index 1159 trigonometric identities (continued) alternate solutions, 577β579 a sin x b cos x c, cofunction, 585β587, 611 definition, 454β455 double-angle, 593β595, 602β603, 605β608 611 Eulerβs formula, 688β689 factoring and, 544 graphical testing, 572β573 graphs and, 506β507 half-angle, 596β598, 604, 611 negative angle, 457β459, 462 periodicity, 456β458, 460, 463 power-reducing, 595β596 product-to-sum, 599 proofs, 573β579 Pythagorean, 456, 460, 463, 575β577 quadratic formula and, 544β545 quotient, 455, 460, 463 reciprocal, 455, 460, 463 summary of, 460, 463, 574 sum-to-product, 599β600 using, 602β608 trimodal data, 855 trivial solutions, 799 tuning forks, 558β562 two-stage paths, 810 typing, 388 U uniform distributions, 846 unit circles, 445β446, 449, 473, 475, 478, 650 unit vectors, 661β664 upper bounds, 255β256 V variability, 857β860 variance, 858 vectors angles between, 671β673, 681 arithmetic, 655β659, 662, 681 components, 674β677, 681 components and magnitudes, 655 direction angles, 662β664 dot products, 670β678 equivalent, 654β655 gravity, 677β678 linear combinations, 662 magnitude, 653, 655 1160 Index vectors (continued) x-coordinate transformations, notation, 653, 655, 662 orthogonal, 673, 681 parallel, 671 projections, 674β677, 681 properties, 659 resultant force, 664β667 Schwarz inequality, 673, 683 unit, 661β664 velocity, 6 |
63 work calculation, 677β678 zero, 658 velocity average, 235 free-fall, 958β959 instantaneous, 234β236 terminal, 908, 959 total distance from, 964β965 vectors, 663 vertical asymptotes, 281β282, 288β289, 950 vertical lines, 37β38, 66, 792 vertical line test, 151β152, 186, 225 vertical shifts, 174, 481β482, 501, 504, 516 vertical stretches and compressions, 177β180, 339, 481, 487, 501 vertices (singular: vertex) directed matrices, 809β810 ellipse, 692, 720 hyperbolas, 701, 720 parabolas, 709, 720 quadratic functions, 163β166, 169, 225 triangle, 415 Very Large Array (Socorro, New Mexico), 713β714 volume, 216, 323 W waves, 493β498, 558β562 whole numbers, 3β4 wind chill, 335 work, 677β678 X x-axis symmetry, 185β186, 188 x-coordinates, 5 180β181 x-intercept form of quadratic functions, 164, 166β169, 225 x-intercept method, 84β86, 94, 112, 127β128, 134, 525β528 x-intercepts ellipses, 694 hyperbolas, 702 parabolas, 163 polynomial functions, 264β265 quadratic functions, 163β165, 169, 225 rational functions, 279β280, 288β289 x-variables, 6 Y y-axis symmetry, 184β185, 188 y-coordinates, 5, 166 y-coordinate transformations, 178β179 y-intercepts ellipses, 694 hyperbolas, 702 parabolas, 163 polynomial functions, 264 quadratic functions, 163β166, 169 rational functions, 279, 288β289 in three dimensions, 792 y-variables, 6 Z z-axis, 790 Zero Product Property, 89 zeros bounds, 254β256 complex, 310β313, 317 complex polynomials, 307β313 conjugate, 309β313 Factor Theorem and, 252β253 multiplicity, 265, 283, 308β309 orbits, 302β304 polynomials, 240, 245β248, 250β257, 265, 308β |
313, 316β317 rational, 250β254, 316 of unity, 298β299 zero vectors, 658 z-values, 894β896, 900 Exponents crcs crs 2 crs cr c s s crs r crdr r cr dr 2 cr 1 cd 1 c db a Algebra Multiplication & Factoring Patterns Difference of Squares: Perfect Squares: Difference of Cubes: u2 v2 u v u v 1 21 2 2 u2 2uv v2 2 u2 2uv v2 2 2 u v u v 1 1 u3 v3 u v u v 1 1 21 21 u2 uv v2 u2 uv v2 2 2 d 0 2 1 Sum of Cubes: u3 v3 c r 1 cr c 0 2 1 Perfect Cubes u3 3u2v 3uv2 v3 3 u3 3u2v 3uv2 v3 The Quadratic Formula If a 0, then the solutions of ax2 bx c 0 are x b Β± 2b2 4ac 2a. Equations and Graphs The solutions of the equation are the x-intercepts of the graph of Natural Logarithms Logarithms to Base b Special Notation 7 0 and any u: For v, w ln v u means eu v ln v ln w vw ln 2 1 1 v wb a v k ln ln 2 ln v ln w k ln v 1 2 w 7 0 For v, and any u: log bv u means bu v log b 1 log bv log bw vw 2 log bv log bw v log b a wb v k log b 1 2 k log b v 2 1 ln v means log e v log v means log 10 v Change of Base Formula log bv ln v ln b The Pythagorean Theorem c2 a2 b2 c b a Geometry Area of a Triangle A 1 2 bh Circle Diameter 2r h b Circumference 2pr r Area pr2 Distance Formula Length of segment PQ d 2 x1 1 x22 2 y1 1 2 y22 Slope x2 Slope of line m y2 x2 PQ, x1 y1 x1 Midpoint Formula Midpoint M of segment PQ x1 y1 x2 2, y2 2 b M a Q (x2, y2) M P ( |
x1, y1) The equation of the straight line through The equation of line with slope m and y-intercept b is x1, y12 1 with slope m is y mx b. y y1 m x x12 1. Rectangular and Parametric Equations for Conic Sections h 1 x r cos t h y r sin t k Circle Center (h, k), radius r2 1 Ellipse Center (h, k) y k b2 1 2 2 2 x h a2 k k h 2 1 Parabola Vertex (h, k) x h 2 4p 2p 2 x a cos t h y b sin t k 0 t 2p 2 1 x t y 1 2 t h 4p 2 k (t any real) Parabola Vertex (h, k) y k 1 2 2 4p 1 x h 2 1 Hyperbola Center (h, k) y k b2 1 2 2 x h a2 2 1 1 2 Hyperbola Center (h, k) x h b2 1 2 2 y k a2 4p 2 h x 1 y t (t any real) x a h cos t y b tan t k 0 t 2p 2 1 x b tan t h y a k cos t 0 t 2p 2 1 Trigonometry If t is a real number and P is the point where the terminal side of an angle of t radians in standard position meets the unit circle, then cos t x-coordinate of P sin t y-coordinate of P tan t sin t cos t csc t 1 sin t sec t 1 cos t cot t cos t sin t Trigonometric Ratios in the Coordinate Plane For any real number t and point (x, y) on the terminal side of an angle of t radians in standard position: (x, y) r y t sin t y r csc t r y x y 0 1 2 cos t x r sec t r x x 0 1 2 tan t y x cot If a 0 and b 7 0, then each of f t 2 1 a sin 1 Periodic Graphs bt c and g t 2 a sin bt c has 1 2 phase shift c b β’ 2 1 2p b β’ amplitude a 0 0 β’ period Right Triangle Trigonometry Special Values opposite sin u opposite hypotenuse tan u opposite adjacent cos u adjacent hypotenuse adjacent Special Right Triangles 45Β° 2 1 45 |
Β° 1 2 30Β° Law of Cosines a2 b2 c2 2bc cos A b2 a2 c2 2ac cos B c2 a2 b2 2ab cos C 3 A 60Β° 1 c U Degrees Radians sin U cos U tan U 0Β° 30Β° 45Β° 60Β° 90 12 2 13 2 1 1 13 2 12 2 1 2 0 0 13 3 1 13 undefined B b a C Law of Sines c a sin A b sin B sin C Area 1 2 ab sin C Heronβs Formula: Area 1s s a 1 21 s b 21 s c where Area Formulas for Triangles Reciprocal Identities Pythagorean Identities Negative Angle Identities Trigonometric Identities sin x 1 csc x csc x 1 sin x cos x 1 sec x sec x 1 cos x cot x 1 tan x tan x 1 cot x sin2x cos2x 1 tan2x 1 sec2x 1 cot2x csc2x sin 1 x 2 sin x cos tan x x 1 1 2 2 cos x tan x Periodicity Identities Cofunction Identities x Β± 2p x Β± 2p sin csc 1 1 x Β± p tan 1 sin x csc x tan x 2 2 2 x Β± 2p x Β± 2p cos sec 1 1 x Β± p cot 1 cos x sec x cot x 2 2 2 sin x cos tan x cot sec x csc cos x sin cot x tan csc x sec Quotient Identities Addition and Subtraction Identities tan x sin x cos x cot x cos x sin x sin sin cos cos sin x cos y cos x sin y sin x cos y cos x sin y cos x cos y sin x sin y cos x cos y sin x sin y tan tan 1 1 x y x y 2 2 tan x tan y 1 tan x tan y tan x tan y 1 tan x tan y Double Angle Identities Half-Angle Identities sin 2x 2 sin x cos x cos 2x 1 2 sin2x sin x 2 Β± B cos 2x cos2x sin2x cos 2x 2 cos2x 1 tan 2x 2 tan x 1 tan2x cos x 2 Β± B 1 cos x 2 1 cos x 2 tan tan x 2 x 2 1 cos x sin x sin x 1 cos x Product-to-Sum Identities Sum-to-Product Identities sin x cos y 1 2 |
1 sin 1 x y sin 1 2 x y 22 sin x sin y 1 2 1 cos cos x cos y 1 2 1 cos x y x y 2 2 1 1 cos cos x y x y 1 1 22 22 cos x sin y 1 2 1 sin 1 x y sin 1 2 x y 22 sin x sin y 2 sin a sin x sin y 2 cos cos x cos y 2 cos cos x cos y 2 sin cos sin a a cos a sins as Ο 6, and so on. If you and your friends carry, Ο, 3Ο 4, Ο 2, Ο 3 backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are 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 (three, four, two, and one) 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 ten machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this? 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 because weights are measured. 1.6 The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. feet, 160 sq. feet, 190 sq. feet, 180 sq. feet, and 210 sq. feet. 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 Cherry Garcia ice cream and two pounds |
(32 ounces chocolate chip cookies). Name data sets that are quantitative discrete, quantitative continuous, and qualitative. Solution 1.7 One Possible Solution: β’ The three cans of soup, two packages of nuts, four kinds of vegetables and two desserts are quantitative discrete data because you count them. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 15 β’ The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure weights as precisely as possible. β’ 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. 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." a. b. the number of pairs of shoes you own the type of car you drive c. where you go on vacation d. e. f. g. the distance it is from your home to the nearest grocery store the number of classes you take per school year. the tuition for your classes the type of calculator you use h. movie ratings i. political party preferences j. weights of sumo wrestlers k. amount of money (in dollars) won playing poker l. number of correct answers on a quiz m. peoplesβ attitudes toward the government n. IQ scores (This may cause some discussion.) Solution 1.9 Items a, e, f, k, and |
l are quantitative discrete; items d, j, and n are quantitative continuous; items b, c, g, h, i, and m are qualitative. 16 CHAPTER 1 | SAMPLING AND DATA 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 Solution 1.10 This pie chart shows the students in each year, which is qualitative data. 1.10 The registrar at State University keeps records of the number of credit hours students complete each semester. The data he collects are summarized in the histogram. The class boundaries are 10 to less than 13, 13 to less than 16, 16 to less than 19, 19 to less than 22, and 22 to less than 25. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 17 Figure 1.4 What type of data does this graph show? 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). 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 40.9% Full-time 4,059 28.6% Part-time 13,296 59.1% Part-time 10,124 71.4% 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. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative data are pie charts and bar graphs. In a pie chart, categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category. In a bar graph, the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal. 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 βbestβ graph depending on the data and the context. Our choice also depends on what we are using the data for. 18 CHAPTER 1 | SAMPLING AND DATA Figure 1.5 (a) (b) Figure 1.6 Percentages That Add to More (or Less) Than 100% Sometimes percentages add up to be more than 100% (or less than 100%). In the graph, the percentages add to more than 100% 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%. Characteristic/Category Full-Time Students Percent 40.9% Students who intend to transfer to a 4-year educational institution 48.6% Students under age 25 TOTAL Table 1.3 De Anza College Spring 2010 61.0% 150.5% This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 19 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% 0.6% 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) Figure 1.8 20 CHAPTER 1 | SAMPLING AND DATA The following graph is the same as the previous graph but the βOther/Unknownβ percent (9.6%) has been included. The βOther/Unknownβ category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0%). 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 Figure 1.10 Pareto Chart With Bars Sorted by Size Pie Charts: No Missing Data The following pie charts have the βOther/Unknownβ category included (since the percentages must add to 100%). 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. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 21 (a) (b) Figure 1.11 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. Any group of n individuals is equally likely to be chosen by |
any other group of n individuals if the simple random sampling technique is used. 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 1.5: ID Name ID Name ID Name 00 01 02 Anselmo Bautista Bayani 11 12 13 King 21 Roquero Legeny 22 Roth Lundquist 23 Rowell 03 Cheng 14 Macierz 24 Salangsang 04 Cuarismo 15 Motogawa 25 Slade 05 Cuningham 16 Okimoto 06 Fontecha 07 Hong 17 18 Patel Price 26 27 28 Stratcher Tallai Tran 08 Hoobler 19 Quizon 29 Wai 09 10 Jiao Khan 20 Reyes 30 Wood Table 1.5 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. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows: 0.94360; 0.99832; 0.14669; 0.51470; 0.40581; 0.73381; 0.04399 22 CHAPTER 1 | SAMPLING AND DATA Lisa reads two-digit groups until she has chosen three class members (that is, she reads 0.94360 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 random numbers 0.94360 and 0.99832 do not contain appropriate two digit numbers. However the third random number, 0.14669, 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 Macierz, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisaβs group will consist of Marcierz, Cuningham |
, and Cuarismo. To generate random numbers: β’ Press MATH. β’ Arrow over to PRB. β’ Press 5:randInt(. Enter 0, 30). β’ Press ENTER for the first random number. β’ Press ENTER two more times for the other 2 random numbers. If there is a repeat press ENTER again. Note: randInt(0, 30, 3) will generate 3 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 take a proportionate number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members 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 departments from your college population, the four departments make up the cluster sample. Divide your college faculty by department. The departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample. To choose a systematic sample, randomly select a starting point and take every nth piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings. You must choose 400 names for the sample. Number the population 1β20,000 and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of 400 names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is |
a simple method. 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 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 23 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. True random sampling is done with replacement. That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done without replacement. Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low. In a college population of 10,000 people, suppose you want to pick a sample of 1,000 randomly for a survey. For any particular sample of 1,000, if you are sampling with replacement, β’ β’ β’ the chance of picking the first person is 1,000 out of 10,000 (0.1000); the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999); the chance of picking the same person again is 1 out of 10,000 (very low). If you are sampling without replacement, β’ β’ the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000); the chance of picking a different second person is 999 out of 9,999 (0.0999); β’ you do not replace the first person before picking the next person. Compare the fractions 999/10,000 and 999/9,999. For accuracy, carry the decimal answers to four |
decimal places. To four decimal places, these numbers are equivalent (0.0999). Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is 25 people, the sample is ten, and you are sampling with replacement for any particular sample, then the chance of picking the first person is ten out of 25, and the chance of picking a different second person is nine out of 25 (you replace the first person). If you sample without replacement, then the chance of picking the first person is ten out of 25, and then the chance of picking the second person (who is different) is nine out of 24 (you do not replace the first person). Compare the fractions 9/25 and 9/24. To four decimal places, 9/25 = 0.3600 and 9/24 = 0.3750. To four decimal places, these numbers are not equivalent. 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. Example 1.11 A study is done to determine the average tuition that San Jose State undergraduate 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 undergraduate San Jose State students is taken by organizing the studentsβ 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 undergraduate 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 undergraduate student in the fall semester has the same probability of being chosen at any stage of the sampling process. 24 CHAPTER 1 | SAMPLING AND DATA 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 1.11 a. stratified; b. systematic; c. simple random; d. cluster; e. convenience 1.11 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. #3 10 9 8 10 9 9 10 9 8 10 #1 5 10 9 9 7 9 7 8 9 8 #2 7 5 10 10 8 9 7 8 7 8 Table 1.6 #4 #5 #6 9 8 6 9 5 10 9 10 Instructions: Use the Random Number Generator to pick samples. 1. Create a stratified sample by column. Pick three quiz scores randomly from each column. β¦ Number each row one through ten. β¦ On your calculator, press Math and arrow over to PRB. β¦ 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. β¦ Repeat for columns two through six. β¦ 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. β¦ Press MATH and arrow over to PRB. β¦ Press 5:randInt( and enter 1,6). Press ENTER. Record the number. Press ENTER and record that number. β¦ The two numbers are for two of the columns. β¦ The quiz scores (20 of them) in these 2 columns are the cluster sample. 3. Create a simple random sample of 15 quiz scores. β¦ Use the numbering one through 60. β¦ Press MATH. Arrow over to PRB. Press 5:randInt( |
and enter 1, 60). This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 25 β¦ Press ENTER 15 times and record the numbers. β¦ Record the quiz scores that correspond to these numbers. β¦ These 15 quiz scores are the systematic sample. 4. Create a systematic sample of 12 quiz scores. β¦ Use the numbering one through 60. β¦ Press MATH. Arrow over to PRB. Press 5:randInt( and enter 1, 60). β¦ Press ENTER. Record the number and the first quiz score. From that number, count ten quiz scores and 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 1.12 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 the average. Solution 1.12 a. stratified; b. cluster; c. stratified; d. systematic; e. simple random; f.convenience 1.12 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 1.13 Suppose ABC College has 10,000 part-time students (the population). We are interested in the average amount of money a part-time student spends on books in the fall term. Asking all 10,000 students is an almost impossible task. Suppose we take two different samples. First, we use convenience sampling and survey ten 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 26 CHAPTER 1 | SAMPLING AND DATA The second sample is taken using a list of senior citizens who take P.E. classes and taking every fifth senior citizen on the list, for a total of ten senior citizens. They spend: $50; $40; $36; $15; $50; $100; $40; $53; $22; $22 It is unlikely that any student is in both samples. 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 1.13 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 1.13 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 1.13 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. 1.13 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 stationβs music concert events. 24 people said theyβd prefer more talk shows, and 176 people said theyβd prefer more music. Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population? 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 university, use all honor students at the university as the sample. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 27 2. To find out the most popular cereal among young people under the age of ten, stand outside a large supermarket for three hours and speak to every twentieth child under age ten 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. 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 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 college 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 Doreenβs and Jungβs samples different. If Doreen and Jung took larger samples (i.e. 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 variability in |
samples cannot be stressed enough. 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 to 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, call-in surveys are invariably biased, because people choose to respond or not. 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 1.7 and Table 1.8 (βfrequencyβ is the number of times a particular face of the die occurs): 28 CHAPTER 1 | SAMPLING AND DATA Face on Die Frequency 1 2 3 4 5 6 Table 1.7 First Experiment (20 rolls) Face on Die Frequency 1 2 3 4 5 6 Table 1.8 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. 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 β’ 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 valid. β’ Self-selected samples: Responses only by people who choose to respond, such as call-in surveys, are often unreliable. β’ 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: crash testing cars or medical testing for rare conditions β’ Undue influence: collecting data or asking questions in a way that influences |
the response β’ 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. β’ 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. β’ 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. β’ Misleading use of data: improperly displayed graphs, incomplete data, or lack of context This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 29 β’ Confounding: When the effects of multiple factors on a response cannot be separated. Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor. 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. For example, the average of the three quiz scores four, six, and nine is 6.3, rounded off to the nearest tenth, because the data are whole numbers. Most answers will be rounded off in this manner. It is not necessary to reduce most fractions in this course. Especially in Probability Topics, the chapter on probability, it 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 (from lowest to highest level): β’ Nominal scale level β’ Ordinal scale level β’ Interval scale level β’ Ratio scale level Data that is measured using a nominal scale is qualitative. 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. Some examples are Sony, Motorola, Nokia, Samsung and Apple. This is just a list and there is no agreed upon order. Some people may favor Apple but that is a matter of opinion. 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 βexcellent,β βgood,β βsatisfactory,β and βunsatisfactory.β 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, 40Β° is equal to 100Β° minus 60Β°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like -10Β° F and -15Β° C exist and are colder than 0. Interval level data can be used in calculations, but one type of comparison cannot be done. 80Β° C is not four times as hot as 20Β° C (nor |
is 80Β° F four times as hot as 20Β° 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. 30 CHAPTER 1 | SAMPLING AND DATA 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 1.9 lists the different data values in ascending order and their frequencies. DATA VALUE FREQUENCY Table 1.9 Frequency Table of Student Work Hours A frequency is the number of times a value of the data occurs. According to Table 1.9, 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 sampleβin this case, 20. Relative frequencies can be written as fractions, percents, or decimals. DATA VALUE FREQUENCY RELATIVE FREQUENCY or 0.15 or 0.25 or 0.15 or 0.30 or 0.10 or 0.05 3 20 5 20 3 20 6 20 2 20 1 20 Table 1.10 Frequency Table of Student Work Hours with Relative Frequencies The sum of the values in the relative frequency column of Table 1.10 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 1.11. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 DATA VALUE FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY CHAPTER 1 | SAMPLING AND DATA 31 20 5 20 3 20 6 20 2 20 1 20 or 0.15 0.15 or 0.25 or 0.15 or 0.30 or 0.10 or 0.05 0.15 + 0.25 = 0.40 0.40 + 0.15 = 0.55 0.55 + 0.30 = 0.85 0.85 + 0.10 = 0.95 0.95 + 0.05 = 1.00 Table 1.11 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 1.12 represents the heights, in inches, of a sample of 100 male semiprofessional soccer players. HEIGHTS (INCHES) FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 59.95β61.95 5 61.95β63.95 3 63.95β65.95 15 65.95β67.95 40 67.95β69.95 17 69.95β71.95 12 71.95β73.95 7 73.95β75.95 1 5 100 3 100 15 100 40 100 17 100 12 100 7 100 1 100 = 0.05 0.05 = 0.03 = 0.15 = 0.40 = 0.17 = 0.12 = 0.07 = 0.01 0.05 + 0.03 = 0.08 0.08 + 0.15 = 0.23 0.23 + 0.40 = 0.63 0.63 + 0.17 = 0.80 0.80 + 0.12 = 0.92 0. |
92 + 0.07 = 0.99 0.99 + 0.01 = 1.00 Total = 100 Total = 1.00 Table 1.12 Frequency Table of Soccer Player Height 32 CHAPTER 1 | SAMPLING AND DATA The data in this table have been grouped into the following intervals: β’ 59.95 to 61.95 inches β’ 61.95 to 63.95 inches β’ 63.95 to 65.95 inches β’ 65.95 to 67.95 inches β’ 67.95 to 69.95 inches β’ 69.95 to 71.95 inches β’ 71.95 to 73.95 inches β’ 73.95 to 75.95 inches NOTE This example is used again in Section 2., where the method used to compute the intervals will be explained. In this sample, there are five players whose heights fall within the interval 59.95β61.95 inches, three players whose heights fall within the interval 61.95β63.95 inches, 15 players whose heights fall within the interval 63.95β65.95 inches, 40 players whose heights fall within the interval 65.95β67.95 inches, 17 players whose heights fall within the interval 67.95β69.95 inches, 12 players whose heights fall within the interval 69.95β71.95, seven players whose heights fall within the interval 71.95β73.95, and one player whose heights fall within the interval 73.95β75.95. All heights fall between the endpoints of an interval and not at the endpoints. Example 1.14 From Table 1.12, find the percentage of heights that are less than 65.95 inches. Solution 1.14 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%. This percentage is the cumulative relative frequency entry in the third row. 1.14 Table 1.13 shows the amount, in inches, of annual rainfall in a sample of towns. Rainfall (Inches) Frequency Relative Frequency Cumulative Relative Frequency 2.95β4.97 4.97β6.99 6.99β9.01 9.01β11.03 11.03β13.05 Table 1.13 6 7 15 8 9 = 0.12 = 0.14 = 0.30 = 0.16 = |
0.18 6 50 7 50 15 50 8 50 9 50 0.12 0.12 + 0.14 = 0.26 0.26 + 0.30 = 0.56 0.56 + 0.16 = 0.72 0.72 + 0.18 = 0.90 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 33 Rainfall (Inches) Frequency Relative Frequency Cumulative Relative Frequency 13.05β15.07 5 = 0.10 5 50 0.90 + 0.10 = 1.00 Total = 50 Total = 1.00 Table 1.13 From Table 1.13, find the percentage of rainfall that is less than 9.01 inches. Example 1.15 From Table 1.12, find the percentage of heights that fall between 61.95 and 65.95 inches. Solution 1.15 Add the relative frequencies in the second and third rows: 0.03 + 0.15 = 0.18 or 18%. 1.15 From Table 1.13, find the percentage of rainfall that is between 6.99 and 13.05 inches. Example 1.16 Use the heights of the 100 male semiprofessional soccer players in Table 1.12. Fill in the blanks and check your answers. a. The percentage of heights that are from 67.95 to 71.95 inches is: ____. b. The percentage of heights that are from 67.95 to 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? 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 1.16 a. 29% b. 36% c. 77% d. 87 e. quantitative continuous f. get rosters from each team and choose a simple random sample from each 34 |
CHAPTER 1 | SAMPLING AND DATA 1.16 From Table 1.13, 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 1.17 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 1.14 was produced: DATA FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 3 4 5 7 10 12 13 15 18 20 19 1 19 3 19 2 19 4 19 2 19 1 19 1 19 1 19 1 19 0.1579 0.2105 0.1579 0.2632 0.4737 0.7895 0.8421 0.8948 0.9474 1.0000 Table 1.14 Frequency of Commuting Distances a. Is the table correct? If it is not correct, what is wrong? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 35 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 1.17 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: 0.1052, 0. |
1579, 0.2105, 0.3684, 0.4737, 0.6316, 0.7368, 0.7895, 0.8421, 0.9474, 1.0000. c. d. 5 19 7 19, 12 19, 7 19 1.17 Table 1.13 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? Example 1.18 Table 1.15 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 231 2001 21,357 2002 11,685 2003 33,819 2004 228,802 2005 88,003 2006 6,605 2007 712 2008 88,011 2009 1,790 2010 320,120 2011 21,953 2012 768 Total 823,356 Table 1.15 Answer the following questions. 36 CHAPTER 1 | SAMPLING AND DATA 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 1.18 a. 97,118 (11.8%) b. 41.6% c. 67,092/823,356 or 0.081 or 8.1 % d. 27.8% e. Quantitative discrete f. Quantitative continuous 1.18 Table 1.16 contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 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 1.16 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. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 37 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 the influence of alcohol? 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. When one variable causes change in another, we call the first variable the explanatory variable. The affected variable is called the response variable. In a randomized experiment, the researcher manipulates values of the explanatory variable and measures the resulting changes in the response variable. The different values of the explanatory variable are called treatments. An experimental unit is a single object or individual to be measured. 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, choosing not to smoke. 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 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. 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 drugs, researchers noted: Results showed that believing one had taken the substance resulted in [performance] times almost as fast as those associated with consuming the drug itself. In contrast, taking the drug 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 treatmentβa 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 which contains no actual medication, then the power of suggestion is no longer a factor. Blinding in a randomized experiment preserves the power of suggestion. 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. Example 1.19 Researchers want to investigate whether taking aspirin regularly reduces the risk of heart attack. Four hundred 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 1.19 The population is men aged 50 to 84. The sample is the 400 men who participated. 1. McClung, M. Collins, D. βBecause I know it |
will!β: placebo effects of an ergogenic aid on athletic performance. Journal of Sport & Exercise Psychology. 2007 Jun. 29(3):382-94. Web. April 30, 2013. 38 CHAPTER 1 | SAMPLING AND DATA 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. Example 1.20 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 subjectβs impression of the maskβs 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 1.20 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 1.21 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 1.21 The explanatory variable is birth order. You cannot randomly assign a personβs 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. 1.21 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. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 39 f. How can blinding be used in this study? Ethics The widespread misuse and misrepresentation of statistical information often gives the field a bad name. Some say that βnumbers donβt lie,β 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 worldβs 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 sexy results that journals found attractive. βIt was a quest for aesthetics, for beautyβinstead of the truth,β he said. He described his behavior as an addiction that drove him to carry out acts of increasingly daring fraud, like a junkie seeking a bigger and better high.[2] The committee investigating Stapel concluded that he is guilty of several practices including: β’ creating datasets, which largely confirmed the prior expectations, β’ altering data |
in existing datasets, β’ changing measuring instruments without reporting the change, and β’ 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 Stapelβs fraud states that, βstatistical flaws frequently revealed a lack of familiarity with elementary statistics.β[3] Many of Stapelβs 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 donβt 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: β’ Risks to participants must be minimized and reasonable with respect to projected benefits. β’ 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. β’ 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 participantβs name from the data record sufficient to protect privacy? Perhaps the personβs 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 2. Yudhijit Bhattacharjee, β |
The Mind of a Con Man,β Magazine, New York Times, April 26, 2013. Available online at: http://www.nytimes.com/2013/04/28/magazine/diederik-stapels-audacious-academic-fraud.html?src=dayp&_r=2& (accessed May 1, 2013). 3. βFlawed Science: The Fraudulent Research Practices of Social Psychologist Diederik Stapel,β Tillburg University, November 28, 2012, http://www.tilburguniversity.edu/upload/064a10cdbce5-4385-b9ff-05b840caeae6_120695_Rapp_nov_2012_UK_web.pdf (accessed May 1, 2013). 40 CHAPTER 1 | SAMPLING AND DATA 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 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 surprisedβand disappointed. There is a website How prevalent (www.retractionwatch.com) (http://www.retractionwatch.com) dedicated to cataloging 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 1.22 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 1.22 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 βrealβ 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. 1.22 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% prefer Brand Y and 42% have no preference between the two brands. Brand X references the study in a commercial saying βMost teens like Brand X as much as or more than Brand Y.β This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 41 1.1 Data Collection Experiment Class Time: Names: Student Learning Outcomes β’ The student will demonstrate the systematic sampling technique. β’ The student will construct relative frequency tables. β’ The student will interpret results and their differences from different data groupings. Movie Survey Ask five classmates from a different class how many movies they saw at the theater last month. Do not include rented movies. 1. Record the data. 2. In class, randomly pick one person. On the class list, mark that personβs name. Move down four names on the class list. Mark that personβs name. Continue doing this until you have marked 12 names. You may need to go back |
to the start of the list. For each marked name record the five data values. You now have a total of 60 data values. 3. For each name marked, record the data. ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ Table 1.17 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 42 CHAPTER 1 | SAMPLING AND DATA Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 7+ Table 1.18 Frequency of Number of Movies Viewed Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0β1 2β3 4β5 6β7+ Table 1.19 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 1. Is one of the tables βmore correctβ than the other? Why or why not? 2. 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? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 43 1.2 Sampling Experiment Class Time: Names: Student Learning Outcomes β’ The student will demonstrate the simple random, systematic, stratified, and cluster sampling techniques. β’ 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 sample you obtained. NOTE The |
. __________ 4. __________ 9. __________ 14. __________ 5. __________ 10. __________ 15. __________ Table 1.21 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 1.22 A Stratified Sample Pick a stratified sample, by city, of 20 restaurants. Use 25% 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 1.23 A Stratified Sample Pick a stratified sample, by entree cost, of 21 restaurants. Use 25% of the restaurants from each stratum. Round to the nearest whole number. 1. Describe your procedure. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 2. Complete the table with your sample. CHAPTER 1 | SAMPLING AND DATA 45 1. __________ 6. __________ 11. __________ 16. __________ 2. __________ 7. __________ 12. __________ 17. __________ 3. __________ 8. __________ 13. __________ 18. __________ 4. __________ 9. __________ 14. __________ 19. __________ 5. __________ 10. __________ 15. __________ 20. __________ 21. __________ Table 1. |
24 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 1.25 46 CHAPTER 1 | SAMPLING AND DATA 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 forest is a continuous RV. a random variable (RV) whose outcomes are measured; the height of trees in the 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 a variable that has an effect on a study even though it is neither an explanatory variable nor a response variable 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 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 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 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 47 Qualitative Data See Data. Quantitative Data See Data. Random Assignment the act of organizing experimental units into treatment groups using random methods Random Sampling being selected. a method of selecting a sample that gives every member of the population an equal chance of 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 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. 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. Systematic Sampling a method for selecting a random sample; list the members of the population. Use simple in the population. Let k = (number of individuals in the random sampling to select a starting point 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 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, 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, 48 CHAPTER 1 | SAMPLING AND DATA 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 off 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. In addition to rounding your answers, you can measure your data using the following four levels of measurement. β’ Nominal scale level: data that cannot be ordered nor can it be used in calculations β’ Ordinal scale level: data that can be ordered; the differences cannot be measured β’ 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. β’ 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. βAn 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.β[4] Ethical violations in statistics are not always easy to spot. Professional 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 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 AIDS antibody drug is currently under study. It is given to patients once the AIDS 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 AIDS 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: 4. Andrew Gelman, βOpen Data and Open Methods,β Ethics and Statistics, http://www.stat.columbia.edu/~gelman/research/ published/ChanceEthics1.pdf (accessed May 1, 2013). This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 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. CHAPTER 1 | SAMPLING AND DATA 49 1. population 2. sample 3. parameter 4. statistic 5. variable 1.2 Data, Sampling, and Variation in Data and Sampling 6. βNumber of times per weekβ is what type of data? a. qualitative; 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. 7. The sampling method was a. simple random; b. systematic; c. stratified; d. cluster 8. βDuration (amount of time)β is what type of data? a. qualitative; b. quantitative discrete; c. quantitative continuous 9. The colors of the houses around the park are what kind of data? a. qualitative; b. quantitative discrete; c. quantitative continuous 10. The population is ______________________ 11. Table 1.26 contains the total number of deaths worldwide as a result of earthquakes from 2000 to 2012. Year Total Number of Deaths 2000 231 2001 21,357 2002 11,685 2003 33,819 2004 228,802 2005 88,003 2006 6,605 2007 712 2008 88,011 2009 1,790 2010 320,120 2011 21,953 2012 768 Total 823,856 Table 1.26 Use Table 1.26 to answer the following questions. a. What is the proportion of deaths between 2007 and 2012? b. What percent of deaths occurred before 2001? 50 CHAPTER 1 | SAMPLING AND DATA 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. For the following four exercises, determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). 12. A group of test subjects is divided into twelve groups; then four of the groups are chosen at random. 13. A market researcher polls every tenth person who walks into a store. 14. The first 50 people who walk into a sporting event are polled on their television preferences. 15. 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 AIDS antibody drug is currently under study. It is given to patients once the AIDS 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 AIDS patients 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 16. Complete the tables using the data provided: Survival Length (in months) Frequency Relative Frequency Cumulative Relative Frequency 0.5β6.5 6.5β12.5 12.5β18.5 18.5β24.5 24.5β30.5 30.5β36.5 36.5β42.5 42.5β48.5 Table 1.27 Researcher A Survival Length (in months) Frequency Relative Frequency Cumulative Relative Frequency 0.5β6.5 6.5β12.5 12.5β18.5 18.5β24.5 24.5β30.5 30.5β36.5 36.5-45.5 Table 1.28 Researcher B This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 51 17. Determine what the key term data refers to in the above example for Researcher A. 18. List two reasons why the data may differ. 19. Can you tell if one researcher is correct and the other one is incorrect? Why? 20. Would you expect the data to be identical? Why or why not? 21. |
How might the researchers gather random data? 22. 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? 23. 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. Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency 0β2 2β4 4β6 6β8 8β10 10β12 Table 1.29 Researcher A 26 30 49 25 12 8 0.17 0.20 0.33 0.17 0.08 0.05 0.17 0.37 0.70 0.87 0.95 1 Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency 0β2 2β4 4β6 6β8 8β10 10β12 Table 1.30 Researcher B 48 51 24 12 11 4 0.32 0.34 0.16 0.08 0.07 0.03 0.32 0.66 0.82 0.90 0.97 1 24. Give a reason why the data may differ. 25. Would the sample size be large enough if the population is the students in the school? 26. Would the sample size be large enough if the population is school-aged children and young adults in the United States? 27. 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? 28. 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 1.31. The second study collected the data in Table 1.32. 52 CHAPTER 1 | SAMPLING AND DATA Group Showed improvement No improvement Deterioration Used program 142 Did not use program 72 Table 1.31 43 110 15 18 Group Showed improvement No improvement Deterioration Used program 105 Did not use program 89 Table 1.32 74 99 19 12 29. Given what you know, which study is correct? 30. 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? 31. Both groups that performed the study concluded that the software works. Is this accurate? 32. The company takes the two studies as proof that their software causes mental improvement in stroke patients. Is this a fair statement? 33. 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 1.31? 34. Is a sample size of 1,000 a reliable measure for a population of 5,000? 35. Is a sample of 500 volunteers a reliable measure for a population of 2,500? 36. 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? 37. Is a sample size of two representative of a population of five? 38. 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 39. 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. Political outlook: extreme left, left-of-center, right-of-center, extreme right 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 21β65 |
year-old women l. Common letter grades: A, B, C, D, and F 1.4 Experimental Design and Ethics 40. 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. 41. 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 childrenβs 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. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 53 HOMEWORK 1.1 Definitions of Statistics, Probability, and Key Terms 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. 42. A fitness center is interested in the mean amount of time a client exercises in the center each week. 43. 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. 44. A cardiologist is interested in the mean recovery period of her patients who have had heart attacks. 45. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance. 46. A politician is interested in the proportion of voters in his district who think he is doing a good job. 47. A marriage counselor is interested in the proportion of clients she counsels who stay married. 48. Political pollsters may be interested in the proportion of people who will vote for a particular cause. 49. 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. 50. 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 51. Consider the following: X = number of days a Lake Tahoe Community College math student is absent In this case, X is an example of a: a. variable. b. population. c. statistic. d. data. 52. The instructorβs sample produces a mean number of days absent of 3.5 days. This value is an example of a: a. parameter. b. data. c. statistic. 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. 53. number of tickets sold to a concert 54. percent of body fat 55. favorite baseball team 56. time in line to buy groceries 57. number of students enrolled at Evergreen Valley College 58. most-watched television show 59. brand of toothpaste 60. distance to the closest movie theatre 61. age of executives in Fortune 500 companies 62. number of competing computer spreadsheet software packages 54 CHAPTER 1 | SAMPLING AND DATA 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. 63. βNumber of times per weekβ is what type of data? a. qualitative b. quantitative discrete c. quantitative continuous 64. βDuration (amount of time)β is what type of data? a. qualitative b. quantitative discrete c. quantitative continuous 65. 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. 66. 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. 67. 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. 68. List some practical difficulties involved in getting accurate results from a telephone survey. 69. List some practical difficulties involved in getting accurate results from a mailed survey. 70. With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey. 71. 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 a. cluster sampling b. c. d. convenience sampling stratified sampling simple random sampling 72. 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: simple random systematic stratified a. b. c. d. cluster 73. 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 airportβs 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 partyβs 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. 74. A βrandom surveyβ was conducted of 3,274 people of the βmicroprocessor generationβ (people born since 1971, the year the microprocessor was invented). It was reported that 48% of those individuals surveyed stated that if they had $2,000 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 1 | SAMPLING AND DATA 55 to spend, they would use it for computer equipment. Also, 66% 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 βgut feeling,β 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 βAmericaβs Smithsonian.β 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. 75. The Gallup-Healthways 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? 76. In advance of the 1936 Presidential Election, a magazine titled Literary Digest 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, George Gallup conducted his own poll of 30,000 prospective voters. His 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? 77. 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 Section 1.2 could explain this connection? 78. YouPolls is a website that allows anyone to create and respond to polls. One question posted April 15 asks: βDo you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?β[5] As of April 25, 11 people responded to this question. Each participant answered βNO!β Which of the potential problems with samples discussed in this module could explain this connection? 79. A scholarly article about response rates begins with the following quote: βDeclining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about |
the validity of estimates drawn from such research.β[6] The Pew Research Center for People and the Press admits: βThe percentage of people we interview β out of all we try to interview β has been declining over the past decade or more.β[7] 5. lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: http://www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013). 6. Scott Keeter et al., βGauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,β Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (http://poq.oxfordjournals.org/content/70/5/759.full) (accessed May 1, 2013). 56 CHAPTER 1 | SAMPLING AND DATA 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 80. 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 0.6 Table 1.33 Part-time Student Course Loads a. Fill in the blanks in Table 1.33. b. What percent of students take exactly two courses? c. What percent of students take one or two courses? 81. 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 1.34. # Flossing per Week Frequency Relative Frequency Cumulative Relative Freq. 0 1 3 6 7 27 18 3 1 0.4500 0.0500 0.0167 0.9333 Table 1.34 Flossing Frequency for Adults with Gum Disease a. Fill in the blanks in Table 1.34. b. What percent of adults flossed six times per week? c. What percent flossed at most three times per week? 82. Nineteen immigrants to the U.S were asked how many years |
, to the nearest year, they have lived in the U.S. 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 1.35 was produced. Data Frequency Relative Frequency Cumulative Relative Frequency 0 2 4 2 3 1 2 19 3 19 1 19 0.1053 0.2632 0.3158 Table 1.35 Frequency of Immigrant Survey Responses 7. Frequently Asked Questions, Pew Research Center for the People & the Press, http://www.people-press.org/methodology/ frequently-asked-questions/#dont-you-have-trouble-getting-people-to-answer-your-polls (accessed May 1, 2013). This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 Data Frequency Relative Frequency Cumulative Relative Frequency CHAPTER 1 | SAMPLING AND DATA 57 5 7 10 12 15 20 3 2 2 2 1 1 3 19 2 19 2 19 2 19 1 19 1 19 0.4737 0.5789 0.6842 0.7895 0.8421 1.0000 Table 1.35 Frequency of Immigrant Survey Responses a. Fix the errors in Table 1.35. Also, explain how someone might have arrived at the incorrect number(s). b. Explain what is wrong with this statement: β47 percent of the people surveyed have lived in the U.S. for 5 years.β c. Fix the statement in b to make it correct. d. What fraction of the people surveyed have lived in the U.S. five or seven years? e. What fraction of the people surveyed have lived in the U.S. at most 12 years? f. What fraction of the people surveyed have lived in the U.S. fewer than 12 years? g. What fraction of the people surveyed have lived in the U.S. from five to 20 years, inclusive? 83. How much time does it take to travel to work? Table 1.36 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 1.36 84. Forbes 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 1.37 shows the ages of the chief executive officers for the first 60 ranked firms. Age Frequency Relative Frequency Cumulative Relative Frequency 40β44 45β49 50β54 55β59 60β64 65β69 70β74 Table 1.37 3 11 13 16 10 6 1 58 CHAPTER 1 | SAMPLING AND DATA 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) Use the following information to answer the next two exercises: Table 1.38 contains data on hurricanes that have made direct hits on the U.S. Between 1851 and 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 Total = 273 0.3993 0.2637 0.2601 0.0110 0.3993 0.6630 0.9890 1.0000 Table 1.38 Frequency of Hurricane Direct Hits 85. What is the relative frequency of direct hits that were category 4 hurricanes? a. 0.0768 b. 0.0659 c. 0.2601 d. Not enough information to calculate 86. What is the |
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