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null hypothesis, because α ≤ p. 41. Reject the null hypothesis, because α ≥ p. 42. H0: μ ≥ 29.0” Ha: μ < 29.0” 43. t19. Because you do not know the population standard deviation, use the t distribution. The degrees of freedom are 19, because df = n – 1. 44. The test statistic is –4.4721 and the p-value is 0.00013 using the calculator function TTEST. 45. With α = 0.05, reject the null hypothesis. 46. With α = 0.05, the p-value is almost zero using the calculator function TTEST, so reject the null hypothesis. 9.5: Additional Information and Full Hypothesis Test Examples 47. The level of significance is five percent. 48. two-tailed 49. one-tailed 50. H0: p = 0.8 Ha: p ≠ 0.8 51. You will use the normal test for a single population proportion because np and nq are both greater than five. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 861 10.1: Comparing Two Independent Population Means with Unknown Population Standard Deviations 52. They are matched (paired), because you interviewed married couples. 53. They are independent, because participants were assigned at random to the groups. 54. They are matched (paired), because you collected data twice from each individual. 55. d = x¯ 1 − x¯ s pooled 2 = 4.8 − 4.2 1.6 = 0.375 This is a small effect size, because 0.375 falls between Cohen’s small (0.2) and medium (0.5) effect sizes. 56. d = x¯ 1 − x¯ s pooled 2 = 5.2 − 4.2 1.6 = 0.625 The effect size is 0.625. By Cohen’s standard, this is a medium effect size, because it falls between the medium (0.5) and large (0.8) effect sizes. 57. p-value < 0.01. 58. You will only reject the null hypothesis if you get a value significantly below the hypothesized mean of 110. 10.2: Comparing Two Independent Population Means with Known Population Standard Deviations ¯ 59. X ¯ 1 − X 2 ; i.e., |
the mean difference in amount spent on textbooks for the two groups ¯ 60. H0: X ¯ Ha: X 2 This could also be written as > 0 ¯ H0: X ¯ Ha 61. Using the calculator function 2-SampTTest, reject the null hypothesis. At the five percent significance level, there is sufficient evidence to conclude that the science students spend more on textbooks than the humanities students. 62. Using the calculator function 2-SampTTest, reject the null hypothesis. At the one percent significance level, there is sufficient evidence to conclude that the science students spend more on textbooks than the humanities students. 10.3: Comparing Two Independent Population Proportions 63. H0: pA = pB Ha: pA ≠ pB 64. pc = = 65 + 78 100 + 100 = 0.715 65. Using the calculator function 2-PropZTest, the p-value = 0.0417. Reject the null hypothesis. At the three percent significance level, here is sufficient evidence to conclude that there is a difference between the proportions of households in the two communities that have cable service. 66. Using the calculator function 2-PropZTest, the p-value = 0.0417. Do not reject the null hypothesis. At the one percent significance level, there is insufficient evidence to conclude that there is a difference between the proportions of households in the two communities that have cable service. 10.4: Matched or Paired Samples d ≥ 0 67. H0: x¯ Ha: x¯ d < 0 68. t = –4.5644. 862 69. df = 30 – 1 = 29. Appendix B 70. Using the calculator function TTEST, the p-value = 0.00004, so reject the null hypothesis. At the five percent level, there is sufficient evidence to conclude that the participants lost weight, on average. 71. A positive t statistic would mean that participants, on average, gained weight over the six months. 11.1: Facts About the Chi-Square Distribution 72. μ = df = 20 σ = 2(d f ) = 40 = 6.32 11.2: Goodness-of-Fit Test 73. Enrolled = 200(0.66) = 132. Not enrolled = 200(0.34) = 68. 74. Observed (O) Expected (E) O – E (O – E)2 132 68 145 – 132 = 13 169 |
55 – 68 = –13 169 Enrolled 145 Not enrolled 55 Table B13 75. df = n – 1 = 2 – 1 = 1. (O − E)2 z 169 132 169 68 = 1.280 = 2.485 76. Using the calculator function Chi-Square GOF Test (in STAT TESTS), the test statistic is 3.7656 and the p-value is 0.0523. Do not reject the null hypothesis. At the five percent significance level, there is insufficient evidence to conclude that high school most recent graduating class distribution of enrolled and not enrolled does not fit that of the national distribution. 77. approximates the normal 78. skewed right 11.3: Test of Independence 79. Cell = Yes Cell = No Total Freshman 250(300) 500 = 150 Senior 250(300) 500 = 150 250(200) 500 250(200) 500 = 100 250 = 100 250 Total 300 200 500 Table B14 = 16.67 80. (100 − 150)2 150 (150 − 100)2 100 (200 − 100)2 150 = 25 = 16.67 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 863 Appendix B (50 − 100)2 100 = 25 81. Chi-square = 16.67 + 25 + 16.67 + 25 = 83.34. df = (r – 1)(c – 1) = 1. 82. p-value = P(Chi-square, 83.34) = 0. Reject the null hypothesis. You could also use the calculator function STAT TESTS Chi-Square Test. 11.4: Test of Homogeneity 83. The table has five rows and two columns. df = (r – 1)(c – 1) = (4)(1) = 4. 11.5: Comparison Summary of the Chi-Square Tests: Goodness-of-Fit, Independence and Homogeneity 84. Using the calculator function (STAT TESTS) Chi-Square Test, the p-value = 0. Reject the null hypothesis. At the five percent significance level, there is sufficient evidence to conclude that the poll responses are independent of the participants’ ethnic group. 85. The expected value of each cell must be at least five. 86. H0: The variables are independent. Ha: The variables are not independent. 87. H0: The populations have the same distribution. Ha: |
The populations do not have the same distribution. 11.6: Test of a Single Variance 88. H0: σ2 ≤ 5 Ha: σ2 > 5 Practice Test 4 12.1 Linear Equations 1. Which of the following equations is/are linear? A. y = –3x B. y = 0.2 + 0.74x C. y = –9.4 – 2x D. A and B E. A, B, and C 2. To complete a painting job requires four hours setup time, plus one hour per 1,000 square feet. How would you express this information in a linear equation? 3. A statistics instructor is paid a per-class fee of $2,000, plus $100 for each student in the class. How would you express this information in a linear equation? 4. A tutoring school requires students to pay a one-time enrollment fee of $500, plus tuition of $3,000 per year. Express this information in an equation. 12.2: Slope and y-intercept of a Linear Equation Use the following information to answer the next four exercises. For the labor costs of doing repairs, an auto mechanic charges a flat fee of $75 per car, plus an hourly rate of $55. 5. What are the independent and dependent variables for this situation? 6. Write the equation and identify the slope and intercept. 7. What is the labor charge for a job that takes 3.5 hours to complete? 8. One job takes 2.4 hours to complete, while another takes 6.3 hours. What is the difference in labor costs for these two jobs? 864 Appendix B 12.3: Scatter Plots 9. Describe the pattern in this scatter plot, and decide whether the X and Y variables would be good candidates for linear regression. Figure B4 10. Describe the pattern in this scatter plot, and decide whether the X and Y variables would be good candidates for linear regression. Figure B5 11. Describe the pattern in this scatter plot, and decide whether the X and Y variables would be good candidates for linear regression. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 865 Figure B6 12. Describe the pattern in this scatter plot, and decide whether the X and Y variables would be good candidates for linear regression. Figure B7 12.4: The Reg |
ression Equation Use the following information to answer the next four exercises. Height (in inches) and weight (in pounds) in a sample of college freshman males have a linear relationship with the following summary statistics: x¯ = 68.4 y¯ =141.6 sx = 4.0 sy = 9.6 r = 0.73 Let Y = weight and X = height, and write the regression equation in the form 866 y^ = a + bx 13. What is the value of the slope? 14. What is the value of the y-intercept? Appendix B 15. Write the regression equation predicting weight from height in this data set, and calculate the predicted weight for someone 68 inches tall. 12.5: Correlation Coefficient and Coefficient of Determination 16. The correlation between body weight and fuel efficiency (measured as miles per gallon) for a sample of 2,012 model cars is –0.56. Calculate the coefficient of determination for this data and explain what it means. 17. The correlation between high school GPA and freshman college GPA for a sample of 200 university students is 0.32. How much variation in freshman college GPA is not explained by high school GPA? 18. Rounded to two decimal places, what correlation between two variables is necessary to have a coefficient of determination of at least 0.50? 12.6: Testing the Significance of the Correlation Coefficient 19. Write the null and alternative hypotheses for a study to determine if two variables are significantly correlated. 20. In a sample of 30 cases, two variables have a correlation of 0.33. Do a t test to see if this result is significant at the α = 0.05 level. Use the formula 21. In a sample of 25 cases, two variables have a correlation of 0.45. Do a t test to see if this result is significant at the α = 0.05 level. Use the formula 12.7: Prediction Use the following information to answer the next two exercises. A study relating the grams of potassium (Y) to the grams of fiber (X) per serving in enriched flour products (bread, rolls, etc.) produced the equation y^ = 25 + 16x 22. For a product with five grams of fiber per serving, what are the expected grams of potassium per serving? 23. Comparing two products, one with three grams of fiber per serving and one with six grams of fiber per serving, what is the expected difference in grams of potassium per |
serving? 12.8: Outliers 24. In the context of regression analysis, what is the definition of an outlier, and what is a rule of thumb to evaluate if a given value in a data set is an outlier? 25. In the context of regression analysis, what is the definition of an influential point, and how does an influential point differ from an outlier? 26. The least squares regression line for a data set is y^ = 5 + 0.3x and the standard deviation of the residuals is 0.4. Does a case with the values x = 2, y = 6.2 qualify as an outlier? 27. The least squares regression line for a data set is y^ = 2.3 − 0.1x and the standard deviation of the residuals is 0.13. Does a case with the values x = 4.1, y = 2.34 qualify as an outlier? 13.1: One-Way ANOVA 28. What are the five basic assumptions to be met if you want to do a one-way ANOVA? 29. You are conducting a one-way ANOVA comparing the effectiveness of four drugs in lowering blood pressure in hypertensive patients. What are the null and alternative hypotheses for this study? 30. What is the primary difference between the independent samples t test and one-way ANOVA? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 867 31. You are comparing the results of three methods of teaching geometry to high school students. The final exam scores X1, X2, X3, for the samples taught by the different methods have the following distributions: X1 ~ N(85, 3.6) X1 ~ N(82, 4.8) X1 ~ N(79, 2.9) Each sample includes 100 students, and the final exam scores have a range of zero–100. Assuming the samples are independent and randomly selected, have the requirements for conducting a one-way ANOVA been met? Explain why or why not for each assumption. 32. You conduct a study comparing the effectiveness of four types of fertilizer to increase crop yield on wheat farms. When examining the sample results, you find that two of the samples have an approximately normal distribution, and two have an approximately uniform distribution. Is this a violation of the assumptions for conducting a one-way ANOVA? 13.2: The F |
Distribution Use the following information to answer the next seven exercises. You are conducting a study of three types of feed supplements for cattle to test their effectiveness in producing weight gain among calves whose feed includes one of the supplements. You have four groups of 30 calves (one is a control group receiving the usual feed, but no supplement). You will conduct a one-way ANOVA after one year to see if there are differences in the mean weight for the four groups. 33. What is SSwithin in this experiment, and what does it mean? 34. What is SSbetween in this experiment, and what does it mean? 35. What are k and i for this experiment? 36. If SSwithin = 374.5 and SStotal = 621.4 for this data, what is SSbetween? 37. What are MSbetween, and MSwithin for this experiment? 38. What is the F statistic for this data? 39. If there had been 35 calves in each group, instead of 30, with the sums of squares remaining the same, would the F statistic be larger or smaller? 13.3: Facts About the F Distribution 40. Which of the following numbers are possible F statistics? A. 2.47 B. 5.95 C. –3.61 D. 7.28 E. 0.97 41. Histograms F1 and F2 below display the distribution of cases from samples from two populations, one distributed F3,15 and one distributed F5,500. Which sample came from which population? 868 Appendix B Figure B8 Figure B9 42. The F statistic from an experiment with k = 3 and n = 50 is 3.67. At α = 0.05, will you reject the null hypothesis? 43. The F statistic from an experiment with k = 4 and n = 100 is 4.72. At α = 0.01, will you reject the null hypothesis? 13.4: Test of Two Variances 44. What assumptions must be met to perform the F test of two variances? 45. You believe there is greater variance in grades given by the math department at your university than in the English department. You collect all the grades for undergraduate classes in the two departments for a semester, compute the variance of each, and conduct an F test of two variances. What are the null and alternative hypotheses for this study? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix |
B 869 Practice Test 4 Solutions 12.1 Linear Equations 1. e. A, B, and C. All three are linear equations of the form y = mx + b. 2. Let y = the total number of hours required, and x the square footage, measured in units of 1,000. The equation is y = x + 4 3. Let y = the total payment, and x the number of students in a class. The equation is y = 100(x) + 2,000 4. Let y = the total cost of attendance, and x the number of years enrolled. The equation is y = 3,000(x) + 500 12.2: Slope and y-intercept of a Linear Equation 5. The independent variable is the hours worked on a car. The dependent variable is the total labor charges to fix a car. 6. Let y = the total charge, and x the number of hours required. The equation is y = 55x + 75 The slope is 55 and the intercept is 75. 7. y = 55(3.5) + 75 = 267.50 8. Because the intercept is included in both equations, while you are only interested in the difference in costs, you do not need to include the intercept in the solution. The difference in number of hours required is 6.3 – 2.4 = 3.9. Multiply this difference by the cost per hour: 55(3.9) = 214.5. The difference in cost between the two jobs is $214.50. 12.3: Scatter Plots 9. The X and Y variables have a strong linear relationship. These variables would be good candidates for analysis with linear regression. 10. The X and Y variables have a strong negative linear relationship. These variables would be good candidates for analysis with linear regression. 11. There is no clear linear relationship between the X and Y variables, so they are not good candidates for linear regression. 12. The X and Y variables have a strong positive relationship, but it is curvilinear rather than linear. These variables are not good candidates for linear regression. 12.4: The Regression Equation 13.73 ⎛ ⎝ 9.6 4.0 ⎞ ⎠ = 1.752 ≈ 1.75 14. a = y¯ − b x¯ = 141.6 − 1.752(68.4) = 21.7632 ≈ 21.76 |
15. y^ = 21.76 + 1.75(68) = 140.76 12.5: Correlation Coefficient and Coefficient of Determination 16. The coefficient of determination is the square of the correlation, or r2. For this data, r2 = (–0.56)2 = 0.3136 ≈ 0.31 or 31 percent. This means that 31 percent of the variation in fuel efficiency can be explained by the bodyweight of the automobile. 17. The coefficient of determination = 0.322 = 0.1024. This is the amount of variation in freshman college GPA that can be explained by high school GPA. The amount that cannot be explained is 1 – 0.1024 = 0.8976 ≈ 0.90. So, about 90 percent of variance in freshman college GPA in this data is not explained by high school GPA. 18. r = r 2 0.5 = 0.707106781 ≈ 0.71 You need a correlation of 0.71 or higher to have a coefficient of determination of at least 0.5. 12.6: Testing the Significance of the Correlation Coefficient 19. H0: ρ = 0 Ha: ρ ≠ 0 870 Appendix B 20.33 30 − 2 1 − 0.332 = 1.85 The critical value for α = 0.05 for a two-tailed test using the t29 distribution is 2.045. Your value is less than this, so you fail to reject the null hypothesis and conclude that the study produced no evidence that the variables are significantly correlated. Using the calculator function tcdf, the p-value is 2tcdf(1.85, 10^99, 29) = 0.0373. Do not reject the null hypothesis and conclude that the study produced no evidence that the variables are significantly correlated. 21.45 25 − 2 1 − 0.452 = 2.417 The critical value for α = 0.05 for a two-tailed test using the t24 distribution is 2.064. Your value is greater than this, so you reject the null hypothesis and conclude that the study produced evidence that the variables are significantly correlated. Using the calculator function tcdf, the p-value is 2tcdf(2.417, 10^99, 24) = 0.0118. Reject the null hypothesis and conclude that the study produced evidence that the variables are significantly correlated. 12.7: Prediction 22. y^ = 25 |
+ 16(5) = 105 23. Because the intercept appears in both predicted values, you can ignore it in calculating a predicted difference score. The difference in grams of fiber per serving is 6 – 3 = 3, and the predicted difference in grams of potassium per serving is (16)(3) = 48. 12.8: Outliers 24. An outlier is an observed value that is far from the least squares regression line. A rule of thumb is that a point more than two standard deviations of the residuals from its predicted value on the least squares regression line is an outlier. 25. An influential point is an observed value in a data set that is far from other points in the data set, in a horizontal direction. Unlike an outlier, an influential point is determined by its relationship with other values in the data set, not by its relationship to the regression line. 26. The predicted value for y is y^ = 5 + 0.3x = 5.6. The value of 6.2 is less than two standard deviations from the predicted value, so it does not qualify as an outlier. Residual for (2, 6.2): 6.2 – 5.6 = 0.6 (0.6 < 2(0.4)) 27. The predicted value for y is y^ = 2.3 – 0.1(4.1) = 1.89. The value of 2.32 is more than two standard deviations from the predicted value, so it qualifies as an outlier. Residual for (4.1, 2.34): 2.32 – 1.89 = 0.43 (0.43 > 2(0.13)) 13.1: One-Way ANOVA 28. 1. Each sample is drawn from a normally distributed population. 2. All samples are independent and randomly selected. 3. The populations from which the samples are drawn have equal standard deviations. 4. The factor is a categorical variable. 5. The response is a numerical variable. 29. H0: μ1 = μ2 = μ3 = μ4 Ha: At least two of the group means μ1, μ2, μ3, μ4 are not equal. 30. The independent samples t test can only compare means from two groups, while one-way ANOVA can compare means of more than two groups. 31. Each sample appears to have been drawn from normally distributed populations, the factor is a categorical variable (method), |
the outcome is a numerical variable (test score), and you were told the samples were independent and randomly selected, so those requirements are met. However, each sample has a different standard deviation, and this suggests that the populations from which they were drawn also have different standard deviations, which is a violation of an assumption for one-way ANOVA. Further statistical testing will be necessary to test the assumption of equal variance before proceeding with the analysis. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 871 32. One of the assumptions for a one-way ANOVA is that the samples are drawn from normally distributed populations. Since two of your samples have an approximately uniform distribution, this casts doubt on whether this assumption has been met. Further statistical testing will be necessary to determine if you can proceed with the analysis. 13.2: The F Distribution 33. SSwithin is the sum of squares within groups, representing the variation in outcome that cannot be attributed to the different feed supplements but due to individual or chance factors among the calves in each group. 34. SSbetween is the sum of squares between groups, representing the variation in outcome that can be attributed to the different feed supplements. 35. k = the number of groups = 4 n1 = the number of cases in group 1 = 30 n = the total number of cases = 4(30) = 120 36. SStotal = SSwithin + SSbetween, so SSbetween = SStotal – SSwithin 621.4 – 374.5 = 246.9 37. The mean squares in an ANOVA are found by dividing each sum of squares by its respective degrees of freedom (df). For SStotal, df = n – 1 = 120 – 1 = 119. For SSbetween, df = k – 1 = 4 – 1 = 3. For SSwithin, df = 120 – 4 = 116. MSbetween = 246.9 = 82.3 3 MSwithin = 374.5 116 = 3.23 38. F = MSbetween MSwithin = 82.3 3.23 = 25.48 39. It would be larger, because you would be dividing by a smaller number. The value of MSbetween would not change with a change of sample size, but the value of MSwithin would be smaller, because you would be dividing by a larger number (dfwithin would be 136, not 116). Dividing a constant by a smaller number produces a larger result. |
13.3: Facts About the F Distribution 40. All but choice c, –3.61. F Statistics are always greater than or equal to 0. 41. As the degrees of freedom increase in an F distribution, the distribution becomes more nearly normal. Histogram F2 is closer to a normal distribution than histogram F1, so the sample displayed in histogram F1 was drawn from the F3,15 population, and the sample displayed in histogram F2 was drawn from the F5,500 population. 42. Using the calculator function Fcdf, p-value = Fcdf(3.67, 1E, 3, 50) = 0.0182. Reject the null hypothesis. 43. Using the calculator function Fcdf, p-value = Fcdf(4.72, 1E, 4, 100) = 0.0016 Reject the null hypothesis. 13.4: Test of Two Variances 44. The samples must be drawn from populations that are normally distributed, and must be drawn from independent populations. 2 ≤ σE 2 2 2 > σE 2 = variance in English grades. 2 = variance in math grades, and σE 45. Let σ M H0: σ M Ha: σ M Practice Final Exam 1 Use the following information to answer the next two exercises. An experiment consists of tossing two, 12-sided dice (the numbers 1–12 are printed on the sides of each die). • Let Event A = both dice show an even number. • Let Event B = both dice show a number greater than eight 1. Events A and B are 872 Appendix B A. Mutually exclusive B. Independent C. Mutually exclusive and independent D. Neither mutually exclusive nor independent 2. Find P(A|B). A. B. C. D. 2 4 16 144 4 16 2 144 3. Which of the following are TRUE when we perform a hypothesis test on matched or paired samples? A. Sample sizes are almost never small. B. Two measurements are drawn from the same pair of individuals or objects. C. Two sample means are compared to each other. D. Answer choices b and c are both true. Use the following information to answer the next two exercises. One hundred eighteen students were asked what type of color their bedrooms were painted: light colors, dark colors, or vibrant colors. The results were tabulated according to gender. Light colors Dark colors Vibrant colors Female Male |
20 10 Table B15 22 30 28 8 4. Find the probability that a randomly chosen student is male or has a bedroom painted with light colors. A. B. C. D. 10 118 68 118 48 118 10 48 5. Find the probability that a randomly chosen student is male given the student’s bedroom is painted with dark colors. A. B. C. 30 118 30 48 22 118 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B D. 30 52 873 Use the following information to answer the next two exercises. We are interested in the number of times a teenager must be reminded to do his or her chores each week. A survey of 40 mothers was conducted. Table B16 shows the results of the survey. x P (x) 0 1 2 3 4 5 2 40 5 40 14 40 7 40 4 40 Table B16 6. Find the probability that a teenager is reminded two times. A. 8 B. C. 8 40 6 40 D. 2 7. Find the expected number of times a teenager is reminded to do his or her chores. A. 15 B. 2.78 C. 1.0 D. 3.13 Use the following information to answer the next two exercises. On any given day, approximately 37.5 percent of the cars parked in the De Anza parking garage are parked crookedly. We randomly survey 22 cars. We are interested in the number of cars that are parked crookedly. 8. For every 22 cars, how many would you expect to be parked crookedly, on average? A. 8.25 B. 11 C. 18 D. 7.5 9. What is the probability that at least 10 of the 22 cars are parked crookedly? A. 0.1263 B. 0.1607 874 C. 0.2870 D. 0.8393 Appendix B 10. Using a sample of 15 Stanford-Binet IQ scores, we wish to conduct a hypothesis test. Our claim is that the mean IQ score on the Stanford-Binet IQ test is more than 100. It is known that the standard deviation of all Stanford-Binet IQ scores is 15 points. Which of the following is the correct distribution to use for the hypothesis test? A. Binomial B. Student's t C. Normal D. Uniform Use the following information to answer the next three exercises. De Anza College keeps statistics on the pass rate of |
students who enroll in math classes. In a sample of 1,795 students enrolled in Math 1A (1st quarter calculus), 1,428 passed the course. In a sample of 856 students enrolled in Math 1B (2nd quarter calculus), 662 passed. In general, are the pass rates of Math 1A and Math 1B statistically the same? Let A = the subscript for Math 1A and B = the subscript for Math 1B. 11. If you were to conduct an appropriate hypothesis test, the alternate hypothesis would be A. Ha: pA = pB B. Ha: pA > pB C. Ho: pA = pB D. Ha: pA ≠ pB 12. The Type I error is to A. conclude that the pass rate for Math 1A is the same as the pass rate for Math 1B when, in fact, the pass rates are different. B. conclude that the pass rate for Math 1A is different than the pass rate for Math 1B when, in fact, the pass rates are the same. C. conclude that the pass rate for Math 1A is greater than the pass rate for Math 1B when, in fact, the pass rate for Math 1A is less than the pass rate for Math 1B. D. conclude that the pass rate for Math 1A is the same as the pass rate for Math 1B when, in fact, they are the same. 13. The correct decision is to A. reject H0. B. not reject H0. C. There is not enough information given to conduct the hypothesis test. Kia, Alejandra, and Iris are runners on the track teams at three different schools. Their running times, in minutes, and the statistics for the track teams at their respective schools, for a one mile run, are given in the table below: Running Time School Average Running Time School Standard Deviation Kia 4.9 Alejandra 4.2 Iris 4.5 Table B17 5.2 4.6 4.9 0.15 0.25 0.12 14. Which student is the BEST when compared to the other runners at her school? A. Kia B. Alejandra C. Iris This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 875 D. Impossible to determine Use the following information to answer the next two exercises. The following adult ski sweater prices |
are from the Gorsuch Ltd. Winter catalog: $212, $292, $278, $199, $280, $236. Assume the underlying sweater price population is approximately normal. The null hypothesis is that the mean price of adult ski sweaters from Gorsuch Ltd. is at least $275. 15. Which of the following is the correct distribution to use for the hypothesis test? A. Normal B. Binomial C. Student's t D. Exponential 16. The hypothesis test A. B. C. is two-tailed. is left-tailed. is right-tailed. D. has no tails. 17. Sara, a statistics student, wanted to determine the mean number of books that college professors have in their office. She randomly selected two buildings on campus and asked each professor in the selected buildings how many books are in his or her office. Sara surveyed 25 professors. The type of sampling selected is A. B. simple random sampling. systematic sampling. C. cluster sampling. D. stratified sampling. 18. A clothing store would use which measure of the center of data when placing orders for the typical middle customer? A. Mean B. Median C. Mode D. IQR 19. In a hypothesis test, the p-value is A. the probability that an outcome of the data will happen purely by chance when the null hypothesis is true. B. called the preconceived alpha. C. compared to beta to decide whether to reject or not reject the null hypothesis. D. Answer choices A and B are both true. Use the following information to answer the next three exercises. A community college offers classes six days a week: Monday through Saturday. Maria conducted a study of the students in her classes to determine how many days per week the students who are in her classes come to campus for classes. In each of her five classes she randomly selected 10 students and asked them how many days they come to campus for classes. Each of her classes are the same size. The results of her survey are summarized in Table B18. Number of Days on Campus Frequency Relative Frequency Cumulative Relative Frequency 1 Table B18 2 876 Appendix B Number of Days on Campus Frequency Relative Frequency Cumulative Relative Frequency 2 3 4 5 6 Table B18 12 10 0 1.24.20.02.98 1 20. Combined with convenience sampling, what other sampling technique did Maria use? A. Simple random B. Systematic C. Cluster D. Stratified 21. How many students come to campus for classes four days a |
week? A. 49 B. 25 C. 30 D. 13 22. What is the 60th percentile for this data? A. 2 B. 3 C. 4 D. 5 Use the following information to answer the next two exercises. The following data are the results of a random survey of 110 reservists called to active duty to increase security at California airports. Number of Dependents Frequency 0 1 2 3 4 Table B19 11 27 33 20 19 23. Construct a 95 percent confidence interval for the true population mean number of dependents of reservists called to active duty to increase security at California airports. A. B. C. D. (1.85, 2.32) (1.80, 2.36) (1.97, 2.46) (1.92, 2.50) This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 877 24. The 95 percent confidence interval above means: A. Five percent of confidence intervals constructed this way will not contain the true population aveage number of dependents. B. We are 95 percent confident the true population mean number of dependents falls in the interval. C. Both of the above answer choices are correct. D. None of the above. 25. X ~ U(4, 10). Find the 30th percentile. A. 0.3000 B. 3 C. 5.8 D. 6.1 26. If X ~ Exp(0.8), then P(x < μ) = — A. 0.3679 B. 0.4727 C. 0.6321 D. cannot be determined 27. The lifetime of a computer circuit board is normally distributed with a mean of 2,500 hours and a standard deviation of 60 hours. What is the probability that a randomly chosen board will last at most 2,560 hours? A. 0.8413 B. 0.1587 C. 0.3461 D. 0.6539 28. A survey of 123 reservists called to active duty as a result of the September 11, 2001, attacks was conducted to determine the proportion that were married. Eighty-six reported being married. Construct a 98 percent confidence interval for the true population proportion of reservists called to active duty that are married. A. B. C. D. (0.6030, 0.7954) (0.6181, 0.7802) (0 |
.5927, 0.8057) (0.6312, 0.7672) 29. Winning times in 26 mile marathons run by world class runners average 145 minutes with a standard deviation of 14 minutes. A sample of the last 10 marathon winning times is collected. Let x = mean winning times for 10 marathons. The distribution for x is A. N ⎛ ⎝145, 14 10 ⎞ ⎠ B. N(145,14) C. D. t9 t10 30. Suppose that Phi Beta Kappa honors the top 1 percent of college and university seniors. Assume that grade point means (GPA) at a certain college are normally distributed with a 2.5 mean and a standard deviation of 0.5. What would be the minimum GPA needed to become a member of Phi Beta Kappa at that college? A. 3.99 B. 1.34 C. 3.00 878 Appendix B D. 3.66 The number of people living on American farms has declined steadily during the 20th century. Here are data on the farm population (in millions of persons) from 1935 to 1980. Year 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 Population 32.1 30.5 24.4 23 19.1 15.6 12.4 9.7 8.9 7.2 Table B20 31. The linear regression equation is y^ = 1166.93 – 0.5868x. What was the expected farm population in millions of persons for 1980? A. 7.2 B. 5.1 C. 6 D. 8 32. In linear regression, which is the best possible SSE? A. 13.46 B. 18.22 C. 24.05 D. 16.33 33. In regression analysis, if the correlation coefficient is close to one, what can be said about the best fit line? A. It is a horizontal line. Therefore, we cannot use it. B. There is a strong linear pattern. Therefore, it is most likely a good model to be used. C. The coefficient correlation is close to the limit. Therefore, it is hard to make a decision. D. We do not have the equation. Therefore, we cannot say anything about it. Use the following information to answer the next three exercises. A study of the career plans of young women and men sent questionnaires to all 722 members of the senior class in the College of Business Administration at the University of Illinois |
. One question asked which major within the business program the student had chosen. Here are the data from the students who responded. Female Male Accounting Administration Economics Finance 68 91 5 61 56 40 6 59 Table B21 Does the data suggest that there is a relationship between the gender of students and their choice of major? 34. The distribution for the test is A. Chi2 8. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 879 Appendix B B. Chi2 3. C. t721. D. N(0, 1). 35. The expected number of females who choose finance is A. 37. B. 61. C. 60. D. 70. 36. The p-value is 0.0127 and the level of significance is 0.05. The conclusion to the test is: A. B. C. D. there is insufficient evidence to conclude that the choice of major and the gender of the student are not independent of each other. there is sufficient evidence to conclude that the choice of major and the gender of the student are not independent of each other. there is sufficient evidence to conclude that students find economics very hard. there is in sufficient evidence to conclude that more females prefer administration than males. 37. An agency reported that the work force nationwide is composed of 10 percent professional, 10 percent clerical, 30 percent skilled, 15 percent service, and 35 percent semiskilled laborers. A random sample of 100 San Jose residents indicated 15 professional, 15 clerical, 40 skilled, 10 service, and 20 semiskilled laborers. At α = 0.10, does the work force in San Jose appear to be consistent with the agency report for the nation? Which kind of test is it? A. Chi2 goodness of fit B. Chi2 test of independence C. Independent groups proportions D. Unable to determine Practice Final Exam 1 Solutions Solutions 1. B independent 2. C 4 16 3. B Two measurements are drawn from the same pair of individuals or objects. 4. B 68 118 5. D 30 52 6. B 8 40 7. B 2.78 8. A 8.25 9. C 0.2870 10. C Normal 11. D Ha: pA ≠ pB 12. B conclude that the pass rate for Math 1A is different than the pass rate for Math 1B when, in fact, the pass rates are the same. 13. B not reject H0 8 |
80 14. C Iris 15. C Student's t 16. B is left-tailed. 17. C cluster sampling 18. B median 19. A the probability that an outcome of the data will happen purely by chance when the null hypothesis is true. Appendix B 20. D stratified 21. B 25 22. C 4 23. A (1.85, 2.32) 24. C Both above are correct. 25. C 5.8 26. C 0.6321 27. A 0.8413 28. A (0.6030, 0.7954) 29. A N ⎛ ⎝145, 14 10 ⎞ ⎠ 30. D 3.66 31. B 5.1 32. A 13.46 33. B There is a strong linear pattern. Therefore, it is most likely a good model to be used. 34. B Chi2 3. 35. D 70 36. B There is sufficient evidence to conclude that the choice of major and the gender of the student are not independent of each other. 37. A Chi2 goodness-of-fit Practice Final Exam 2 1. A study was done to determine the proportion of teenagers that own a car. The population proportion of teenagers that own a car is the A. statistic. B. parameter. C. population. D. variable. Use the following information to answer the next two exercises. value frequency 0 1 1 4 Table B22 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 881 value frequency 2 3 6 7 9 4 Table B22 2. The box plot for the data is Figure B10 3. If six were added to each value of the data in the table, the 15th percentile of the new list of values is would be A. six B. one C. seven D. eight Use the following information to answer the next two exercises. Suppose that the probability of a drought in any independent year is 20 percent. Out of those years in which a drought occurs, the probability of water rationing is 10 percent. However, in any year, the probability of water rationing is 5 percent. 4. What is the probability of both a drought and water rationing occurring? A. 0.05 B. 0.01 C. 0.02 D. 0.30 5. Which of the following is true? A. Drought and water rationing are independent events. |
B. Drought and water rationing are mutually exclusive events. C. None of the above. 882 Appendix B Use the following information to answer the next two exercises. Suppose that a survey yielded the following data: gender apple pumpkin pecan female male 40 20 10 30 Table B23 Favorite Pie 30 10 6. Suppose that one individual is randomly chosen. The probability that the person’s favorite pie is apple or the person is male is — A. B. C. D. 40 60 60 140 120 140 100 140 7. Suppose H0 is favorite pie and gender are independent. The p-value is — A. ≈ 0 B. 1 C. 0.05 D. Cannot be determined Use the following information to answer the next two exercises. Let’s say that the probability that an adult watches the news at least once per week is 0.60. We randomly survey 14 people. Of interest is the number of people who watch the news at least once per week. 8. Which of the following statements is FALSE? A. X ~ B(14 0.60) B. The values for x are {1, 2, 3,... 14}. C. μ = 8.4 D. P(X = 5) = 0.0408 9. Find the probability that at least six adults watch the news at least once per week. A. 6 14 B. 0.8499 C. 0.9417 D. 0.6429 10. The following histogram is most likely to be a result of sampling from which distribution? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 883 Figure B11 A. Chi-square with df = 6 B. Exponential C. Uniform D. Binomial 11. The ages of campus day and evening students is known to be normally distributed. A sample of six campus day and evening students reported their ages (in years) as {18, 35, 27, 45, 20, 20}. What is the error bound for the 90 percent confidence interval of the true average age? A. 11.2 B. 22.3 C. 17.5 D. 8.7 12. If a normally distributed random variable has µ = 0 and σ = 1, then 97.5 percent of the population values lie above A. –1.96 B. 1.96 C. 1 D. –1 Use the following information to |
answer the next three exercises. The amount of money a customer spends in one trip to the supermarket is known to have an exponential distribution. Suppose the average amount of money a customer spends in one trip to the supermarket is $72. 13. What is the probability that one customer spends less than $72 in one trip to the supermarket? A. 0.6321 B. 0.5000 C. 0.3714 884 D. 1 Appendix B 14. How much money altogether would you expect the next five customers to spend in one trip to the supermarket (in dollars)? A. 72 B. 722 5 C. 5184 D. 360 15. If you want to find the probability that the mean amount of money 50 customers spend in one trip to the supermarket is less than $60, the distribution to use is A. N(72, 72) B. N ⎛ ⎝72, 72 50 ⎞ ⎠ C. Exp(72) ⎛ D. Exp ⎝ ⎞ ⎠ 1 72 Use the following information to answer the next three exercises. The amount of time it takes a fourth grader to carry out the trash is uniformly distributed in the interval from one to 10 minutes. 16. What is the probability that a randomly chosen fourth grader takes more than seven minutes to take out the trash? A. B. C. D. 3 9 7 9 3 10 7 10 17. Which graph best shows the probability that a randomly chosen fourth grader takes more than six minutes to take out the trash, given that he or she has already taken more than three minutes? Figure B12 18. We should expect a fourth grader to take how many minutes to take out the trash? A. 4.5 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B B. 5.5 C. 5 D. 10 885 Use the following information to answer the next three exercises. At the beginning of the quarter, the amount of time a student waits in line at the campus cafeteria is normally distributed with a mean of five minutes and a standard deviation of 1.5 minutes. 19. What is the 90th percentile of waiting times in minutes? A. 1.28 B. 90 C. 7.47 D. 6.92 20. The median waiting time in minutes for one student is A. 5 B. 50 C. 2.5 D. 1. |
5 21. Find the probability that the average wait time for ten students is at most 5.5 minutes. A. 0.6301 B. 0.8541 C. 0.3694 D. 0.1459 22. A sample of 80 software engineers in Silicon Valley is taken, and it is found that 20 percent of them earn approximately $50,000 per year. A point estimate for the true proportion of engineers in Silicon Valley who earn $50,000 per year is A. 16 B. 0.2 C. 1 D. 0.95 23. If P(Z < zα) = 0.1587 where Z ~ N(0, 1), then α is equal to A. –1 B. 0.1587 C. 0.8413 D. 1 24. A professor tested 35 students to determine their entering skills. At the end of the term, after completing the course, the same test was administered to the same 35 students to study their improvement. This would be a test of A. B. independent groups two proportions C. matched pairs, dependent groups D. exclusive groups A math exam was given to all the third-grade children attending ABC School. Two random samples of scores were taken. 886 Appendix B n ¯ x Boys 55 82 Girls 60 86 Table B24 s 5 7 25. Which of the following correctly describes the results of a hypothesis test of the claim, “There is a difference between the mean scores obtained by third-grade girls and boys at the 5 percent level of significance”? A. Do not reject H0. There is insufficient evidence to conclude that there is a difference in the mean scores. B. Do not reject H0. There is sufficient evidence to conclude that there is a difference in the mean scores. C. Reject H0. There is insufficient evidence to conclude that there is no difference in the mean scores. D. Reject H0. There is sufficient evidence to conclude that there is a difference in the mean scores. 26. In a survey of 80 males, 45 had played an organized sport growing up. Of the 70 females surveyed, 25 had played an organized sport growing up. We are interested in whether the proportion for males is higher than the proportion for females. The correct conclusion is that A. There is insufficient information to conclude that the proportion for males is the same as the proportion for females. B. There is insufficient information to conclude that the proportion for males is not the same as the proportion for females. |
C. There is sufficient evidence to conclude that the proportion for males is higher than the proportion for females. D. There is not enough information to make a conclusion. 27. From past experience, a statistics teacher has found that the average score on a midterm is 81, with a standard deviation of 5.2. This term, a class of 49 students had a standard deviation of 5 on the midterm. Do the data indicate that we should reject the teacher’s claim that the standard deviation is 5.2? Use α = 0.05. A. Yes B. No C. Not enough information given to solve the problem 28. Three loading machines are being compared. Ten samples were taken for each machine. Machine I took an average of 31 minutes to load packages, with a standard deviation of two minutes. Machine II took an average of 28 minutes to load packages, with a standard deviation of 1.5 minutes. Machine III took an average of 29 minutes to load packages, with a standard deviation of one minute. Find the p-value when testing that the average loading times are the same. A. p-value is close to zero B. p-value is close to one C. Not enough information given to solve the problem Use the following information to answer the next three exercises. A corporation has offices in different parts of the country. It has gathered the following information concerning the number of bathrooms and the number of employees at seven sites: Number of employees x 650 730 810 900 102 107 1150 Number of bathrooms y 40 50 54 61 82 110 121 Table B25 29. Is the correlation between the number of employees and the number of bathrooms significant? A. Yes B. No C. Not enough information to answer question This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix B 887 30. The linear regression equation is A. ŷ = 0.0094 − 79.96x B. ŷ = 79.96 + 0.0094x C. ŷ = 79.96 − 0.0094x D. ŷ = −0.0094 + 79.96x 31. If a site has 1,150 employees, approximately how many bathrooms should it have? A. 69 B. 91 C. 91,954 D. We should not be estimating here. 32. Suppose that a sample of size 10 was collected, with x¯ = 4.4 and s = 1.4. H |
0: σ2 = 1.6 vs. Ha: σ2 ≠ 1.6. Which graph best describes the results of the test? Figure B13 Sixty-four backpackers were asked the number of days since their latest backpacking trip. The number of days is given in Table B26. # of days 1 2 3 4 5 6 7 8 Frequency 5 9 6 12 7 10 5 10 Table B26 33. Conduct an appropriate test to determine if the distribution is uniform. A. The p-value is > 0.10. There is insufficient information to conclude that the distribution is not uniform. B. The p-value is < 0.01. There is sufficient information to conclude the distribution is not uniform. C. The p-value is between 0.01 and 0.10, but without alpha (α) there is not enough information. D. There is no such test that can be conducted. 34. Which of the following statements is true when using one-way ANOVA? A. The populations from which the samples are selected have different distributions. B. The sample sizes are large. C. The test is to determine if the different groups have the same means. D. There is a correlation between the factors of the experiment. 888 Appendix B Practice Final Exam 2 Solutions Solutions 1. B parameter. 2. A 3. C seven 4. C 0.02 5. C none of the above 6. D 100 140 7. A ≈ 0 8. B The values for x are: {1, 2, 3,... 14} 9. C 0.9417. 10. D binomial 11. D 8.7 12. A –1.96 13. A 0.6321 14. D 360 15. B N ⎛ ⎝72, 72 50 ⎞ ⎠ 16. A 3 9 17. D 18. B 5.5 19. D 6.92 20. A 5 21. B 0.8541 22. B 0.2 23. A –1. 24. C matched pairs, dependent groups. 25. D Reject H0. There is sufficient evidence to conclude that there is a difference in the mean scores. 26. C there is sufficient evidence to conclude that the proportion for males is higher than the proportion for females. 27. B no 28. B p-value is close to 1. 29. B No 30. C y^ = 79.96x |
– 0.0094 31. D We should not be estimating here. 32. A 33. A The p-value is > 0.10. There is insufficient information to conclude that the distribution is not uniform. 34. C The test is to determine if the different groups have the same means. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix C 889 APPENDIX C: DATA SETS Lap Times The following tables provide lap times from Terri Vogel's log book. Times are recorded in seconds for 2.5-mile laps completed in a series of races and practice runs. Lap 1 Lap 2 Lap 3 Lap 4 Lap 5 Lap 6 Lap 7 Race 1 Race 2 Race 3 Race 4 Race 5 Race 6 Race 7 Race 8 Race 9 135 134 129 125 133 130 132 127 132 Race 10 135 Race 11 132 Race 12 134 Race 13 128 Race 14 132 Race 15 136 Race 16 129 Race 17 134 Race 18 129 Race 19 130 Race 20 131 130 131 128 125 132 130 131 128 130 131 131 130 127 131 129 129 131 129 129 128 131 131 127 126 132 130 133 127 127 131 132 130 128 131 129 129 132 130 129 130 132 129 127 125 132 129 131 130 128 132 131 130 128 131 129 128 131 130 129 128 130 128 130 124 131 129 134 128 126 130 130 131 128 132 129 128 132 133 129 129 131 128 127 125 130 130 134 126 127 131 129 130 129 130 129 129 132 133 129 130 133 129 129 125 132 129 131 128 124 130 129 130 128 130 129 129 132 127 128 130 Table C1 Race Lap Times (in seconds) Lap 1 Lap 2 Lap 3 Lap 4 Lap 5 Lap 6 Lap 7 Practice 1 Practice 2 Practice 3 142 140 130 143 135 133 180 134 130 137 133 128 134 128 135 134 128 133 172 131 133 Table C2 Practice Lap Times (in seconds) 890 Appendix C Lap 1 Lap 2 Lap 3 Lap 4 Lap 5 Lap 6 Lap 7 Practice 4 Practice 5 Practice 6 Practice 7 Practice 8 Practice 9 141 140 142 139 143 135 Practice 10 131 Practice 11 143 Practice 12 132 Practice 13 149 Practice 14 133 Practice 15 138 136 138 142 137 136 134 130 139 133 144 132 136 137 136 139 135 134 133 128 139 131 144 137 133 136 137 138 135 133 133 129 138 129 139 133 133 136 135 129 137 134 132 127 138 128 138 134 132 136 134 129 134 133 132 128 137 127 138 130 131 145 134 127 135 |
132 133 127 138 126 137 131 131 Table C2 Practice Lap Times (in seconds) Stock Prices The following table lists initial public offering (IPO) stock prices for all 1999 stocks that at least doubled in value during the first day of trading. $17.00 $23.00 $14.00 $16.00 $12.00 $26.00 $20.00 $22.00 $14.00 $15.00 $22.00 $18.00 $18.00 $21.00 $21.00 $19.00 $15.00 $21.00 $18.00 $17.00 $15.00 $25.00 $14.00 $30.00 $16.00 $10.00 $20.00 $12.00 $16.00 $17.44 $16.00 $14.00 $15.00 $20.00 $20.00 $16.00 $17.00 $16.00 $15.00 $15.00 $19.00 $48.00 $16.00 $18.00 $9.00 $18.00 $18.00 $20.00 $8.00 $20.00 $17.00 $14.00 $11.00 $16.00 $19.00 $15.00 $21.00 $12.00 $8.00 $16.00 $13.00 $14.00 $15.00 $14.00 $13.41 $28.00 $21.00 $17.00 $28.00 $17.00 $19.00 $16.00 $17.00 $19.00 $18.00 $17.00 $15.00 $14.00 $21.00 $12.00 $18.00 $24.00 $15.00 $23.00 $14.00 $16.00 $12.00 $24.00 $20.00 $14.00 $14.00 $15.00 $14.00 $19.00 $16.00 $38.00 $20.00 Table C3 IPO Offer Prices This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix C 891 $24.00 $16.00 $8.00 $18.00 $17.00 $16.00 $15.00 $7.00 $19.00 $ |
12.00 $8.00 $23.00 $12.00 $18.00 $20.00 $21.00 $34.00 $16.00 $26.00 $14.00 Table C3 IPO Offer Prices References Data compiled by Jay R. Ritter of University of Florida using data from Securities Data Co. and Bloomberg. 892 Appendix C This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix D 893 APPENDIX D: GROUP AND PARTNER PROJECTS Univariate Data Student Learning Objectives • The student will design and carry out a survey. • The student will analyze and graphically display the results of the survey. Instructions As you complete each task below, check it off. Answer all questions in your summary. ____ Decide what data you are going to study. Here are two examples, but you may NOT use them: number of M&M's per bag, number of pencils students have in their backpacks. ____ Are your data discrete or continuous? How do you know? ____ Decide how you are going to collect the data (for instance, buy 30 bags of M&M's; collect data from the World Wide Web). ____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random (using a random number generator) sampling. Do not use convenience sampling. Which method did you use? Why did you pick that method? ____ Conduct your survey. Your data size must be at least 30. ____ Summarize your data in a chart with columns showing data value, frequency, relative frequency and cumulative relative frequency. Answer the following (rounded to two decimal places): a. b. x¯ = _____ s = _____ c. First quartile = _____ d. Median = _____ e. 70th percentile = _____ ____ What value is two standard deviations above the mean? ____ What value is 1.5 standard deviations below the mean? ____ Construct a histogram displaying your data. ____ In complete sentences, describe the shape of your graph. ____ Do you notice any potential outliers? If so, what values are they? Show your work in how you used the potential outlier formula to determine whether or not the values might be outliers. ____ Construct a box plot displaying your data. ____ Does the middle 50% of the data appear to be |
concentrated together or spread apart? Explain how you determined this. ____ Looking at both the histogram and the box plot, discuss the distribution of your data. Assignment Checklist You need to turn in the following typed and stapled packet, with pages in the following order: ____ Cover sheet: name, class time, and name of your study 894 Appendix D ____ Summary page: This should contain paragraphs written with complete sentences. It should include answers to all the questions above. It should also include statements describing the population under study, the sample, a parameter or parameters being studied, and the statistic or statistics produced. ____ URL for data, if your data are from the World Wide Web ____ Chart of data, frequency, relative frequency, and cumulative relative frequency ____ Page(s) of graphs: histogram and box plot Continuous Distributions and Central Limit Theorem Student Learning Objectives • The student will collect a sample of continuous data. • The student will attempt to fit the data sample to various distribution models. • The student will validate the central limit theorem. Instructions As you complete each task below, check it off. Answer all questions in your summary. Part I: Sampling ____ Decide what continuous data you are going to study. (Here are two examples, but you may NOT use them: the amount of money a student spent on college supplies this term, or the length of time distance telephone call lasts.) ____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random (using a random number generator) sampling. Do not use convenience sampling. What method did you use? Why did you pick that method? ____ Conduct your survey. Gather at least 150 pieces of continuous, quantitative data. ____ Define (in words) the random variable for your data. X = _______ ____ Create two lists of your data: (1) unordered data, (2) in order of smallest to largest. ____ Find the sample mean and the sample standard deviation (rounded to two decimal places). a. b. x¯ = ______ s = ______ ____ Construct a histogram of your data containing five to ten intervals of equal width. The histogram should be a representative display of your data. Label and scale it. Part II: Possible Distributions ____ Suppose that X followed the following theoretical distributions. Set up each distribution using the appropriate information from your data. ____ Uniform: X ~ U ____________ Use the lowest |
and highest values as a and b. ____ Normal: X ~ N ____________ Use x¯ to estimate for μ and s to estimate for σ. ____ Must your data fit one of the above distributions? Explain why or why not. ____ Could the data fit two or three of the previous distributions (at the same time)? Explain. ____ Calculate the value k(an X value) that is 1.75 standard deviations above the sample mean. k = _________ (rounded to two decimal places) Note: k = x¯ + (1.75)s ____ Determine the relative frequencies (RF) rounded to four decimal places. NOTE RF = frequency total number surveyed a. RF(X < k) = ______ b. RF(X > k) = ______ c. RF(X = k) = ______ This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix D NOTE 895 You should have one page for the uniform distribution, one page for the exponential distribution, and one page for the normal distribution. ____ State the distribution: X ~ _________ ____ Draw a graph for each of the three theoretical distributions. Label the axes and mark them appropriately. ____ Find the following theoretical probabilities (rounded to four decimal places). a. P(X < k) = ______ b. P(X > k) = ______ c. P(X = k) = ______ ____ Compare the relative frequencies to the corresponding probabilities. Are the values close? ____ Does it appear that the data fit the distribution well? Justify your answer by comparing the probabilities to the relative frequencies, and the histograms to the theoretical graphs. Part III: CLT Experiments ______ From your original data (before ordering), use a random number generator to pick 40 samples of size five. For each sample, calculate the average. ______ On a separate page, attached to the summary, include the 40 samples of size five, along with the 40 sample averages. ______ List the 40 averages in order from smallest to largest. ______ Define the random variable, X ¯, in words. X ¯ = _______________ ______ State the approximate theoretical distribution of X ______ Base this on the mean and standard deviation from your original data. ______ Construct a histogram displaying your data. Use five to six intervals of equal width. Label and scale it.  |
¯. X ¯ ~ ______________ ¯ (an X ¯ value) that is 1.75 standard deviations above the sample mean. k ¯ = _____ (rounded to Calculate the value k two decimal places) Determine the relative frequencies (RF) rounded to four decimal places. a. RF( X ¯ < k ¯ ) = _______ b. RF( X ¯ > k ¯ ) = _______ c. RF( X ¯ = k ¯ ) = _______ Find the following theoretical probabilities (rounded to four decimal places). a. P( X ¯ < k ¯ ) = _______ b. P( X ¯ > k ¯ ) = _______ c. P( X ¯ = k ¯ ) = _______ ______ Draw the graph of the theoretical distribution of X. ______ Compare the relative frequencies to the probabilities. Are the values close? ______ Does it appear that the data of averages fit the distribution of X probabilities to the relative frequencies, and the histogram to the theoretical graph. In three to five complete sentences for each, answer the following questions. Give thoughtful explanations. ______ In summary, do your original data seem to fit the uniform, exponential, or normal distributions? Answer why or why not for each distribution. If the data do not fit any of those distributions, explain why. ¯ well? Justify your answer by comparing the 896 Appendix D ______ What happened to the shape and distribution when you averaged your data? In theory, what should have happened? In theory, would “itâ€? always happen? Why or why not? ¯ ______ Were the relative frequencies compared to the theoretical probabilities closer when comparing the X or X distributions? Explain your answer. Assignment Checklist You need to turn in the following typed and stapled packet, with pages in the following order: ____ Cover sheet: name, class time, and name of your study ____ Summary pages: These should contain several paragraphs written with complete sentences that describe the experiment, including what you studied and your sampling technique, as well as answers to all of the questions previously asked questions ____ URL for data, if your data are from the World Wide Web ____ Pages, one for each theoretical distribution, with the distribution stated, the graph, and the probability questions answered ____ Pages of the data requested ____ All graphs required Hypot |
hesis Testing-Article Student Learning Objectives • The student will identify a hypothesis testing problem in print. • The student will conduct a survey to verify or dispute the results of the hypothesis test. • The student will summarize the article, analysis, and conclusions in a report. Instructions As you complete each task, check it off. Answer all questions in your summary. ____Find an article in a newspaper, magazine, or on the internet which makes a claim about ONE population mean or ONE population proportion. The claim may be based upon a survey that the article was reporting on. Decide whether this claim is the null or alternate hypothesis. ____Copy or print out the article and include a copy in your project, along with the source. ____State how you will collect your data. (Convenience sampling is not acceptable.) ____Conduct your survey. You must have more than 50 responses in your sample. When you hand in your final project, attach the tally sheet or the packet of questionnaires that you used to collect data. Your data must be real. ____State the statistics that are a result of your data collection: sample size, sample mean, and sample standard deviation, OR sample size and number of successes. ____Make two copies of the appropriate solution sheet. ____Record the hypothesis test on the solution sheet, based on your experiment. Do a DRAFT solution first on one of the solution sheets and check it over carefully. Have a classmate check your solution to see if it is done correctly. Make your decision using a 5% level of significance. Include the 95% confidence interval on the solution sheet. ____Create a graph that illustrates your data. This may be a pie or bar graph or may be a histogram or box plot, depending on the nature of your data. Produce a graph that makes sense for your data and gives useful visual information about your data. You may need to look at several types of graphs before you decide which is the most appropriate for the type of data in your project. ____Write your summary (in complete sentences and paragraphs, with proper grammar and correct spelling) that describes the project. The summary MUST include: a. Brief discussion of the article, including the source b. Statement of the claim made in the article (one of the hypotheses). c. Detailed description of how, where, and when you collected the data, including the sampling technique; did you use cluster, stratified, systematic, or simple random sampling (using a random number generator)? |
As previously mentioned, convenience sampling is not acceptable. d. Conclusion about the article claim in light of your hypothesis test; this is the conclusion of your hypothesis test, stated in words, in the context of the situation in your project in sentence form, as if you were writing this conclusion for a non-statistician. e. Sentence interpreting your confidence interval in the context of the situation in your project This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix D 897 Assignment Checklist Turn in the following typed (12 point) and stapled packet for your final project: ____Cover sheet containing your name(s), class time, and the name of your study ____Summary, which includes all items listed on summary checklist ____Solution sheet neatly and completely filled out. The solution sheet does not need to be typed. ____Graphic representation of your data, created following the guidelines previously discussed; include only graphs which are appropriate and useful. ____Raw data collected AND a table summarizing the sample data (n, x¯ and s; or x, n, and p’, as appropriate for your hypotheses); the raw data does not need to be typed, but the summary does. Hand in the data as you collected it. (Either attach your tally sheet or an envelope containing your questionnaires.) Bivariate Data, Linear Regression, and Univariate Data Student Learning Objectives • The students will collect a bivariate data sample through the use of appropriate sampling techniques. • The student will attempt to fit the data to a linear model. • The student will determine the appropriateness of linear fit of the model. • The student will analyze and graph univariate data. Instructions 1. As you complete each task below, check it off. Answer all questions in your introduction or summary. 2. Check your course calendar for intermediate and final due dates. 3. Graphs may be constructed by hand or by computer, unless your instructor informs you otherwise. All graphs must be neat and accurate. 4. All other responses must be done on the computer. 5. Neatness and quality of explanations are used to determine your final grade. Part I: Bivariate Data Introduction ____State the bivariate data your group is going to study. Here are two examples, but you may NOT use them: height vs. weight and age vs. running distance. ____Describe your sampling technique in detail. Use cluster |
, stratified, systematic, or simple random sampling (using a random number generator) sampling. Convenience sampling is NOT acceptable. ____Conduct your survey. Your number of pairs must be at least 30. ____Print out a copy of your data. Analysis ____On a separate sheet of paper construct a scatter plot of the data. Label and scale both axes. ____State the least squares line and the correlation coefficient. ____On your scatter plot, in a different color, construct the least squares line. ____Is the correlation coefficient significant? Explain and show how you determined this. ____Interpret the slope of the linear regression line in the context of the data in your project. Relate the explanation to your data, and quantify what the slope tells you. ____Does the regression line seem to fit the data? Why or why not? If the data does not seem to be linear, explain if any other model seems to fit the data better. ____Are there any outliers? If so, what are they? Show your work in how you used the potential outlier formula in the Linear Regression and Correlation chapter (since you have bivariate data) to determine whether or not any pairs might be outliers. 898 Appendix D Part II: Univariate Data In this section, you will use the data for ONE variable only. Pick the variable that is more interesting to analyze. For example: if your independent variable is sequential data such as year with 30 years and one piece of data per year, your x-values might be 1971, 1972, 1973, 1974, …, 2000. This would not be interesting to analyze. In that case, choose to use the dependent variable to analyze for this part of the project. _____Summarize your data in a chart with columns showing data value, frequency, relative frequency, and cumulative relative frequency. _____Answer the following question, rounded to two decimal places: a. Sample mean = ______ b. Sample standard deviation = ______ c. First quartile = ______ d. Third quartile = ______ e. Median = ______ f. 70th percentile = ______ g. Value that is 2 standard deviations above the mean = ______ h. Value that is 1.5 standard deviations below the mean = ______ _____Construct a histogram displaying your data. Group your data into six to ten intervals of equal width. Pick regularly spaced intervals that make sense in relation to your data. For example, do NOT group data |
by age as 20-26,27-33,34-40,41-47,48-54,55-61... Instead, maybe use age groups 19.5-24.5, 24.5-29.5,... or 19.5-29.5, 29.5-39.5, 39.5-49.5,... _____In complete sentences, describe the shape of your histogram. _____Are there any potential outliers? Which values are they? Show your work and calculations as to how you used the potential outlier formula in Descriptive Statistics (since you are now using univariate data) to determine which values might be outliers. _____Construct a box plot of your data. _____Does the middle 50% of your data appear to be concentrated together or spread out? Explain how you determined this. _____Looking at both the histogram AND the box plot, discuss the distribution of your data. For example: how does the spread of the middle 50% of your data compare to the spread of the rest of the data represented in the box plot; how does this correspond to your description of the shape of the histogram; how does the graphical display show any outliers you may have found; does the histogram show any gaps in the data that are not visible in the box plot; are there any interesting features of your data that you should point out. Due Dates • Part I, Intro: __________ (keep a copy for your records) • Part I, Analysis: __________ (keep a copy for your records) • Entire Project, typed and stapled: __________ ____ Cover sheet: names, class time, and name of your study ____ Part I: label the sections “Introâ€? and “Analysis.â€? ____ Part II: ____ Summary page containing several paragraphs written in complete sentences describing the experiment, including what you studied and how you collected your data. The summary page should also include answers to ALL the questions asked above. ____ All graphs requested in the project ____ All calculations requested to support questions in data ____ Description: what you learned by doing this project, what challenges you had, how you overcame the challenges NOTE Include answers to ALL questions asked, even if not explicitly repeated in the items above. This OpenStax book is available for free at http://cnx.org/ |
content/col30309/1.8 Appendix E 899 APPENDIX E: SOLUTION SHEETS Hypothesis Testing With One Sample Class Time: __________________________ Name: _____________________________________ a. H0: _______ b. Ha: _______ c. In words, clearly state what your random variable X ¯ or P′ represents. d. State the distribution to use for the test. e. What is the test statistic? f. What is the p-value? In one or two complete sentences, explain what the p-value means for this problem. g. Use the previous information to sketch a picture of this situation. Clearly, label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Figure E1 h. Indicate the correct decision (reject or do not reject the null hypothesis), the reason for it and write appropriate conclusions using complete sentences." i. Alpha: _______ ii. Decision: _______ iii. Reason for decision: _______ iv. Conclusion: _______ i. Construct a 95 percent confidence interval for the true mean or proportion. Sketch of the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. 900 Appendix E Figure E2 Hypothesis Testing With Two Samples Class Time: __________________________ Name: _____________________________________ a. H0: _______ b. Ha: _______ c. ¯ In words, clearly state what your random variable X ¯ 1 − X, P′1 − P′2 2 ¯ or X d represents. d. State the distribution to use for the test. e. What is the test statistic? f. What is the p-value? In one to two complete sentences, explain what the p-value means for this problem. g. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Figure E3 h. Indicate the correct decision (reject or do not reject the null hypothesis), and write appropriate conclusions using complete sentences. i. Alpha: _______ ii. Decision: _______ iii. Reason for decision: _______ iv. Conclusion: _______ i. In complete sentences, explain how you determined which distribution to use. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix E 901 The Chi-Square Distribution Class Time: ________________________ |
__ Name: ____________________________________ a. H0: _______ b. Ha: _______ c. What are the degrees of freedom? d. State the distribution to use for the test. e. What is the test statistic? f. What is the p-value? In one to two complete sentences, explain what the p-value means for this problem. g. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Figure E4 h. Indicate the correct decision (reject or do not reject the null hypothesis) and write appropriate conclusions, using complete sentences. i. Alpha: _______ ii. Decision: _______ iii. Reason for decision: _______ iv. Conclusion: _______ F Distribution and One-Way ANOVA Class Time: __________________________ Name: ____________________________________ a. H0: _______ b. Ha: _______ c. df(n) = ______ df(d) = _______ d. State the distribution to use for the test. e. What is the test statistic? f. What is the p-value? g. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p-value. 902 Appendix E Figure E5 h. Indicate the correct decision (reject or do not reject the null hypothesis) and write appropriate conclusions, using complete sentences. a. Alpha: _______ b. Decision: _______ c. Reason for decision: _______ d. Conclusion: _______ This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix F 903 APPENDIX F: MATHEMATICAL PHRASES, SYMBOLS, AND FORMULAS English Phrases Written Mathematically When the English says: Interpret this as: X is at least 4. The minimum of X is 4. X is no less than 4 is greater than or equal to 4. X ≥ 4 X is at most 4. The maximum of X is 4. X is no more than 4. X is less than or equal to 4. X does not exceed 4. X is greater than 4. X is more than 4. X exceeds 4. X is less than 4. There are fewer X than 4. X is 4. X is equal to 4. X is |
the same as 4. X is not 4. X is not equal to 4. X is not the same as 4. X is different than 4. Table F1 Formulas Formula 1: Factorial 904 Appendix F n! = n(n − 1)(n − 2)...(1) 0! = 1 Formula 2: Combinations ⎞ ⎠ = n ⎛ ⎝ r n! (n − r)!r! Formula 3: Binomial Distribution X ~ B(n, p) P(X = x) = n ⎛ ⎝ x ⎞ ⎠p x qn − x, for x = 0, 1, 2,..., n Formula 4: Geometric Distribution X ~ G(p) P(X = x) = q x − 1 p, for x = 1, 2, 3,... Formula 5: Hypergeometric Distribution X ~ H(r, b, n) P(X = x) = ⎛ ⎜ ⎝ ⎛ ⎝ ⎞ ⎠ ⎞ ⎠ ⎞ ⎟ ⎠ Formula 6: Poisson Distribution X ~ P(μ) P(X = x) = −μ μ x e x! Formula 7: Uniform Distribution X ~ U(a, b) f (X Formula 8: Exponential Distribution X ~ Exp(m) f (x) = me−mx m > 0, x ≥ 0 Formula 9: Normal Distribution X ~ N(μ, σ 2) − (x − μ)2 2σ 2 f (x) = 1 σ 2π e, – ∞ < x < ∞ Formula 10: Gamma Function This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 905 Appendix F 0 Γ(z) = ∫ ∞ xz − 1 e−x dx (m + 1) = m! for m, a nonnegative integer otherwise: Γ(a + 1) = aΓ(a) Formula 11: Student's t-distribution ⎛ ⎝1 + x2 n ⎞ ⎠ − (n + 1) 2 nπΓ ⎞ ⎠ ⎛ ⎝ ~ td f f (x(0, 1), Y ~ Χd f 2, |
n = degrees of freedom Formula 12: Chi-Square Distribution 2 X ~ Χd f f (x) = x 2 n − 2 − = positive integer and degrees of freedom Formula 13: F Distribution X ~ Fd f (n), d f (d) d f (n) = degrees of freedom for the numerator d f (d) = degrees of freedom for the denominator f (x) = Γ(u + v 2 ) 2)Γ( v Γ(u 2) (u 2 u 2 x (u v) − 1) [1 + (u v)x −0.5(u + v) ] X = Yu Wv, Y, W are chi-square Symbols and Their Meanings Chapter (1st used) Symbol Spoken Meaning Sampling and Data Sampling and Data Descriptive Statistics π Q1 Table F2 Symbols and their Meanings The square root of same Pi 3.14159… (a specific number) Quartile one the first quartile 906 Appendix F Chapter (1st used) Symbol Spoken Meaning Descriptive Statistics Descriptive Statistics Descriptive Statistics Descriptive Statistics Descriptive Statistics Descriptive Statistics Descriptive Statistics Q2 Q3 IQR ¯ x μ s sx sx 2 s2 s x Quartile two Quartile three interquartile range x-bar mu s the second quartile the third quartile Q3 – Q1 = IQR sample mean population mean sample standard deviation s squared sample variance Descriptive Statistics σ σ x σx sigma population standard deviation Descriptive Statistics 2 σ 2 σ x sigma squared population variance Descriptive Statistics Probability Topics Probability Topics Probability Topics Σ {} S A capital sigma sum brackets S Event A set notation sample space event A Probability Topics P(A) probability of A probability of A occurring Probability Topics P(A|B) probability of A given B prob. of A occurring given B has occurred Probability Topics P(A OR B) prob. of A or B prob. of A or B or both occurring Probability Topics P(A AND B) prob. of A and B prob. of both A and B occurring (same time) Probability Topics Probability Topics Probability Topics Probability Topics A′ P(A') G1 P(G1) A-prime, complement of A complement of A, not A prob. of |
complement of A same green on first pick same prob. of green on first pick same Discrete Random Variables PDF prob. distribution function same Discrete Random Variables X X the random variable X Discrete Random Variables X ~ the distribution of X Discrete Random Variables B binomial distribution Discrete Random Variables G geometric distribution Discrete Random Variables H hypergeometric dist. Discrete Random Variables P Poisson dist. same same same same same Discrete Random Variables Discrete Random Variables Discrete Random Variables λ ≥ ≤ Lambda average of Poisson distribution greater than or equal to same less than or equal to same same Discrete Random Variables = equal to Table F2 Symbols and their Meanings This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix F 907 Chapter (1st used) Symbol Spoken Discrete Random Variables ≠ not equal to Meaning same f(x) pdf U Exp k f of x function of x prob. density function same uniform distribution same exponential distribution same k critical value f(x) = f of x equals same Continuous Random Variables Continuous Random Variables Continuous Random Variables Continuous Random Variables Continuous Random Variables Continuous Random Variables Continuous Random Variables The Normal Distribution The Normal Distribution The Normal Distribution m decay rate (for exp. dist.) m N z Z normal distribution z-score standard normal dist. The Central Limit Theorem CLT Central Limit Theorem The Central Limit Theorem ¯ X X-bar the random variable X-bar The Central Limit Theorem μ x mean of X the average of X The Central Limit Theorem μ x¯ mean of X-bar the average of X-bar The Central Limit Theorem σ x standard deviation of X same The Central Limit Theorem σ x¯ standard deviation of X-bar same The Central Limit Theorem ΣX The Central Limit Theorem Σx sum of X sum of x confidence level confidence interval error bound for a mean error bound for a proportion Student's t-distribution degrees of freedom student t with a/2 area in right tail Confidence Intervals Confidence Intervals Confidence Intervals Confidence Intervals Confidence Intervals Confidence Intervals Confidence Intervals Confidence Intervals Confidence Intervals CL CI EBM EBP t df t α 2 ^ p′ ; p ^ q′ ; q Table F2 Symbols and their Meanings p-prime |
; p-hat sample proportion of success q-prime; q-hat sample proportion of failure same same same same same same same same same same same same same 908 Appendix F Chapter (1st used) Symbol Spoken Meaning Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing Hypothesis Testing H0 Ha H1 α β X1 − X2 μ1 − μ2 P′1 − P′2 H-naught, H-sub 0 null hypothesis H-a, H-sub a H-1, H-sub 1 alpha beta alternate hypothesis alternate hypothesis probability of Type I error probability of Type II error X1-bar minus X2-bar difference in sample means mu-1 minus mu-2 difference in population means P1-prime minus P2-prime difference in sample proportions Hypothesis Testing p1 − p2 p1 minus p2 difference in population proportions Chi-Square Distribution Chi-Square Distribution Chi-Square Distribution Linear Regression and Correlation Linear Regression and Correlation Linear Regression and Correlation Linear Regression and Correlation Linear Regression and Correlation Linear Regression and Correlation Χ 2 O E Ky-square Observed Expected Chi-square Observed frequency Expected frequency y = a + bx y equals a plus b-x equation of a line ^ y r ε SSE 1.9s y-hat estimated value of y correlation coefficient same error same Sum of Squared Errors same 1.9 times s cut-off value for outliers F-Distribution and ANOVA F F-ratio F-ratio Table F2 Symbols and their Meanings This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix G 909 APPENDIX G: NOTES FOR THE TI-83, 83+, 84, 84+ CALCULATORS Quick Tips Legend • represents a button press • [ ] represents yellow command or green letter behind a key • < > represents items on the screen To adjust the contrast Press, then hold to increase the contrast or to decrease the contrast. To capitalize letters and words Press to get one capital letter, or press, then to set all button presses to capital letters. You can return to the top-level button values by pressing again. To correct a mistake If you hit a wrong button, press and start again. To write in scientific |
notation Numbers in scientific notation are expressed on the TI-83, 83+, 84, and 84+ using E notation, such that... • 4.321 E 4 = 4.321×104 • 4.321 E –4 = 4.321×10–4 To transfer programs or equations from one calculator to another Both calculators: Insert your respective end of the link cable cable and press, then [LINK]. Calculator receiving information 1. Use the arrows to navigate to and select <RECEIVE>. 2. Press. Calculator sending information 1. Press the appropriate number or letter. 910 Appendix G 2. Use the up and down arrows to access the appropriate item. 3. Press to select the item to transfer. 4. Press the right arrow to navigate to and select <TRANSMIT>. 5. Press NOTE. ERROR 35 LINK generally means that the cables have not been inserted far enough. Both calculators—Insert your respective end of the link cable, press Manipulating One-Variable Statistics, then [QUIT] to exit when done. NOTE These directions are for entering data using the built-in statistical program. Data Frequency –2 –1 0 1 3 10 3 4 5 8 Table G1 Sample Data We are manipulating onevariable statistics. To begin 1. Turn on the calculator. 2. Access statistics mode. 3. Select <4:ClrList> to clear data from lists, if desired., then. 4. Enter the list [L1] to be cleared., [L1],. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix G 911 5. Display the last instruction., [ENTRY]. 6. Continue clearing any remaining lists in the same fashion, if desired.,, [L2], 7. Access statistics mode. 8. Select <1:Edit...>. 9. Enter data. Data values go into [L1]. (You may need to arrow over to [L1]). ◦ Type in a data value and enter it. For negative numbers, use the negate – key at the bottom of the keypad.,,. ◦ Continue in the same manner until all data values are entered. 10. In [L2], enter the frequencies for each data value in [L1]. ◦ Type in a frequency and enter it. If a data value appears only once, the frequency is 1.,. ◦ Continue in |
the same manner until all data values are entered. 11. Access statistics mode. 12. Navigate to <CALC>. 13. Access <1:1-var Stats>. 14. Indicate that the data is in [L1]..., [L1],, 15....and indicate that the frequencies are in [L2]., [L2],. 16. The statistics should be displayed. You may arrow down to get remaining statistics. Repeat as necessary. Drawing Histograms NOTE We will assume that the data are already entered. 912 Appendix G We will construct two histograms with the built-in [STAT PLOT] application. In the first method, we will use the default ZOOM. The second method will involve customizing a new graph. 1. Access graphing mode., [STAT PLOT]. 2. Select <1:plot 1> to access plotting - first graph. 3. Use the arrows to navigate to <ON> to turn on Plot 1. <ON>,. 4. Use the arrows to go to the histogram picture and select the histogram. 5. Use the arrows to navigate to <Xlist>. 6. If [L1] is not selected, select it., [L1],. 7. Use the arrows to navigate to <Freq>. 8. Assign the frequencies to [L2]., [L2],. 9. Go back to access other graphs., [STAT PLOT]. 10. Use the arrows to turn off the remaining plots. 11. Be sure to deselect or clear all equations before graphing. To deselect equations 1. Access the list of equations. 2. Select each equal sign (=).. 3. Continue until all equations are deselected. To clear equations 1. Access the list of equations. 2. Use the arrow keys to navigate to the right of each equal sign (=) and clear them. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix G 913. 3. Repeat until all equations are deleted. To draw default histogram 1. Access the ZOOM menu. 2. Select <9:ZoomStat>. 3. The histogram will display with a window automatically set. To draw a custom histogram 1. Access window mode to set the graph parameters. 2. ◦ Xmin = –2.5 ◦ Xmax = 3.5 |
◦ Xscl = 1 (width of bars) ◦ Ymin = 0 ◦ Ymax = 10 ◦ Yscl = 1 (spacing of tick marks on y-axis) ◦ Xres = 1 3. Access graphing mode to see the histogram. To draw box plots 1. Access graphing mode., [STAT PLOT]. 2. Select <1:Plot 1> to access the first graph. 3. Use the arrows to select <ON> and turn on Plot 1. 4. Use the arrows to select the box plot picture and enable it. 5. Use the arrows to navigate to <Xlist>. 914 Appendix G 6. If [L1] is not selected, select it., [L1],. 7. Use the arrows to navigate to <Freq>. 8. Indicate that the frequencies are in [L2]., [L2],. 9. Go back to access other graphs., [STAT PLOT]. 10. Be sure to deselect or clear all equations before graphing using the method mentioned above. 11. View the box plot., [STAT PLOT]. Linear Regression Sample Data The following data are real. The percent of declared ethnic minority students at De Anza College for selected years from 1970–1995 is indicated in the following table. Year Student Ethnic Minority Percentage 1970 14.13% 1973 12.27% 1976 14.08% 1979 18.16% 1982 27.64% 1983 28.72% 1986 31.86% 1989 33.14% 1992 45.37% 1995 53.1% Table G2 The independent variable is Year, while the independent variable is Student Ethnic Minority Percentage. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix G 915 Figure G1 Student Ethnic Minority Percentage By hand, verify the scatterplot above. NOTE The TI-83 has a built-in linear regression feature, which allows the data to be edited. The x-values will be in [L1]; the y-values in [L2]. To enter data and perform linear regression 1. ON Turns calculator on. 2. Before accessing this program, be sure to turn off all plots. ◦ Access graphing mode., [STAT PLOT]. ◦ Turn off all plots.,. 3. Round to three decimal places. ◦ Access the mode menu., [STAT PLOT]. ◦ |
Navigate to <Float> and then to the right until you reach <3>. ◦ All numbers will be rounded to three decimal places until changed.. 4. Enter statistics mode and clear lists [L1] and [L2], as described previously. 916 Appendix G,. 5. Enter editing mode to insert values for x and y.,. 6. Enter each value. Press to continue. To display the correlation coefficient 1. Access the catalog., [CATALOG]. 2. Arrow down and select <DiagnosticOn>....,,. 3. r and r 2 will be displayed during regression calculations. 4. Access linear regression. 5. Select the form of y = a + bx..,. The display will show the following information LinReg • y = a + bx • a = –3176.909 • b = 1.617 r2 = 0.924 • • r = 0.961 This means the Line of Best Fit (Least Squares Line) is: • y = –3176.909 + 1.617x • % = –3176.909 + 1.617 (year #) The correlation coefficient is r = 0.961. To see the scatter plot 1. Access graphing mode., [STAT PLOT]. 2. Select <1:Plot 1> To access plotting - first graph. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix G 917 3. Navigate and select <ON> to turn on <1:Plot 1>. <ON>. 4. Navigate to the first picture. 5. Select the scatter plot. 6. Navigate to <Xlist>. 7. If [L1] is not selected, press, then [L1] to select it. 8. Confirm that the data values are in [L1]. <ON>,. 9. Navigate to <Ylist>. 10. Select that the frequencies are in [L2]., [L2], 11. Go back to access other graphs., [STAT PLOT] 12. Use the arrows to turn off the remaining plots. 13. Access window mode to set the graph parameters. ◦ Xmin = 1970 ◦ Xmax = 2000 ◦ Xscl = 10 (spacing of tick marks on x-axis) ◦ Ymin = − 0.05 ◦ |
Ymax = 60 ◦ Yscl = 10 (spacing of tick marks on y-axis) ◦ Xres = 1 14. Be sure to deselect or clear all equations before graphing, using the instructions above. 15. Press the graph button to see the scatter plot. To see the regression graph 1. Access the equation menu. The regression equation will be put into Y1. 918 Appendix G 2. Access the vars menu and navigate to <5: Statistics>.,. 3. Navigate to <EQ>. 4. <1: RegEQ> contains the regression equation which will be entered in Y1. 5. Press the graphing mode button. The regression line will be superimposed over the scatter plot. To see the residuals and use them to calculate the critical point for an outlier 1. Access the list. <RESID> will be an item on the menu. Navigate to it., [LIST], then <RESID>. 2. Press enter twice to view the list of residuals. Use the arrows to select them.,. 3. The critical point for an outlier is 1.9V SSE n − 2, where ◦ n = number of pairs of data ◦ SSE = sum of the squared errors ◦ ∑ ⎛ ⎝residual2⎞ ⎠ 4. Store the residuals in [L3].,, [L3],. 5. Calculate the (residual)2 n − 2. Note that n − 2 = 8., [L3],,, then. 6. Store this value in [L4].,, [L4],. 7. Calculate the critical value using the equation aboveV],, [LIST], [L4],,, then. 8. Verify that the calculator displays 7.642669563. This is the critical value. 9. Compare the absolute value of each residual value in [L3] to 7.64. If the absolute value is greater than 7.64, then the (x, y) corresponding point is an outlier. In this case, none of the points is an outlier. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix G 919 To obtain estimates of y for various x-values There are various ways to determine estimates for "y." |
One way is to substitute values for "x" in the equation. Another way on the graph of the regression line. is to use the TI-83, 83+, 84, 84+ instructions for distributions and tests Distributions Access DISTR for Distributions. For technical assistance, visit the Texas Instruments website at http://www.ti.com (http://www.ti.com) and enter your calculator model into the search box. Binomial Distribution • binompdf(n,p,x) corresponds to P(X = x) • binomcdf(n,p,x) corresponds to P(X ≤ x) • To see a list of all probabilities for x: 0, 1,..., n, leave off the "x" parameter. Poisson Distribution • poissonpdf(λ,x) corresponds to P(X = x) • poissoncdf(λ,x) corresponds to P(X ≤ x) Continuous Distributions (general) • −∞ uses the value –1EE99 for left bound • +∞ uses the value 1EE99 for right bound Normal Distribution • normalpdf(x,μ,σ) yields a probability density function value, only useful to plot the normal curve, in which case "x" is the variable • normalcdf(left bound, right bound, μ, σ) corresponds to P(left bound < X < right bound) • normalcdf(left bound, right bound) corresponds to P(left bound < Z < right bound) – standard normal • invNorm(p,μ,σ) yields the critical value, k: P(X < k) = p • invNorm(p) yields the critical value, k: P(Z < k) = p for the standard normal Student's t-Distribution • tpdf(x,df) yields the probability density function value, only useful to plot the student-t curve, in which case "x" is the variable) • tcdf(left bound, right bound, df) corresponds to P(left bound < t < right bound) Chi-square Distribution • Χ2pdf(x,df) yields the probability density function value, only useful to plot the chi2 curve, in which case "x" is the variable • Χ2cdf(left bound, right bound, df) corresponds to P(left bound < Χ2 < right bound) F Distribution • Fpdf(x,df |
num,dfdenom) yields the probability density function value, only useful to plot the F curve, in which case "x" is the variable • Fcdf(left bound,right bound,dfnum,dfdenom) corresponds to P(left bound < F < right bound) Tests and Confidence Intervals Access STAT and TESTS. For the confidence intervals and hypothesis tests, you may enter the data into the appropriate lists and press DATA to have the calculator find the sample means and standard deviations. Or, you may enter the sample means and sample standard deviations directly by pressing STAT once in the appropriate tests. Confidence Intervals 920 Appendix G • ZInterval is the confidence interval for mean when σ is known. • TInterval is the confidence interval for mean when σ is unknown; s estimates σ. • 1-PropZInt is the confidence interval for proportion. NOTE The confidence levels should be given as percents (e.g., enter "95" or ".95" for a 95 percent confidence level). Hypothesis Tests • Z-Test is the hypothesis test for single mean when σ is known. • T-Test is the hypothesis test for single mean when σ is unknown; s estimates σ. • 2-SampZTest is the hypothesis test for two independent means when both σs are known. • 2-SampTTest is the hypothesis test for two independent means when both σs are unknown. • 1-PropZTest is the hypothesis test for a single proportion. • 2-PropZTest is the hypothesis test for two proportions. • Χ2-Test is the hypothesis test for independence. • Χ2GOF-Test is the hypothesis test for goodness-of-fit (TI-84+ only). • LinRegTTEST is the hypothesis test for Linear Regression (TI-84+ only). NOTE Input the null hypothesis value in the row below "Inpt." For a test of a single mean, "μ∅" represents the null hypothesis. For a test of a single proportion, "p∅" represents the null hypothesis. Enter the alternate hypothesis on the bottom row. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Appendix H 921 APPENDIX H: TABLES The module contains links to government site tables used in statistics. NOTE When you are finished with the table link, use |
the back button on your browser to return here. Tables (NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, January 3, 2009) • Student t table (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3672.htm) • Normal table (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm) • Chi-Square table (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm) • F-table (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm) • All four tables (http://www.itl.nist.gov/div898/handbook/eda/section3/eda367.htm) can be accessed by going to http://www.itl.nist.gov/div898/handbook/eda/section3/eda367.htm 95% Critical Values of the Sample Correlation Coefficient Table • 95% Critical Values of the Sample Correlation Coefficient 922 Appendix H This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Index 923 INDEX A alternative hypothesis, 524 analysis of variance, 781 and, 235, 235, 235 average, 47, 439 B Bernoulli trials, 293 binomial distribution, 430, 490, 529, 552 binomial experiment, 293 binomial probability distribution, 265, 293 bivariate, 696 Blinding, 38 blinding, 47 box plot, 131 Box plots, 102 box-and-whisker plots, 102 box-whisker plots, 102 C categorical data, 10 categorical variable, 47 Categorical variables, 7 central limit theorem, 413, 415, 422, 439 central limit theorem for means, 417 central limit theorem for sums, 419 chi-square distribution, 638 cluster sampling, 47 coefficient of correlation, 728 coefficient of determination, 706 Cohen’s d, 590 complement, 185 conditional probability, 185, 225, 336, 356 confidence interval, 460, 472 confidence interval (CI |
), 490, 552 confidence intervals, 477 Confidence intervals, 523 confidence level, 461, 477 confidence level (CL), 490 contingency table, 203, 225, 649, 667 continuity correction factor, 430 continuous, 10 continuous random variable, 47, 340 control group, 38, 47 convenience sampling, 47 critical value, 388 cumulative distribution function, 329 cumulative distribution function (CDF), 341 Cumulative relative frequency, 30 cumulative relative frequency, 47 D data, 5, 47 Data, 7 decay parameter, 356 degrees of freedom (df), 490, 585, 610 dependent events, 225 descriptive statistics, 6 discrete, 10 discrete random variable, 47 double-blind experiment, 38 double-blinding, 47 E Empirical Rule, 382 empirical rule, 460 equally likely, 225 error bound, 477 error bound for a population mean, 461 error bound for a population mean (EBM), 490 error bound for a population proportion (EBP), 490 event, 225 expected value, 257, 293 expected values, 639 experiment, 225 Experimental Probability of Event A, 184 experimental unit, 47 explanatory variable, 47 exponential distribution, 340, 356, 426, 439 F F distribution, 763 F ratio, 763 first quartile, 93, 131 frequency, 30, 47, 83, 131 frequency polygon, 131 frequency table, 131 G geometric distribution, 274, 293 geometric experiment, 271, 293 H histogram, 83, 131 hypergeometric experiment, 295, 293 hypergeometric probability, 276, 293 hypotheses, 524 hypothesis, 552 hypothesis test, 532, 553 hypothesis testing, 552 hypothesis testing., 524 I independent, 198 independent events, 189, 225 inferential statistics, 6, 460, 490 informed consent, 40, 47 institutional review board, 47 Institutional Review Boards (IRB), 40 interquartile range, 94, 131 interval, 131 interval scale, 29 L law of large numbers, 423 level of measurement, 29 level of significance of the test, 531, 552 lurking variable, 47 lurking variables, 37 M margin of error, 461 margin of error for a population mean, 461 mathematical models, 6, 47 mean, 7, 106, 131, 257, 294, 414, 417, 423, 439 mean of a probability distribution, 294 median, 94, 106, 131 memoryless property, 356 midpoint, 131 mode, 108, 131 multivariate, 696 mutually exclusive, 192, 199, |
225 N nominal scale, 29 nonsampling error, 47 normal distribution, 399, 439, 472, 490, 529, 552 normally distributed, 415, 419, 529 924 Index the AND event, 225 the complement event, 225 the conditional probability of one event GIVEN another event, 225 the law of large numbers, 294 the OR event, 225 the OR of two events, 225 Theoretical Probability of Event A, 184 treatments, 48 tree diagram, 210, 225 two-way table, 203 Type 1 error, 552 Type 2 error, 552 Type I error, 526, 531 Type II error, 526 U unfair, 184 uniform distribution, 356, 423, 439 Use the following information to answer the next three exercises, 234 V validity, 48 variable, 7, 49 variable (random variable), 610 variance, 118, 132, 781 variances, 762 Variation, 26 Venn diagram, 217, 225 Z z-score, 399, 472 null hypothesis, 524, 531 numerical Variable, 47 Numerical variables, 7 O observational studies, 38 observational study, 47 observed values, 639 one-way ANOVA, 781 or, 235, 235, 235 ordinal scale, 29 outcome, 225 outlier, 73, 95, 131, 728 P p-value, 529, 532, 552 paired data set, 91, 131 parameter, 47, 460, 490 Pearson, 7 percentile, 131 percentiles, 93 placebo, 38, 47 plus-four confidence interval, 490 point estimate, 460, 490 Poisson distribution, 356 Poisson probability distribution, 279, 295, 294 pooled proportion, 595, 610 population, 7, 27, 48 population variance, 657 potential outlier, 717 Probability, 7, 182 probability, 48, 225 probability density function, 326 probability distribution function, 255 probability distribution function (PDF), 294 proportion, 7, 48 Q Qualitative data, 10 qualitative data, 48 quantitative continuous data, 10 Quantitative data, 10 quantitative data, 48 quantitative discrete data, 10 quartiles, 93, 131 Quartiles, 94 R random assignment, 37, 48 random sampling, 48 Random variable, 586 Random Variable, 593 random variable (RV), 294 random variables, 255 ratio scale, 29 relative frequency, 30, 48, 83, 131 reliability, 48 replacement, 189 representative sample, 7, 48 response variable, 48 S sample, 7, 48 sample mean |
, 415 sample size, 415 sample space, 198, 211, 225 samples, 27 sampling, 7 sampling bias, 48 sampling distribution, 109, 439 sampling error, 48 sampling variability of a statistic, 119 sampling with replacement, 48, 225 sampling without replacement, 48, 225 simple random sample, 529 simple random sampling, 48 skewed, 131 standard deviation, 116, 131, 472, 490, 529, 529, 530, 552, 584, 610 standard deviation of a discrete probability distribution, 258 standard deviation of a probability distribution, 294 standard error, 584 standard error of the mean, 415, 439 standard normal distribution, 399 statistic, 48 statistical models, 48 statistics, 5 stratified sampling, 48 Student's t-distribution, 472, 490, 529, 529, 552 sum of squared errors (SSE), 702 survey, 48 surveys, 38 systematic sampling, 48 T test for homogeneity, 654 test of a single variance, 657 test of independence, 649 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8point symmetric to (−2, 3) about the origin. We summarize and generalize this process below. Reflections To reflect a point (x, y) about the: x-axis, replace y with −y. y-axis, replace x with −x. origin, replace x with −x and y with −y. 1.1.3 Distance in the Plane Another important concept in Geometry is the notion of length. If we are going to unite Algebra and Geometry using the Cartesian Plane, then we need to develop an algebraic understanding of what distance in the plane means. Suppose we have two points, P (x0, y0) and Q (x1, y1), in the plane. By the distance d between P and Q, we mean the length of the line segment joining P with Q. (Remember, given any two distinct points in the plane, there is a unique line containing both 1.1 Sets of Real Numbers and the Cartesian Coordinate Plane 11 points.) Our goal now is to create an algebraic formula to compute the distance between these two points. Consider the generic situation below on the left. Q (x1, y1) Q (x1, y1) d d P (x0, y0) P (x0, y0) |
(x1, y0) With a little more imagination, we can envision a right triangle whose hypotenuse has length d as drawn above on the right. From the latter figure, we see that the lengths of the legs of the triangle are |x1 − x0| and |y1 − y0| so the Pythagorean Theorem gives us |x1 − x0|2 + |y1 − y0|2 = d2 (x1 − x0)2 + (y1 − y0)2 = d2 (Do you remember why we can replace the absolute value notation with parentheses?) By extracting the square root of both sides of the second equation and using the fact that distance is never negative, we get Equation 1.1. The Distance Formula: The distance d between the points P (x0, y0) and Q (x1, y1) is: d = (x1 − x0)2 + (y1 − y0)2 It is not always the case that the points P and Q lend themselves to constructing such a triangle. If the points P and Q are arranged vertically or horizontally, or describe the exact same point, we cannot use the above geometric argument to derive the distance formula. It is left to the reader in Exercise 35 to verify Equation 1.1 for these cases. Example 1.1.4. Find and simplify the distance between P (−2, 3) and Q(1, −3). Solution. d = = = So the distance is 3 √ 5. = 3 √ (x1 − x0)2 + (y1 − y0)2 (1 − (−2))2 + (−3 − 3)2 9 + 36 √ 5 12 Relations and Functions Example 1.1.5. Find all of the points with x-coordinate 1 which are 4 units from the point (3, 2). Solution. We shall soon see that the points we wish to find are on the line x = 1, but for now we’ll just view them as points of the form (1, y). Visually, y 3 2 1 −1 −2 −3 (3, 2) distance is 4 units 2 3 x (1, y) We require that the distance from (3, 2) to (1, y) be 4. The Distance Formula, Equation 1.1, yields d = (x1 − |
x0)2 + (y1 − y0)2 (1 − 3)2 + (y − 2)2 4 + (y − 2)2 4 + (y − 2)22 4 = 4 = 42 = 16 = 4 + (y − 2)2 12 = (y − 2)2 √ (y − 2)2 = 12 2 12 √ 3 √ y = 2 ± 2 3 squaring both sides extracting the square root We obtain two answers: (1, 2 + 2 why there are two answers. √ 3) and (1, 2 − 2 √ 3). The reader is encouraged to think about Related to finding the distance between two points is the problem of finding the midpoint of the line segment connecting two points. Given two points, P (x0, y0) and Q (x1, y1), the midpoint M of P and Q is defined to be the point on the line segment connecting P and Q whose distance from P is equal to its distance from Q. 1.1 Sets of Real Numbers and the Cartesian Coordinate Plane 13 Q (x1, y1) M P (x0, y0) If we think of reaching M by going ‘halfway over’ and ‘halfway up’ we get the following formula. Equation 1.2. The Midpoint Formula: The midpoint M of the line segment connecting P (x0, y0) and Q (x1, y1) is: M = x0 + x1 2, y0 + y1 2 If we let d denote the distance between P and Q, we leave it as Exercise 36 to show that the distance between P and M is d/2 which is the same as the distance between M and Q. This suffices to show that Equation 1.2 gives the coordinates of the midpoint. Example 1.1.6. Find the midpoint of the line segment connecting P (−2, 3) and Q(1, −3). Solution. M = = = x0 + x1, 2 (−2) + 1 2 y0 + y1 2, 3 + (−3 The midpoint is − 1 2, 0. We close with a more abstract application of the Midpoint Formula. We will revisit the following example in Exercise 72 in Section 2.1. Example 1.1.7 |
. If a = b, prove that the line y = x equally divides the line segment with endpoints (a, b) and (b, a). Solution. To prove the claim, we use Equation 1.2 to find the midpoint Since the x and y coordinates of this point are the same, we find that the midpoint lies on the line y = x, as required. 14 Relations and Functions 1.1.4 Exercises 1. Fill in the chart below: Set of Real Numbers Interval Notation Region on the Real Number Line {x | − 1 ≤ x < 5} {x | − 5 < x ≤ 0} {x | x ≤ 3} {x | x ≥ −3} [0, 3) (−3, 3) (−∞, 9) 7 7 2 5 4 In Exercises 2 - 7, find the indicated intersection or union and simplify if possible. Express your answers in interval notation. 2. (−1, 5] ∩ [0, 8) 3. (−1, 1) ∪ [0, 6] 4. (−∞, 4] ∩ (0, ∞) 5. (−∞, 0) ∩ [1, 5] 6. (−∞, 0) ∪ [1, 5] 7. (−∞, 5] ∩ [5, 8) In Exercises 8 - 19, write the set using interval notation. 8. {x | x = 5} 9. {x | x = −1} 10. {x | x = −3, 4} 1.1 Sets of Real Numbers and the Cartesian Coordinate Plane 15 11. {x | x = 0, 2} 12. {x | x = 2, −2} 13. {x | x = 0, ±4} 14. {x | x ≤ −1 or x ≥ 1} 15. {x | x < 3 or x ≥ 2} 16. {x | x ≤ −3 or x > 0} 17. {x | x ≤ 5 or x = 6} 18. {x | x > 2 or x = ±1} 19. {x | − 3 < x < 3 or x = 4} 20. Plot and label the points A(−3, −7), B(1.3, −2), C(π, √ 10), D(0, |
8), E(−5.5, 0), F (−8, 4), G(9.2, −7.8) and H(7, 5) in the Cartesian Coordinate Plane given below9 −8 −7 −6 −5 −4 −3 −2 −1 −2 −3 −4 −5 −6 −7 −8 −9 21. For each point given in Exercise 20 above Identify the quadrant or axis in/on which the point lies. Find the point symmetric to the given point about the x-axis. Find the point symmetric to the given point about the y-axis. Find the point symmetric to the given point about the origin. 16 Relations and Functions In Exercises 22 - 29, find the distance d between the points and the midpoint M of the line segment which connects them. 22. (1, 2), (−3, 5) 23. (3, −10), (−1, 2) 24. 26. 1 2 24 5 √, − 45, 11 −, 5 12, √ √ 28. 2. 19 5 √ 25. − 2 3 27. √ 2, − √ 8, − 12 20, 27. 29. (0, 0), (x, y) 30. Find all of the points of the form (x, −1) which are 4 units from the point (3, 2). 31. Find all of the points on the y-axis which are 5 units from the point (−5, 3). 32. Find all of the points on the x-axis which are 2 units from the point (−1, 1). 33. Find all of the points of the form (x, −x) which are 1 unit from the origin. 34. Let’s assume for a moment that we are standing at the origin and the positive y-axis points due North while the positive x-axis points due East. Our Sasquatch-o-meter tells us that Sasquatch is 3 miles West and 4 miles South of our current position. What are the coordinates of his position? How far away is he from us? If he runs 7 miles due East what would his new position be? 35. Verify the Distance Formula 1.1 for the cases when: (a) The points are arranged vertically. (Hint: Use P (a, y0) and Q(a, y1).) (b) |
The points are arranged horizontally. (Hint: Use P (x0, b) and Q(x1, b).) (c) The points are actually the same point. (You shouldn’t need a hint for this one.) 36. Verify the Midpoint Formula by showing the distance between P (x1, y1) and M and the distance between M and Q(x2, y2) are both half of the distance between P and Q. 37. Show that the points A, B and C below are the vertices of a right triangle. (a) A(−3, 2), B(−6, 4), and C(1, 8) (b) A(−3, 1), B(4, 0) and C(0, −3) 38. Find a point D(x, y) such that the points A(−3, 1), B(4, 0), C(0, −3) and D are the corners of a square. Justify your answer. 39. Discuss with your classmates how many numbers are in the interval (0, 1). 40. The world is not flat.12 Thus the Cartesian Plane cannot possibly be the end of the story. Discuss with your classmates how you would extend Cartesian Coordinates to represent the three dimensional world. What would the Distance and Midpoint formulas look like, assuming those concepts make sense at all? 12There are those who disagree with this statement. Look them up on the Internet some time when you’re bored. 1.1 Sets of Real Numbers and the Cartesian Coordinate Plane 17 1.1.5 Answers 1. Set of Real Numbers Interval Notation Region on the Real Number Line {x | − 1 ≤ x < 5} [−1, 5) {x | 0 ≤ x < 3} {x | 2 < x ≤ 7} [0, 3) (2, 7] {x | − 5 < x ≤ 0} (−5, 0] {x | − 3 < x < 3} (−3, 3) {x | 5 ≤ x ≤ 7} [5, 7] {x | x ≤ 3} (−∞, 3] {x | x < 9} (−∞, 9) {x | x > 4} (4, ∞) {x | x ≥ −3} [−3, ∞) 5 3 7 0 3 7 3 9 − |
1 0 2 −5 −3 5 4 −3 2. (−1, 5] ∩ [0, 8) = [0, 5] 3. (−1, 1) ∪ [0, 6] = (−1, 6] 4. (−∞, 4] ∩ (0, ∞) = (0, 4] 5. (−∞, 0) ∩ [1, 5] = ∅ 6. (−∞, 0) ∪ [1, 5] = (−∞, 0) ∪ [1, 5] 7. (−∞, 5] ∩ [5, 8) = {5} 8. (−∞, 5) ∪ (5, ∞) 9. (−∞, −1) ∪ (−1, ∞) 10. (−∞, −3) ∪ (−3, 4) ∪ (4, ∞) 11. (−∞, 0) ∪ (0, 2) ∪ (2, ∞) 12. (−∞, −2) ∪ (−2, 2) ∪ (2, ∞) 13. (−∞, −4) ∪ (−4, 0) ∪ (0, 4) ∪ (4, ∞) 18 Relations and Functions 14. (−∞, −1] ∪ [1, ∞) 15. (−∞, ∞) 16. (−∞, −3] ∪ (0, ∞) 17. (−∞, 5] ∪ {6} 18. {−1} ∪ {1} ∪ (2, ∞) 19. (−3, 3) ∪ {4} 20. The required points A(−3, −7), B(1.3, −2), C(π, √ 10), D(0, 8), E(−5.5, 0), F (−8, 4), G(9.2, −7.8), and H(7, 5) are plotted in the Cartesian Coordinate Plane below. y D(0, 8) H(7, 5) √ 10) C(π (−8, 4) E(−5.5, 0) −9 −8 −7 −6 −5 −4 −3 −2 −(1.3, −2) −1 −2 −3 − |
4 −5 −6 −7 −8 −9 A(−3, −7) G(9.2, −7.8) 1.1 Sets of Real Numbers and the Cartesian Coordinate Plane 19 21. (a) The point A(−3, −7) is (b) The point B(1.3, −2) is in Quadrant III symmetric about x-axis with (−3, 7) symmetric about y-axis with (3, −7) symmetric about origin with (3, 7) in Quadrant IV symmetric about x-axis with (1.3, 2) symmetric about y-axis with (−1.3, −2) symmetric about origin with (−1.3, 2) (c) The point C(π, √ 10) is (d) The point D(0, 8) is in Quadrant I symmetric about x-axis with (π, − symmetric about y-axis with (−π, symmetric about origin with (−π, − √ √ √ 10) 10) 10) on the positive y-axis symmetric about x-axis with (0, −8) symmetric about y-axis with (0, 8) symmetric about origin with (0, −8) (e) The point E(−5.5, 0) is (f) The point F (−8, 4) is on the negative x-axis symmetric about x-axis with (−5.5, 0) symmetric about y-axis with (5.5, 0) symmetric about origin with (5.5, 0) in Quadrant II symmetric about x-axis with (−8, −4) symmetric about y-axis with (8, 4) symmetric about origin with (8, −4) (g) The point G(9.2, −7.8) is in Quadrant IV symmetric about x-axis with (9.2, 7.8) symmetric about y-axis with (−9.2, −7.8) symmetric about origin with (−9.2, 7.8) (h) The point H(7, 5) is in Quadrant I symmetric about x-axis with (7, −5) symmetric about y-axis with (−7, 5) symmetric about origin with (−7, −5) 22. |
d = 5, M = −1, 7 2 26, M = 1, 3 2 24. d = √ √ √ 26. d = 28. d = 30. (3 + 74, M = 13 10, − 13 √ 10 83, M = 4 √ 2 5, 5 √ √ 7, −1), (3 − √ 32. (−1 + 3, 0), (−1 − 7, −1) √ 3, 0) 34. (−3, −4), 5 miles, (4, −4) 3 29. d = √ 23. d = 4 √ 25. d = 27. d = 3 37 10, M = (1, −4, − x2 + y2, M = x 5 31. (0, 3) √ 33 37. (a) The distance from A to B is |AB| = 52, and the distance from B to C is |BC| =, we are guaranteed by the converse of the Pythagorean Theorem that the triangle is a right triangle. 13, the distance from A to C is |AC| = 652 132 65. Since √ √ = √ + √ 522 √ √ (b) Show that |AC|2 + |BC|2 = |AB|2 20 1.2 Relations Relations and Functions From one point of view,1 all of Precalculus can be thought of as studying sets of points in the plane. With the Cartesian Plane now fresh in our memory we can discuss those sets in more detail and as usual, we begin with a definition. Definition 1.4. A relation is a set of points in the plane. Since relations are sets, we can describe them using the techniques presented in Section 1.1.1. That is, we can describe a relation verbally, using the roster method, or using set-builder notation. Since the elements in a relation are points in the plane, we often try to describe the relation graphically or algebraically as well. Depending on the situation, one method may be easier or more convenient to use than another. As an example, consider the relation R = {(−2, 1), (4, 3), (0, −3)}. As written, R is described using the roster method. Since R consists of points in the plane, we follow our instinct and plot the |
points. Doing so produces the graph of R. y 4 3 2 1 (−2, 1) (4, 3) −4 −3 −2 −1 1 2 3 4 x −1 −2 −3 −4 (0, −3) The graph of R. In the following example, we graph a variety of relations. Example 1.2.1. Graph the following relations. 1. A = {(0, 0), (−3, 1), (4, 2), (−3, 2)} 2. HLS1 = {(x, 3) | − 2 ≤ x ≤ 4} 3. HLS2 = {(x, 3) | − 2 ≤ x < 4} 4. V = {(3, y) | y is a real number} 5. H = {(x, y) | y = −2} 6. R = {(x, y) | 1 < y ≤ 3} 1Carl’s, of course. 1.2 Relations Solution. 21 1. To graph A, we simply plot all of the points which belong to A, as shown below on the left. 2. Don’t let the notation in this part fool you. The name of this relation is HLS1, just like the name of the relation in number 1 was A. The letters and numbers are just part of its name, just like the numbers and letters of the phrase ‘King George III’ were part of George’s name. In words, {(x, 3) | − 2 ≤ x ≤ 4} reads ‘the set of points (x, 3) such that −2 ≤ x ≤ 4.’ All of these points have the same y-coordinate, 3, but the x-coordinate is allowed to vary between −2 and 4, inclusive. Some of the points which belong to HLS1 include some friendly points like: (−2, 3), (−1, 3), (0, 3), (1, 3), (2, 3), (3, 3), and (4, 3). However, HLS1 also contains the points (0.829, 3), − 5 It is impossible2 to list all of these points, which is why the variable x is used. Plotting several friendly representative points should convince you that HLS1 describes the horizontal line segment from the point (−2, 3) up to and including the point (4, 3). |
π, 3), and so on. 6, 34 −3 −2 −1 1 2 3 4 x −4 −3 −2 −1 1 2 3 4 x The graph of A The graph of HLS1 3. HLS2 is hauntingly similar to HLS1. In fact, the only difference between the two is that instead of ‘−2 ≤ x ≤ 4’ we have ‘−2 ≤ x < 4’. This means that we still get a horizontal line segment which includes (−2, 3) and extends to (4, 3), but we do not include (4, 3) because of the strict inequality x < 4. How do we denote this on our graph? It is a common mistake to make the graph start at (−2, 3) end at (3, 3) as pictured below on the left. The problem with this graph is that we are forgetting about the points like (3.1, 3), (3.5, 3), (3.9, 3), (3.99, 3), and so forth. There is no real number that comes ‘immediately before’ 4, so to describe the set of points we want, we draw the horizontal line segment starting at (−2, 3) and draw an open circle at (4, 3) as depicted below on the right. 2Really impossible. The interested reader is encouraged to research countable versus uncountable sets. 22 Relations and Functions 4 −3 −2 −1 1 2 3 4 x −4 −3 −2 −1 1 2 3 4 x This is NOT the correct graph of HLS2 The graph of HLS2 4. Next, we come to the relation V, described as the set of points (3, y) such that y is a real number. All of these points have an x-coordinate of 3, but the y-coordinate is free to be whatever it wants to be, without restriction.3 Plotting a few ‘friendly’ points of V should convince you that all the points of V lie on the vertical line4 x = 3. Since there is no restriction on the y-coordinate, we put arrows on the end of the portion of the line we draw to indicate it extends indefinitely in both directions. The graph of V is below on the left. 5. Though written slightly differently, the relation H = |
{(x, y) | y = −2} is similar to the relation V above in that only one of the coordinates, in this case the y-coordinate, is specified, leaving x to be ‘free’. Plotting some representative points gives us the horizontal line y = −2. y 4 3 2 1 −1 −2 −3 −4 1 2 3 4 x y −4 −3 −2 −1 1 2 3 4 x −1 −2 −3 −4 The graph of H The graph of V 6. For our last example, we turn to R = {(x, y) | 1 < y ≤ 3}. As in the previous example, x is free to be whatever it likes. The value of y, on the other hand, while not completely free, is permitted to roam between 1 and 3 excluding 1, but including 3. After plotting some5 friendly elements of R, it should become clear that R consists of the region between the horizontal 3We’ll revisit the concept of a ‘free variable’ in Section 8.1. 4Don’t worry, we’ll be refreshing your memory about vertical and horizontal lines in just a moment! 5The word ‘some’ is a relative term. It may take 5, 10, or 50 points until you see the pattern. 1.2 Relations 23 lines y = 1 and y = 3. Since R requires that the y-coordinates be greater than 1, but not equal to 1, we dash the line y = 1 to indicate that those points do not belong to R. y 4 3 2 1 −4 −3 −2 −1 1 2 3 4 x The graph of R The relations V and H in the previous example lead us to our final way to describe relations: algebraically. We can more succinctly describe the points in V as those points which satisfy the equation ‘x = 3’. Most likely, you have seen equations like this before. Depending on the context, ‘x = 3’ could mean we have solved an equation for x and arrived at the solution x = 3. In this case, however, ‘x = 3’ describes a set of points in the plane whose x-coordinate is 3. Similarly, the relation H above can be described by the equation ‘y = −2’. At some point in your mathematical upbringing, you probably |
learned the following. Equations of Vertical and Horizontal Lines The graph of the equation x = a is a vertical line through (a, 0). The graph of the equation y = b is a horizontal line through (0, b). Given that the very simple equations x = a and y = b produced lines, it’s natural to wonder what shapes other equations might yield. Thus our next objective is to study the graphs of equations in a more general setting as we continue to unite Algebra and Geometry. 1.2.1 Graphs of Equations In this section, we delve more deeply into the connection between Algebra and Geometry by focusing on graphing relations described by equations. The main idea of this section is the following. The Fundamental Graphing Principle The graph of an equation is the set of points which satisfy the equation. That is, a point (x, y) is on the graph of an equation if and only if x and y satisfy the equation. Here, ‘x and y satisfy the equation’ means ‘x and y make the equation true’. It is at this point that we gain some insight into the word ‘relation’. If the equation to be graphed contains both x and y, then the equation itself is what is relating the two variables. More specifically, in the next two examples, we consider the graph of the equation x2 + y3 = 1. Even though it is not specifically 24 Relations and Functions spelled out, what we are doing is graphing the relation R = {(x, y) | x2 + y3 = 1}. The points (x, y) we graph belong to the relation R and are necessarily related by the equation x2 + y3 = 1, since it is those pairs of x and y which make the equation true. Example 1.2.2. Determine whether or not (2, −1) is on the graph of x2 + y3 = 1. Solution. We substitute x = 2 and y = −1 into the equation to see if the equation is satisfied. Hence, (2, −1) is not on the graph of x2 + y3 = 1. (2)2 + (−1)3? = 1 3 = 1 We could spend hours randomly guessing and checking to see if points are on the graph of the equation. A more systematic approach is outlined |
in the following example. Example 1.2.3. Graph x2 + y3 = 1. Solution. To efficiently generate points on the graph of this equation, we first solve for y x2 + y3 = 1 y3 = 1 − x2 y3 = 3√ y = 3√ We now substitute a value in for x, determine the corresponding value y, and plot the resulting point (x, y). For example, substituting x = −3 into the equation yields 1 − x2 1 − x2 3 y = 3 1 − x2 = 3 1 − (−3)2 = 3√ −8 = −2, so the point (−3, −2) is on the graph. Continuing in this manner, we generate a table of points which are on the graph of the equation. These points are then plotted in the plane as shown below. y y x −2 −3 √ −2 − 3 −2 3 (x, y) (−3, −2) √ 3 (−2, − 3 3) (−1, 0) (0, 1) (1, 0) √ (2, − 3 3) (3, −2) 3 2 1 −4 −3 −2 −1 1 2 3 4 x −1 −2 −3 Remember, these points constitute only a small sampling of the points on the graph of this equation. To get a better idea of the shape of the graph, we could plot more points until we feel comfortable 1.2 Relations 25 ‘connecting the dots’. Doing so would result in a curve similar to the one pictured below on the far left. y 3 2 1 −4 −3 −2 −1 −1 1 2 3 4 x −2 −3 Don’t worry if you don’t get all of the little bends and curves just right − Calculus is where the art of precise graphing takes center stage. For now, we will settle with our naive ‘plug and plot’ approach to graphing. If you feel like all of this tedious computation and plotting is beneath you, then you can reach for a graphing calculator, input the formula as shown above, and graph. Of all of the points on the graph of an equation, the places where the graph crosses or touches the axes hold special significance. These are called the intercepts of the graph. Intercepts |
come in two distinct varieties: x-intercepts and y-intercepts. They are defined below. Definition 1.5. Suppose the graph of an equation is given. A point on a graph which is also on the x-axis is called an x-intercept of the graph. A point on a graph which is also on the y-axis is called an y-intercept of the graph. In our previous example the graph had two x-intercepts, (−1, 0) and (1, 0), and one y-intercept, (0, 1). The graph of an equation can have any number of intercepts, including none at all! Since x-intercepts lie on the x-axis, we can find them by setting y = 0 in the equation. Similarly, since y-intercepts lie on the y-axis, we can find them by setting x = 0 in the equation. Keep in mind, intercepts are points and therefore must be written as ordered pairs. To summarize, Finding the Intercepts of the Graph of an Equation Given an equation involving x and y, we find the intercepts of the graph as follows: x-intercepts have the form (x, 0); set y = 0 in the equation and solve for x. y-intercepts have the form (0, y); set x = 0 in the equation and solve for y. Another fact which you may have noticed about the graph in the previous example is that it seems to be symmetric about the y-axis. To actually prove this analytically, we assume (x, y) is a generic point on the graph of the equation. That is, we assume x2 + y3 = 1 is true. As we learned in Section 1.1, the point symmetric to (x, y) about the y-axis is (−x, y). To show that the graph is 26 Relations and Functions symmetric about the y-axis, we need to show that (−x, y) satisfies the equation x2 + y3 = 1, too. Substituting (−x, y) into the equation gives (−x)2 + (y)3? = 1 x2 + y3 = 1 Since we are assuming the original equation x2 + y3 = 1 is true, we have shown that (−x |
, y) satisfies the equation (since it leads to a true result) and hence is on the graph. In this way, we can check whether the graph of a given equation possesses any of the symmetries discussed in Section 1.1. We summarize the procedure in the following result. Testing the Graph of an Equation for Symmetry To test the graph of an equation for symmetry about the y-axis − substitute (−x, y) into the equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the y-axis. about the x-axis – substitute (x, −y) into the equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the x-axis. about the origin - substitute (−x, −y) into the equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the origin. Intercepts and symmetry are two tools which can help us sketch the graph of an equation analytically, as demonstrated in the next example. Example 1.2.4. Find the x- and y-intercepts (if any) of the graph of (x − 2)2 + y2 = 1. Test for symmetry. Plot additional points as needed to complete the graph. Solution. To look for x-intercepts, we set y = 0 and solve (x − 2)2 + y2 = 1 (x − 2)2 + 02 = 1 (x − 2)2 = 1 √ (x − 2)2 = 1 x − 2 = ±1 extract square roots x = 2 ± 1 x = 3, 1 We get two answers for x which correspond to two x-intercepts: (1, 0) and (3, 0). Turning our attention to y-intercepts, we set x = 0 and solve 1.2 Relations 27 (x − 2)2 + y2 = 1 (0 − 2)2 + y2 = 1 4 + y2 = 1 y2 = −3 Since there is no real number which squares to a negative number (Do you remember why?), we are forced to conclude that the graph has no y-intercepts. Plotting the data we have so far, we get y 1 −1 (1, 0) (3, 0) 1 2 3 4 x Moving along to symmetry, we can immediately dismiss the possibility |
that the graph is symmetric about the y-axis or the origin. If the graph possessed either of these symmetries, then the fact that (1, 0) is on the graph would mean (−1, 0) would have to be on the graph. (Why?) Since (−1, 0) would be another x-intercept (and we’ve found all of these), the graph can’t have y-axis or origin symmetry. The only symmetry left to test is symmetry about the x-axis. To that end, we substitute (x, −y) into the equation and simplify (x − 2)2 + y2 = 1? = 1 (x − 2)2 + (−y)2 (x − 2)2 + y2 = 1 Since we have obtained our original equation, we know the graph is symmetric about the x-axis. This means we can cut our ‘plug and plot’ time in half: whatever happens below the x-axis is reflected above the x-axis, and vice-versa. Proceeding as we did in the previous example, we obtain y 1 −1 1 2 3 4 x 28 Relations and Functions A couple of remarks are in order. First, it is entirely possible to choose a value for x which does not correspond to a point on the graph. For example, in the previous example, if we solve for y as is our custom, we get y = ± 1 − (x − 2)2. Upon substituting x = 0 into the equation, we would obtain y = ± 1 − (0 − 2)3, which is not a real number. This means there are no points on the graph with an x-coordinate of 0. When this happens, we move on and try another point. This is another drawback of the ‘plug-and-plot’ approach to graphing equations. Luckily, we will devote much of the remainder of this book to developing techniques which allow us to graph entire families of equations quickly.6 Second, it is instructive to show what would have happened had we tested the equation in the last example for symmetry about the y-axis. Substituting (−x, y) into the equation yields (x − 2)2 + y2 =. (−x − 2)2 + y2 ((−1)(x + 2))2 + y2 (x + 2)2 + y2 This last equation does not appear |
to be equivalent to our original equation. However, to actually prove that the graph is not symmetric about the y-axis, we need to find a point (x, y) on the graph whose reflection (−x, y) is not. Our x-intercept (1, 0) fits this bill nicely, since if we substitute (−1, 0) into the equation we get (x − 2)2 + y2 (−1 − 2)2 + 02? = 1 = 1 9 = 1. This proves that (−1, 0) is not on the graph. 6Without the use of a calculator, if you can believe it! 1.2 Relations 1.2.2 Exercises In Exercises 1 - 20, graph the given relation. 1. {(−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9)} 2. {(−2, 0), (−1, 1), (−1, −1), (0, 2), (0, −2), (1, 3), (1, −3)} 29 3. {(m, 2m) | m = 0, ±1, ±2} 4. 6 k, k | k = ±1, ±2, ±3, ±4, ±5, ±6 5. n, 4 − n2 | n = 0, ±1, ±2 6. √ j, j | j = 0, 1, 4, 9 7. {(x, −2) | x > −4} 8. {(x, 3) | x ≤ 4} 9. {(−1, y) | y > 1} 10. {(2, y) | y ≤ 5} 11. {(−2, y) | − 3 < y ≤ 4} 12. {(3, y) | − 4 ≤ y < 3} 13. {(x, 2) | − 2 ≤ x < 3} 14. {(x, −3) | − 4 < x ≤ 4} 15. {(x, y) | x > −2} 16. {(x, y) | x ≤ 3} 17. {(x, y) | y < 4} 18. {(x, y) | x ≤ 3, y < 2} 19. |
{(x, y) | x > 0, y < 4} 20. {(x, y } In Exercises 21 - 30, describe the given relation using either the roster or set-builder method. 21. y 4 3 2 1 −4 −3 −2 −1 −1 1 x Relation A 22. y 3 2 1 −4 −3 −2 −1 1 2 3 4 x Relation B 30 23. y 5 4 3 2 1 −1 −2 −3 1 2 3 x Relations and Functions 24. y 3 2 1 −3 −2 −1 −1 x −2 −3 −4 Relation C Relation D 25. 27. y 3 2 1 −4 −3 −2 −1 1 2 3 4 x Relation E y 3 2 1 26. 28. y 4 3 2 1 −3 −2 −1 1 2 3 x Relation F y 3 2 1 −3 −2 −1 −1 1 2 3 x −4 −3 −2 −1 −1 1 2 3 x −2 −3 Relation G −2 −3 Relation H 1.2 Relations 29. y 5 4 3 2 1 −1 −1 1 2 3 4 5 x Relation I In Exercises 31 - 36, graph the given line. 31. x = −2 33. y = 3 35. x = 0 31 30. y 2 1 −4 −3 −2 −1 −1 1 2 3 4 5 x −2 −3 Relation J 32. x = 3 34. y = −2 36. y = 0 Some relations are fairly easy to describe in words or with the roster method but are rather difficult, if not impossible, to graph. Discuss with your classmates how you might graph the relations given in Exercises 37 - 40. Please note that in the notation below we are using the ellipsis,..., to denote that the list does not end, but rather, continues to follow the established pattern indefinitely. For the relations in Exercises 37 and 38, give two examples of points which belong to the relation and two points which do not belong to the relation. 37. {(x, y) | x is an odd integer, and y is an even integer.} 38. {(x, 1) | x is an irrational number } 39. {(1, 0), (2, 1), (4, 2 |
), (8, 3), (16, 4), (32, 5),...} 40. {..., (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9),...} For each equation given in Exercises 41 - 52: Find the x- and y-intercept(s) of the graph, if any exist. Follow the procedure in Example 1.2.3 to create a table of sample points on the graph of the equation. Plot the sample points and create a rough sketch of the graph of the equation. Test for symmetry. If the equation appears to fail any of the symmetry tests, find a point on the graph of the equation whose reflection fails to be on the graph as was done at the end of Example 1.2.4 32 Relations and Functions 41. y = x2 + 1 43. y = x3 − x 45. y = √ x − 2 47. 3x − y = 7 49. (x + 2)2 + y2 = 16 51. 4y2 − 9x2 = 36 42. y = x2 − 2x − 8 44. y = x3 4 − 3x √ 46. y = 2 x + 4 − 2 48. 3x − 2y = 10 50. x2 − y2 = 1 52. x3y = −4 The procedures which we have outlined in the Examples of this section and used in Exercises 41 - 52 all rely on the fact that the equations were “well-behaved”. Not everything in Mathematics is quite so tame, as the following equations will show you. Discuss with your classmates how you might approach graphing the equations given in Exercises 53 - 56. What difficulties arise when trying to apply the various tests and procedures given in this section? For more information, including pictures of the curves, each curve name is a link to its page at www.wikipedia.org. For a much longer list of fascinating curves, click here. 53. x3 + y3 − 3xy = 0 Folium of Descartes 54. x4 = x2 + y2 Kampyle of Eudoxus 55. y2 = x3 + 3x2 Tschirnhausen cubic 56. (x2 + y2 |
)2 = x3 + y3 Crooked egg 57. With the help of your classmates, find examples of equations whose graphs possess symmetry about the x-axis only symmetry about the y-axis only symmetry about the origin only symmetry about the x-axis, y-axis, and origin Can you find an example of an equation whose graph possesses exactly two of the symmetries listed above? Why or why not? 1.2 Relations 1.2.3 Answers 1. 5. −3 −2 −2 −1 −1 1 2 x −2 −3 −4 y 4 3 2 1 −2 −1 1 2 x 33 2. 4. y 3 2 1 −2 −1 −1 1 2 x −2 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 6 34 7. Relations and Functions y 8. −4 −3 −2 −1 −3 −4 −3 −2 −1 1 2 3 4 x 9. y 101 1 2 x 11. y 4 3 2 1 −3 −2 −1 −1 x −2 −3 5 4 3 2 1 −1 −2 −3 1 2 3 x 12. y 3 2 1 −1 −2 −3 −4 1 2 3 x 1.2 Relations 35 14. 16. 18. 13. 15. 17. y 3 2 1 −4 −3 −2 −2 −1 −1 1 2 3 x −2 −3 y 4 3 2 1 −3 −2 −1 1 2 3 x y −4 −3 −2 −1 −1 1 2 3 4 x −2 −3 y y 3 2 1 −1 −2 −3 3 2 1 −1 −2 −3 1 2 3 x 1 2 3 x 19. y 20. y 4 3 2 1 −2 −1 1 x 36 Relations and Functions 21. A = {(−4, −1), (−2, 1), (0, 3), (1, 4)} 22. B = {(x, 3) | x ≥ −3} 23. C = {(2, y) | y > −3} 24. D = {(−2, y) | − 4 ≤ y < 3} 25. E = {(x, 2) | − 4 ≤ x < 3} 26. F = {(x, y) | y ≥ 0} 27. G |
= {(x, y) | x > −2} 28. H = {(x, y) | − 3 < x ≤ 2} 29. I = {(x, y) | x ≥ 0,y ≥ 0} 30. J = {(x, y) | − 4 < x < 5, −3 < y < 2} 31. 33. 35. y 3 2 1 −3 −2 −1 −1 x −2 −3 The line x = −2 y 3 2 1 −3 −2 −1 1 2 3 x The line y = 3 y 3 2 1 32. y 3 2 1 −1 −2 −3 1 2 3 x The line x = 3 34. y −3 −2 −1 −1 1 2 3 x −2 −3 The line y = −2 36. y 3 2 1 −3 −2 −1 −1 1 2 3 x −3 −2 −1 −1 1 2 3 x −2 −3 −2 −3 The line x = 0 is the y-axis The line y = 0 is the x-axis 1.2 Relations 41. y = x2 + 1 37 42. y = x2 − 2x − 8 The graph has no x-intercepts x-intercepts: (4, 0), (−2, 0) y-intercept: (0, 1) x −2 −1 0 1 2 y (x, y) 5 (−2, 5) 2 (−1, 2) (0, 1) 1 (1, 2) 2 (2, 5) 5 y 5 4 3 2 1 −2−1 1 2 x The graph is not symmetric about the x-axis (e.g. (2, 5) is on the graph but (2, −5) is not) The graph is symmetric about the y-axis The graph is not symmetric about the origin (e.g. (2, 5) is on the graph but (−2, −5) is not) y-intercept: (0, −8) y 7 0 (x, y) x (−3, 7) −3 −2 (−2, 0) −1 −5 (−1, −5) (0, −8) (1, −9) (2, −8) (3, −5) (4, 0) (5, 7) 0 −8 1 −9 2 − |
8 3 −3−2−1 −2 −3 −4 −5 −6 −7 −8 −9 The graph is not symmetric about the x-axis (e.g. (−3, 7) is on the graph but (−3, −7) is not) The graph is not symmetric about the y-axis (e.g. (−3, 7) is on the graph but (3, 7) is not) The graph is not symmetric about the origin (e.g. (−3, 7) is on the graph but (3, −7) is not) 38 43. y = x3 − x x-intercepts: (−1, 0), (0, 0), (1, 0) y-intercept: (0, 0) x y (x, y) −2 −6 (−2, −6) (−1, 0) −1 (0, 0) 0 (1, 0) 1 (2, 62−1 −1 1 2 x −2 −3 −4 −5 −6 The graph is not symmetric about the x-axis. (e.g. (2, 6) is on the graph but (2, −6) is not) The graph is not symmetric about the y-axis. (e.g. (2, 6) is on the graph but (−2, 6) is not) The graph is symmetric about the origin. Relations and Functions 44. y = x3 4 − 3x x-intercepts: ±2 √ 3, 0, (0, 0) y-intercept: (0, 0) x y −4 −4 9 −3 4 −2 4 11 −1 4 0 0 1 − 11 4 2 −4 3 − 9 4 4 4 (x, y) (−4, −4) −3, 9 4 (−2, 4) −1, 11 4 (0, 0) 1, − 11 4 (2, −4) 3, − 9 4 (4, 4) y 4 3 2 1 −4−3−2−1 −1 1 2 3 4 x −2 −3 −4 The graph is not symmetric about the x-axis (e.g. (−4, −4) is on the graph but (−4, 4) is not) The graph is not symmetric about the y-axis (e.g. (−4, −4 |
) is on the graph but (4, −4) is not) The graph is symmetric about the origin 1.2 Relations 45. y = √ x − 2 x-intercept: (2, 0) The graph has no y-intercepts √ 46-intercept: (−3, 0) y-intercept: (0, 2) 39 y (x, y) 0 (2, 0) 1 (3, 1) (6, 2) 2 3 (11, 3) x 2 3 6 11 10 11 x The graph is not symmetric about the x-axis (e.g. (3, 1) is on the graph but (3, −1) is not) The graph is not symmetric about the y-axis (e.g. (3, 1) is on the graph but (−3, 1) is not) The graph is not symmetric about the origin (e.g. (3, 1) is on the graph but (−3, −1) is not) x −4 −3 −2 −1 0 1 y −2 0 2 − 2 −2, 3 − 2 −2, 2 5 − 2 −2, y (x, y) (−4, −2) (−3, 00, 24−3−2−1 −1 1 2 x −2 −3 The graph is not symmetric about the x-axis (e.g. (−4, −2) is on the graph but (−4, 2) is not) The graph is not symmetric about the y-axis (e.g. (−4, −2) is on the graph but (4, −2) is not) The graph is not symmetric about the origin (e.g. (−4, −2) is on the graph but (4, 2) is not) 40 Relations and Functions 47. 3x − y = 7 Re-write as: y = 3x − 7. 48. 3x − 2y = 10 Re-write as: y = 3x−10 2. x-intercept: ( 7 3, 0) y-intercept: (0, −7) x y (x, y) −2 −13 (−2, −13) −1 −10 (−1, −10) (0, −7) (1, −4) (2, −1) (3, 2) 0 −7 1 |
−4 2 −1 2 3 y 3 2 1 −2−1 −1 1 2 3 x −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 x-intercepts: 10 3, 0 y-intercept: (0, −5) x y −2 −8 −1 − 13 2 0 −5 1 − 7 2 2 −2 (x, y) (−2, −8) −1, − 13 2 (0, −5) 1, − 7 2 (2, −2) y 2 1 −3−2−1 −1 1 2 3 4 x −2 −3 −4 −5 −6 −7 −8 −9 The graph is not symmetric about the x-axis (e.g. (3, 2) is on the graph but (3, −2) is not) The graph is not symmetric about the y-axis (e.g. (3, 2) is on the graph but (−3, 2) is not) The graph is not symmetric about the origin (e.g. (3, 2) is on the graph but (−3, −2) is not) The graph is not symmetric about the x-axis (e.g. (2, −2) is on the graph but (2, 2) is not) The graph is not symmetric about the y-axis (e.g. (2, −2) is on the graph but (−2, −2) is not) The graph is not symmetric about the origin (e.g. (2, −2) is on the graph but (−2, 2) is not) 1.2 Relations 41 49. (x + 2)2 + y2 = 16 Re-write as y = ± 16 − (x + 2)2. x-intercepts: (−6, 0), (2, 0) y-intercepts: 0, ±2 3 √ y x −6 0 √ −4 ±2 −2 ±4 √ 0 ±2 0 2 (x, y) (−6, 0) √ 3 −4, ±2 3 3 √ (−2, ±4) 3 0, ±2 (2, 0) y 5 4 3 2 1 50. x2 − y2 = 1 √ Re-write as: y = ± x2 − 1 |
. x-intercepts: (−1, 0), (1, 0) The graph has no y-intercepts x −3 ± −2 ± −x, y) √ √ 8 (−3, ± 3 (−2, ± 8) 3) (−1, 0) (1, 0) √ √ (2, ± (3, ± 3) 8) 3 8 −7−6−5−4−3−2−1 −1 1 2 3 x −2 −3 −4 −5 The graph is symmetric about the x-axis The graph is not symmetric about the y-axis (e.g. (−6, 0) is on the graph but (6, 0) is not) The graph is not symmetric about the origin (e.g. (−6, 0) is on the graph but (6, 0) is not) y 3 2 1 −3−2−1 −1 1 2 3 x −2 −3 The graph is symmetric about the x-axis The graph is symmetric about the y-axis The graph is symmetric about the origin 42 51. 4y2 − 9x2 = 36 Re-write as: y = ± √ 9x2+36 2. The graph has no x-intercepts y-intercepts: (0, ±3) Relations and Functions 52. x3y = −4 Re-write as: y = − 4 x3. The graph has no x-intercepts The graph has no y-intercepts x y √ −4 ±3 √ −2 ±3 ±3 √ √ 0 2 ±3 4 ±3 √ √ (x, y) 5 −4, ±3 2 −2, ±3 (0, ±3) √ √ 2, ±3 4, ±3 2 5 2 5 5 2 x −2 −1 − 1 2 1 y 1 2 4 32 2 −32 ( 1 1 −4 2 − 1 2 (x, y) (−2, 1 2 ) (−1, 4) (− 1 2, 32) 2, −32) (1, −4) (24−3−2−1 −1 1 2 3 4 x −2 −3 −4 −5 −6 −7 y 32 4 −4 −2 −1 1 x 2 The graph is symmetric about the x-axis −32 The graph |
is symmetric about the y-axis The graph is symmetric about the origin The graph is not symmetric about the x-axis (e.g. (1, −4) is on the graph but (1, 4) is not) The graph is not symmetric about the y-axis (e.g. (1, −4) is on the graph but (−1, −4) is not) The graph is symmetric about the origin 1.3 Introduction to Functions 43 1.3 Introduction to Functions One of the core concepts in College Algebra is the function. There are many ways to describe a function and we begin by defining a function as a special kind of relation. Definition 1.6. A relation in which each x-coordinate is matched with only one y-coordinate is said to describe y as a function of x. Example 1.3.1. Which of the following relations describe y as a function of x? 1. R1 = {(−2, 1), (1, 3), (1, 4), (3, −1)} 2. R2 = {(−2, 1), (1, 3), (2, 3), (3, −1)} Solution. A quick scan of the points in R1 reveals that the x-coordinate 1 is matched with two different y-coordinates: namely 3 and 4. Hence in R1, y is not a function of x. On the other hand, every x-coordinate in R2 occurs only once which means each x-coordinate has only one corresponding y-coordinate. So, R2 does represent y as a function of x. Note that in the previous example, the relation R2 contained two different points with the same y-coordinates, namely (1, 3) and (2, 3). Remember, in order to say y is a function of x, we just need to ensure the same x-coordinate isn’t used in more than one point.1 To see what the function concept means geometrically, we graph R1 and R2 in the plane2 −1 −1 1 2 3 x −2 −1 −1 1 2 3 x The graph of R1 The graph of R2 The fact that the x-coordinate 1 is matched with two different y-coordinates in R1 presents itself |
graphically as the points (1, 3) and (1, 4) lying on the same vertical line, x = 1. If we turn our attention to the graph of R2, we see that no two points of the relation lie on the same vertical line. We can generalize this idea as follows Theorem 1.1. The Vertical Line Test: A set of points in the plane represents y as a function of x if and only if no two points lie on the same vertical line. 1We will have occasion later in the text to concern ourselves with the concept of x being a function of y. In this case, R1 represents x as a function of y; R2 does not. 44 Relations and Functions It is worth taking some time to meditate on the Vertical Line Test; it will check to see how well you understand the concept of ‘function’ as well as the concept of ‘graph’. Example 1.3.2. Use the Vertical Line Test to determine which of the following relations describes y as a function of x. y 4 3 2 1 −1 1 x −1 The graph of R The graph of S Solution. Looking at the graph of R, we can easily imagine a vertical line crossing the graph more than once. Hence, R does not represent y as a function of x. However, in the graph of S, every vertical line crosses the graph at most once, so S does represent y as a function of x. In the previous test, we say that the graph of the relation R fails the Vertical Line Test, whereas the graph of S passes the Vertical Line Test. Note that in the graph of R there are infinitely many vertical lines which cross the graph more than once. However, to fail the Vertical Line Test, all you need is one vertical line that fits the bill, as the next example illustrates. Example 1.3.3. Use the Vertical Line Test to determine which of the following relations describes y as a function of x1 1 x −1 −1 1 x −1 The graph of S1 The graph of S2 1.3 Introduction to Functions 45 Solution. Both S1 and S2 are slight modifications to the relation S in the previous example whose graph we determined passed the Vertical Line Test. In both S1 and S2, it is the addition of the point (1, 2) which threatens to cause trouble. In S1, there |
is a point on the curve with x-coordinate 1 just below (1, 2), which means that both (1, 2) and this point on the curve lie on the vertical line x = 1. (See the picture below and the left.) Hence, the graph of S1 fails the Vertical Line Test, so y is not a function of x here. However, in S2 notice that the point with x-coordinate 1 on the curve has been omitted, leaving an ‘open circle’ there. Hence, the vertical line x = 1 crosses the graph of S2 only at the point (1, 2). Indeed, any vertical line will cross the graph at most once, so we have that the graph of S2 passes the Vertical Line Test. Thus it describes y as a function of x1 −1 x −1 1 x −1 S1 and the line x = 1 The graph of G for Ex. 1.3.4 Suppose a relation F describes y as a function of x. The sets of x- and y-coordinates are given special names which we define below. Definition 1.7. Suppose F is a relation which describes y as a function of x. The set of the x-coordinates of the points in F is called the domain of F. The set of the y-coordinates of the points in F is called the range of F. We demonstrate finding the domain and range of functions given to us either graphically or via the roster method in the following example. Example 1.3.4. Find the domain and range of the function F = {(−3, 2), (0, 1), (4, 2), (5, 2)} and of the function G whose graph is given above on the right. Solution. The domain of F is the set of the x-coordinates of the points in F, namely {−3, 0, 4, 5} and the range of F is the set of the y-coordinates, namely {1, 2}. To determine the domain and range of G, we need to determine which x and y values occur as coordinates of points on the given graph. To find the domain, it may be helpful to imagine collapsing the curve to the x-axis and determining the portion of the x-axis that gets covered. This is called projecting the curve to the x-axis. Before we start projecting, |
we need to pay attention to two 46 Relations and Functions subtle notations on the graph: the arrowhead on the lower left corner of the graph indicates that the graph continues to curve downwards to the left forever more; and the open circle at (1, 3) indicates that the point (1, 3) isn’t on the graph, but all points on the curve leading up to that point are. y project down 1 1 x −1 −1 1 x −1 project up The graph of G The graph of G We see from the figure that if we project the graph of G to the x-axis, we get all real numbers less than 1. Using interval notation, we write the domain of G as (−∞, 1). To determine the range of G, we project the curve to the y-axis as follows: y 4 3 2 1 project left project right y 4 3 2 1 −1 1 x −1 −1 1 x −1 The graph of G The graph of G Note that even though there is an open circle at (1, 3), we still include the y value of 3 in our range, since the point (−1, 3) is on the graph of G. We see that the range of G is all real numbers less than or equal to 4, or, in interval notation, (−∞, 4]. 1.3 Introduction to Functions 47 All functions are relations, but not all relations are functions. Thus the equations which described the relations in Section1.2 may or may not describe y as a function of x. The algebraic representation of functions is possibly the most important way to view them so we need a process for determining whether or not an equation of a relation represents a function. (We delay the discussion of finding the domain of a function given algebraically until Section 1.4.) Example 1.3.5. Determine which equations represent y as a function of x. 1. x3 + y2 = 1 2. x2 + y3 = 1 3. x2y = 1 − 3y Solution. For each of these equations, we solve for y and determine whether each choice of x will determine only one corresponding value of y. 1. 2. 3. x3 + y2 = 1 √ y2 = 1 − x3 y2 = y = ± 1 − x3 √ 1 − x3 extract square roots 1 − 03 = ±1, so that (0, |
1) If we substitute x = 0 into our equation for y, we get y = ± and (0, −1) are on the graph of this equation. Hence, this equation does not represent y as a function of x. √ x2 + y3 = 1 3 y3 = 1 − x2 y3 = 3√ y = 3√ 1 − x2 1 − x2 For every choice of x, the equation y = 3√ equation describes y as a function of x. 1 − x2 returns only one value of y. Hence, this x2y = 1 − 3y x2y + 3y = 1 y x2 + 3 = 1 y = 1 x2 + 3 factor For each choice of x, there is only one value for y, so this equation describes y as a function of x. We could try to use our graphing calculator to verify our responses to the previous example, but we immediately run into trouble. The calculator’s “Y=” menu requires that the equation be of the form ‘y = some expression of x’. If we wanted to verify that the first equation in Example 1.3.5 48 Relations and Functions does not represent y as a function of x, we would need to enter two separate expressions into the calculator: one for the positive square root and one for the negative square root we found when solving the equation for y. As predicted, the resulting graph shown below clearly fails the Vertical Line Test, so the equation does not represent y as a function of x. Thus in order to use the calculator to show that x3 + y2 = 1 does not represent y as a function of x we needed to know analytically that y was not a function of x so that we could use the calculator properly. There are more advanced graphing utilities out there which can do implicit function plots, but you need to know even more Algebra to make them work properly. Do you get the point we’re trying to make here? We believe it is in your best interest to learn the analytic way of doing things so that you are always smarter than your calculator. 1.3 Introduction to Functions 49 1.3.1 Exercises In Exercises 1 - 12, determine whether or not the relation represents y as a function of x. Find the domain and range of those relations which are functions. 1. {(−3, 9), (−2, 4 |
), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9)} 2. {(−3, 0), (1, 6), (2, −3), (4, 2), (−5, 6), (4, −9), (6, 2)} 3. {(−3, 0), (−7, 6), (5, 5), (6, 4), (4, 9), (3, 0)} 4. {(1, 2), (4, 4), (9, 6), (16, 8), (25, 10), (36, 12),...} 5. {(x, y) | x is an odd integer, and y is an even integer} 6. {(x, 1) | x is an irrational number} 7. {(1, 0), (2, 1), (4, 2), (8, 3), (16, 4), (32, 5),... } 8. {..., (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9),... } 9. {(−2, y) | − 3 < y < 4} 10. {(x, 3) | − 2 ≤ x < 4} 11. {x, x2 | x is a real number} 12. {x2, x | x is a real number} In Exercises 13 - 32, determine whether or not the relation represents y as a function of x. Find the domain and range of those relations which are functions. 13. y 4 3 2 1 14. y 4 3 2 1 −4 −3 −2 −1 1 x −1 −4 −3 −2 −1 1 x −1 50 15. 17. 19. 21. y 5 4 3 2 1 −2 −4 −3 −2 −1 1 2 3 4 5 x −1 −2 −3−2−1 −1 −2 −3 −4 −5 Relations and Functions 16. 18. 20. 22. y 3 2 1 −3 −2 −1 1 2 3 x −1 −2 −3 y 4 3 2 1 −4 −3 −2 −5 −4 −3 −2 −1 1 2 3 x −1 −2 y 5 4 3 2 1 −5− |
4−3−2−1 −1 1 2 3 4 5 x −2 −3 −4 −5 1.3 Introduction to Functions 51 23. 25. 27. 29. y 5 4 3 2 1 −5−4−3−2−1 −1 1 2 3 4 5 x −2 −3 −4 −5 y 4 3 2 1 24. y 5 4 3 2 1 −1 −1 −2 −3 −4 −5 262 −1 1 2 x −2 −1 1 2 x y 4 3 2 1 28. y 4 3 2 1 −2 −1 1 2 x −2 −1 1 2 x y 2 1 30. y 2 1 −2 −1 1 2 x −3 −2 −1 1 2 3 x −1 −2 −1 −2 52 31. y 2 1 Relations and Functions 32. y 2 1 −2 −1 1 2 x −2 −1 1 2 x −1 −2 −1 −2 In Exercises 33 - 47, determine whether or not the equation represents y as a function of x. 33. y = x3 − x 36. x2 − y2 = 1 39. x = y2 + 4 42. y = √ 4 − x2 34. y = 37. y = √ x − 2 x x2 − 9 40. y = x2 + 4 43. x2 − y2 = 4 45. 2x + 3y = 4 46. 2xy = 4 35. x3y = −4 38. x = −6 41. x2 + y2 = 4 44. x3 + y3 = 4 47. x2 = y2 48. Explain why the population P of Sasquatch in a given area is a function of time t. What would be the range of this function? 49. Explain why the relation between your classmates and their email addresses may not be a function. What about phone numbers and Social Security Numbers? The process given in Example 1.3.5 for determining whether an equation of a relation represents y as a function of x breaks down if we cannot solve the equation for y in terms of x. However, that does not prevent us from proving that an equation fails to represent y as a function of x. What we really need is two points with the same x-coordinate and different y-coordinates which both satisfy the equation so that the graph of the relation would fail the Vertical Line |
Test 1.1. Discuss with your classmates how you might find such points for the relations given in Exercises 50 - 53. 50. x3 + y3 − 3xy = 0 52. y2 = x3 + 3x2 51. x4 = x2 + y2 53. (x2 + y2)2 = x3 + y3 1.3 Introduction to Functions 53 1.3.2 Answers 1. Function 2. Not a function domain = {−3, −2, −1, 0, 1, 2,3} range = {0, 1, 4, 9} 3. Function domain = {−7, −3, 3, 4, 5, 6} range = {0, 4, 5, 6, 9} 4. Function domain = {1, 4, 9, 16, 25, 36,...} = {x | x is a perfect square} range = {2, 4, 6, 8, 10, 12,...} = {y | y is a positive even integer} 5. Not a function 6. Function domain = {x | x is irrational} range = {1} 7. Function 8. Function domain = {x|x = 2n for some whole number n} range = {y | y is any whole number} domain = {x | x is any integer} range = y | y = n2 for some integer n 9. Not a function 10. Function 11. Function domain = (−∞, ∞) range = [0, ∞) domain = [−2, 4), range = {3} 12. Not a function 13. Function 14. Not a function domain = {−4, −3, −2, −1, 0, 1} range = {−1, 0, 1, 2, 3, 4} 15. Function domain = (−∞, ∞) range = [1, ∞) 17. Function domain = [2, ∞) range = [0, ∞) 19. Not a function 16. Not a function 18. Function domain = (−∞, ∞) range = (0, 4] 20. Function domain = [−5, −3) ∪ (−3, 3) range = (−2, −1) ∪ [0, 4) 54 Relations and Functions 21. Function domain = [−2, ∞) range = |
[−3, ∞) 23. Function domain = [−5, 4) range = [−4, 4) 25. Function domain = (−∞, ∞) range = (−∞, 4] 27. Function domain = [−2, ∞) range = (−∞, 3] 29. Function domain = (−∞, 0] ∪ (1, ∞) range = (−∞, 1] ∪ {2} 31. Not a function 22. Not a function 24. Function domain = [0, 3) ∪ (3, 6] range = (−4, −1] ∪ [0, 4] 26. Function domain = (−∞, ∞) range = (−∞, 4] 28. Function domain = (−∞, ∞) range = (−∞, ∞) 30. Function domain = [−3, 3] range = [−2, 2] 32. Function domain = (−∞, ∞) range = {2} 33. Function 34. Function 35. Function 36. Not a function 37. Function 38. Not a function 39. Not a function 40. Function 41. Not a function 42. Function 45. Function 43. Not a function 44. Function 46. Function 47. Not a function 1.4 Function Notation 55 1.4 Function Notation In Definition 1.6, we described a function as a special kind of relation − one in which each xcoordinate is matched with only one y-coordinate. In this section, we focus more on the process by which the x is matched with the y. If we think of the domain of a function as a set of inputs and the range as a set of outputs, we can think of a function f as a process by which each input x is matched with only one output y. Since the output is completely determined by the input x and the process f, we symbolize the output with function notation: ‘f (x)’, read ‘f of x.’ In other words, f (x) is the output which results by applying the process f to the input x. In this case, the parentheses here do not indicate multiplication, as they do elsewhere in Algebra. This can cause confusion if the context is not clear, so you must read carefully. This relationship is typically visualized using a diagram similar to the one below. f x |
Domain (Inputs) y = f (x) Range (Outputs) The value of y is completely dependent on the choice of x. For this reason, x is often called the independent variable, or argument of f, whereas y is often called the dependent variable. As we shall see, the process of a function f is usually described using an algebraic formula. For example, suppose a function f takes a real number and performs the following two steps, in sequence 1. multiply by 3 2. add 4 If we choose 5 as our input, in step 1 we multiply by 3 to get (5)(3) = 15. In step 2, we add 4 to our result from step 1 which yields 15 + 4 = 19. Using function notation, we would write f (5) = 19 to indicate that the result of applying the process f to the input 5 gives the output 19. In general, if we use x for the input, applying step 1 produces 3x. Following with step 2 produces 3x + 4 as our final output. Hence for an input x, we get the output f (x) = 3x + 4. Notice that to check our formula for the case x = 5, we replace the occurrence of x in the formula for f (x) with 5 to get f (5) = 3(5) + 4 = 15 + 4 = 19, as required. 56 Relations and Functions Example 1.4.1. Suppose a function g is described by applying the following steps, in sequence 1. add 4 2. multiply by 3 Determine g(5) and find an expression for g(x). Solution. Starting with 5, step 1 gives 5 + 4 = 9. Continuing with step 2, we get (3)(9) = 27. To find a formula for g(x), we start with our input x. Step 1 produces x + 4. We now wish to multiply this entire quantity by 3, so we use a parentheses: 3(x + 4) = 3x + 12. Hence, g(x) = 3x + 12. We can check our formula by replacing x with 5 to get g(5) = 3(5) + 12 = 15 + 12 = 27. Most of the functions we will encounter in College Algebra will be described using formulas like the ones we developed for f (x) and g(x) above. Evaluating formulas using this function notation is a |
key skill for success in this and many other Math courses. Example 1.4.2. Let f (x) = −x2 + 3x + 4 1. Find and simplify the following. (a) f (−1), f (0), f (2) (b) f (2x), 2f (x) (c) f (x + 2), f (x) + 2, f (x) + f (2) 2. Solve f (x) = 4. Solution. 1. (a) To find f (−1), we replace every occurrence of x in the expression f (x) with −1 f (−1) = −(−1)2 + 3(−1) + 4 = −(1) + (−3) + 4 = 0 Similarly, f (0) = −(0)2 + 3(0) + 4 = 4, and f (2) = −(2)2 + 3(2) + 4 = −4 + 6 + 4 = 6. (b) To find f (2x), we replace every occurrence of x with the quantity 2x f (2x) = −(2x)2 + 3(2x) + 4 = −(4x2) + (6x) + 4 = −4x2 + 6x + 4 The expression 2f (x) means we multiply the expression f (x) by 2 2f (x) = 2 −x2 + 3x + 4 = −2x2 + 6x + 8 1.4 Function Notation 57 (c) To find f (x + 2), we replace every occurrence of x with the quantity x + 2 f (x + 2) = −(x + 2)2 + 3(x + 2) + 4 = − x2 + 4x + 4 + (3x + 6) + 4 = −x2 − 4x − 4 + 3x + 6 + 4 = −x2 − x + 6 To find f (x) + 2, we add 2 to the expression for f (x) f (x) + 2 = −x2 + 3x + 4 + 2 = −x2 + 3x + 6 From our work above, we see f (2) = 6 so that f (x) + f (2) = −x2 + 3x + 4 + |
6 = −x2 + 3x + 10 2. Since f (x) = −x2 + 3x + 4, the equation f (x) = 4 is equivalent to −x2 + 3x + 4 = 4. Solving we get −x2 + 3x = 0, or x(−x + 3) = 0. We get x = 0 or x = 3, and we can verify these answers by checking that f (0) = 4 and f (3) = 4. A few notes about Example 1.4.2 are in order. First note the difference between the answers for f (2x) and 2f (x). For f (2x), we are multiplying the input by 2; for 2f (x), we are multiplying the output by 2. As we see, we get entirely different results. Along these lines, note that f (x + 2), f (x) + 2 and f (x) + f (2) are three different expressions as well. Even though function notation uses parentheses, as does multiplication, there is no general ‘distributive property’ of function notation. Finally, note the practice of using parentheses when substituting one algebraic expression into another; we highly recommend this practice as it will reduce careless errors. Suppose now we wish to find r(3) for r(x) = 2x x2−9. Substitution gives r(3) = 2(3) (3)2 − 9 = 6 0, which is undefined. (Why is this, again?) The number 3 is not an allowable input to the function r; in other words, 3 is not in the domain of r. Which other real numbers are forbidden in this formula? We think back to arithmetic. The reason r(3) is undefined is because substitution results in a division by 0. To determine which other numbers result in such a transgression, we set the denominator equal to 0 and solve x2 − 9 = 0 x2 = 9 √ x2 = x = ±3 √ 9 extract square roots 58 Relations and Functions As long as we substitute numbers other than 3 and −3, the expression r(x) is a real number. Hence, we write our domain in interval notation1 as (−∞, −3) ∪ (−3, 3) ∪ ( |
3, ∞). When a formula for a function is given, we assume that the function is valid for all real numbers which make arithmetic sense when substituted into the formula. This set of numbers is often called the implied domain2 of the function. At this stage, there are only two mathematical sins we need to avoid: division by 0 and extracting even roots of negative numbers. The following example illustrates these concepts. Example 1.4.3. Find the domain3 of the following functions. 1. g(x) = √ 4 − 3x 3. f (x) = 2 4x x − 3 1 − 5. r(t) = 4 √ t + 3 6 − Solution. √ 2. h(x) = 5 4 − 3x 4. F (x) = √ 4 2x + 1 x2 − 1 6. I(x) = 3x2 x 1. The potential disaster for g is if the radicand4 is negative. To avoid this, we set 4 − 3x ≥ 0. 3, the expression From this, we get 3x ≤ 4 or x ≤ 4 4 − 3x ≥ 0, and the formula g(x) returns a real number. Our domain is −∞, 4 3 3. What this shows is that as long as x ≤ 4. 2. The formula for h(x) is hauntingly close to that of g(x) with one key difference − whereas the expression for g(x) includes an even indexed root (namely a square root), the formula for h(x) involves an odd indexed root (the fifth root). Since odd roots of real numbers (even negative real numbers) are real numbers, there is no restriction on the inputs to h. Hence, the domain is (−∞, ∞). 3. In the expression for f, there are two denominators. We need to make sure neither of them is 0. To that end, we set each denominator equal to 0 and solve. For the ‘small’ denominator, we get x − 3 = 0 or x = 3. For the ‘large’ denominator 1See the Exercises for Section 1.1. 2or, ‘implicit domain’ 3The word ‘implied’ is, well, implied. 4The ‘radicand’ is the expression ‘inside’ the radical |
. 1.4 Function Notation 59 1 − 4x x − 3 = 0 1 = (1)(x − 3) = 4x x − 3 4x x − 3 x − 3 = 4x −3 = 3x −1 = x (x − 3) clear denominators So we get two real numbers which make denominators 0, namely x = −1 and x = 3. Our domain is all real numbers except −1 and 3: (−∞, −1) ∪ (−1, 3) ∪ (3, ∞). 4. In finding the domain of F, we notice that we have two potentially hazardous issues: not only do we have a denominator, we have a fourth (even-indexed) root. Our strategy is to determine the restrictions imposed by each part and select the real numbers which satisfy both conditions. To satisfy the fourth root, we require 2x + 1 ≥ 0. From this we get 2x ≥ −1 or x ≥ − 1 2. Next, we round up the values of x which could cause trouble in the denominator by setting the denominator equal to 0. We get x2 − 1 = 0, or x = ±1. Hence, in order for a real number x to be in the domain of F, x ≥ − 1 2 but x = ±1. In interval notation, this set is − 1 2, 1 ∪ (1, ∞). 5. Don’t be put off by the ‘t’ here. It is an independent variable representing a real number, just like x does, and is subject to the same restrictions. As in the previous problem, we have double danger here: we have a square root and a denominator. To satisfy the square root, we need a non-negative radicand so we set t + 3 ≥ 0 to get t ≥ −3. Setting the denominator equal to zero gives 6 − t + 3 = 6. Squaring both sides gives t + 3 = 36, or t = 33. Since we squared both sides in the course of solving this equation, we need to check our answer.5 Sure enough, when t = 33, 6 − 36 = 0, so t = 33 will cause problems in the denominator. At last we can find the domain of r: we need t ≥ −3, but t = 33. Our final answer is [ |
−3, 33) ∪ (33, ∞). t + 3 = 0, or. It’s tempting to simplify I(x) = 3x2 x = 3x, and, since there are no longer any denominators, claim that there are no longer any restrictions. However, in simplifying I(x), we are assuming x = 0, since 0 0 is undefined.6 Proceeding as before, we find the domain of I to be all real numbers except 0: (−∞, 0) ∪ (0, ∞). It is worth reiterating the importance of finding the domain of a function before simplifying, as evidenced by the function I in the previous example. Even though the formula I(x) simplifies to 5Do you remember why? Consider squaring both sides to ‘solve’ 6More precisely, the fraction 0 0 is an ‘indeterminant form’. Calculus is required tame such beasts. √ t + 1 = −2. 60 Relations and Functions 3x, it would be inaccurate to write I(x) = 3x without adding the stipulation that x = 0. It would be analogous to not reporting taxable income or some other sin of omission. 1.4.1 Modeling with Functions The importance of Mathematics to our society lies in its value to approximate, or model real-world phenomenon. Whether it be used to predict the high temperature on a given day, determine the hours of daylight on a given day, or predict population trends of various and sundry real and mythical beasts,7 Mathematics is second only to literacy in the importance humanity’s development.8 It is important to keep in mind that anytime Mathematics is used to approximate reality, there are always limitations to the model. For example, suppose grapes are on sale at the local market for $1.50 per pound. Then one pound of grapes costs $1.50, two pounds of grapes cost $3.00, and so forth. Suppose we want to develop a formula which relates the cost of buying grapes to the amount of grapes being purchased. Since these two quantities vary from situation to situation, we assign them variables. Let c denote the cost of the grapes and let g denote the amount of grapes purchased. To find the cost c of the grapes, we multiply the amount of grapes g by the price $1.50 dollars per pound to |
get c = 1.5g In order for the units to be correct in the formula, g must be measured in pounds of grapes in which case the computed value of c is measured in dollars. Since we’re interested in finding the cost c given an amount g, we think of g as the independent variable and c as the dependent variable. Using the language of function notation, we write c(g) = 1.5g where g is the amount of grapes purchased (in pounds) and c(g) is the cost (in dollars). For example, c(5) represents the cost, in dollars, to purchase 5 pounds of grapes. In this case, c(5) = 1.5(5) = 7.5, so it would cost $7.50. If, on the other hand, we wanted to find the amount of grapes we can purchase for $5, we would need to set c(g) = 5 and solve for g. In this case, c(g) = 1.5g, so solving c(g) = 5 is equivalent to solving 1.5g = 5 Doing so gives g = 5 1.5 = 3.3. This means we can purchase exactly 3.3 pounds of grapes for $5. Of course, you would be hard-pressed to buy exactly 3.3 pounds of grapes,9 and this leads us to our next topic of discussion, the applied domain10 of a function. Even though, mathematically, c(g) = 1.5g has no domain restrictions (there are no denominators and no even-indexed radicals), there are certain values of g that don’t make any physical sense. For example, g = −1 corresponds to ‘purchasing’ −1 pounds of grapes.11 Also, unless the ‘local market’ mentioned is the State of California (or some other exporter of grapes), it also doesn’t make much sense for g = 500,000,000, either. So the reality of the situation limits what g can be, and 7See Sections 2.5, 11.1, and 6.5, respectively. 8In Carl’s humble opinion, of course... 9You could get close... within a certain specified margin of error, perhaps. 10or, ‘explicit domain’ 11Maybe this means returning a pound of grapes |
? 1.4 Function Notation 61 these limits determine the applied domain of g. Typically, an applied domain is stated explicitly. In this case, it would be common to see something like c(g) = 1.5g, 0 ≤ g ≤ 100, meaning the number of pounds of grapes purchased is limited from 0 up to 100. The upper bound here, 100 may represent the inventory of the market, or some other limit as set by local policy or law. Even with this restriction, our model has its limitations. As we saw above, it is virtually impossible to buy exactly 3.3 pounds of grapes so that our cost is exactly $5. In this case, being sensible shoppers, we would most likely ‘round down’ and purchase 3 pounds of grapes or however close the market scale can read to 3.3 without being over. It is time for a more sophisticated example. Example 1.4.4. The height h in feet of a model rocket above the ground t seconds after lift-off is given by h(t) = −5t2 + 100t, 0, if 0 ≤ t ≤ 20 t > 20 if 1. Find and interpret h(10) and h(60). 2. Solve h(t) = 375 and interpret your answers. Solution. 1. We first note that the independent variable here is t, chosen because it represents time. Secondly, the function is broken up into two rules: one formula for values of t between 0 and 20 inclusive, and another for values of t greater than 20. Since t = 10 satisfies the inequality 0 ≤ t ≤ 20, we use the first formula listed, h(t) = −5t2 + 100t, to find h(10). We get h(10) = −5(10)2 + 100(10) = 500. Since t represents the number of seconds since lift-off and h(t) is the height above the ground in feet, the equation h(10) = 500 means that 10 seconds after lift-off, the model rocket is 500 feet above the ground. To find h(60), we note that t = 60 satisfies t > 20, so we use the rule h(t) = 0. This function returns a value of 0 regardless of what value is substituted in for t, so |
h(60) = 0. This means that 60 seconds after lift-off, the rocket is 0 feet above the ground; in other words, a minute after lift-off, the rocket has already returned to Earth. 2. Since the function h is defined in pieces, we need to solve h(t) = 375 in pieces. For 0 ≤ t ≤ 20, h(t) = −5t2 + 100t, so for these values of t, we solve −5t2 + 100t = 375. Rearranging terms, we get 5t2 − 100t + 375 = 0, and factoring gives 5(t − 5)(t − 15) = 0. Our answers are t = 5 and t = 15, and since both of these values of t lie between 0 and 20, we keep both solutions. For t > 20, h(t) = 0, and in this case, there are no solutions to 0 = 375. In terms of the model rocket, solving h(t) = 375 corresponds to finding when, if ever, the rocket reaches 375 feet above the ground. Our two answers, t = 5 and t = 15 correspond to the rocket reaching this altitude twice – once 5 seconds after launch, and again 15 seconds after launch.12 12What goes up... 62 Relations and Functions The type of function in the previous example is called a piecewise-defined function, or ‘piecewise’ function for short. Many real-world phenomena, income tax formulas13 for example, are modeled by such functions. By the way, if we wanted to avoid using a piecewise function in Example 1.4.4, we could have used h(t) = −5t2 + 100t on the explicit domain 0 ≤ t ≤ 20 because after 20 seconds, the rocket is on the ground and stops moving. In many cases, though, piecewise functions are your only choice, so it’s best to understand them well. Mathematical modeling is not a one-section topic. It’s not even a one-course topic as is evidenced by undergraduate and graduate courses in mathematical modeling being offered at many universities. Thus our goal in this section cannot possibly be to tell you the whole story. What we can do is get you started. As we study new classes of functions, we will see what phenomena they can be used |
to model. In that respect, mathematical modeling cannot be a topic in a book, but rather, must be a theme of the book. For now, we have you explore some very basic models in the Exercises because you need to crawl to walk to run. As we learn more about functions, we’ll help you build your own models and get you on your way to applying Mathematics to your world. 13See the Internal Revenue Service’s website 1.4 Function Notation 1.4.2 Exercises 63 In Exercises 1 - 10, find an expression for f (x) and state its domain. 1. f is a function that takes a real number x and performs the following three steps in the order given: (1) multiply by 2; (2) add 3; (3) divide by 4. 2. f is a function that takes a real number x and performs the following three steps in the order given: (1) add 3; (2) multiply by 2; (3) divide by 4. 3. f is a function that takes a real number x and performs the following three steps in the order given: (1) divide by 4; (2) add 3; (3) multiply by 2. 4. f is a function that takes a real number x and performs the following three steps in the order given: (1) multiply by 2; (2) add 3; (3) take the square root. 5. f is a function that takes a real number x and performs the following three steps in the order given: (1) add 3; (2) multiply by 2; (3) take the square root. 6. f is a function that takes a real number x and performs the following three steps in the order given: (1) add 3; (2) take the square root; (3) multiply by 2. 7. f is a function that takes a real number x and performs the following three steps in the order given: (1) take the square root; (2) subtract 13; (3) make the quantity the denominator of a fraction with numerator 4. 8. f is a function that takes a real number x and performs the following three steps in the order given: (1) subtract 13; (2) take the square root; (3) make the quantity the denominator of a fraction with numerator 4. 9. f is a function that takes a |
real number x and performs the following three steps in the order given: (1) take the square root; (2) make the quantity the denominator of a fraction with numerator 4; (3) subtract 13. 10. f is a function that takes a real number x and performs the following three steps in the order given: (1) make the quantity the denominator of a fraction with numerator 4; (2) take the square root; (3) subtract 13. In Exercises 11 - 18, use the given function f to find and simplify the following: f (3) f (4x) f (x − 4) f (−1) 4f (x) f (x) − 4 f 3 2 f (−x) f x2 64 Relations and Functions 11. f (x) = 2x + 1 13. f (x) = 2 − x2 15. f (x) = x x − 1 17. f (x) = 6 12. f (x) = 3 − 4x 14. f (x) = x2 − 3x + 2 16. f (x) = 2 x3 18. f (x) = 0 In Exercises 19 - 26, use the given function f to find and simplify the following: f (2) 2f (a) f 2 a 19. f (x) = 2x − 5 21. f (x) = 2x2 − 1 2x + 1 23. f (x) = 25. f (x) = √ x 2 f (−2) f (a + 2) f (a) 2 f (2a) f (a) + f (2) f (a + h) 20. f (x) = 5 − 2x 22. f (x) = 3x2 + 3x − 2 24. f (x) = 117 26. f (x) = 2 x In Exercises 27 - 34, use the given function f to find f (0) and solve f (x) = 0 27. f (x) = 2x − 1 29. f (x) = 2x2 − 6 31. f (x) = √ x + 4 28. f (x) = 3 − 2 5 x 30. f (x) = x2 − x − 12 √ 32. f (x) = 1 − 2 |
x 33. f (x) = 3 4 − x 35. Let f (x) = (a) f (−4) (d) f (3.001) 34. f (x) = 3x2 − 12x 4 − x2 √ x + 5 9 − x2 −x + 5 x ≤ −3 if if −3 < x ≤ 3 x > 3 if Compute the following function values. (b) f (−3) (e) f (−3.001) (c) f (3) (f) f (2) 1.4 Function Notation 65 36. Let f (x) = √ x2 1 − x2 x x ≤ −1 if if −1 < x ≤ 1 x > 1 if Compute the following function values. (a) f (4) (d) f (0) (b) f (−3) (e) f (−1) (c) f (1) (f) f (−0.999) In Exercises 37 - 62, find the (implied) domain of the function. 37. f (x) = x4 − 13x3 + 56x2 − 19 38. f (x) = x2 + 4 39. f (x) = 41. f (x) = 43. f (x) = 45. f (x) = x − 2 x + 1 2x x2 + 3 x + 4 x2 − 36 √ 3 − x 47. f (x) = 9x √ x + 3 49. f (x) = √ 6x − 2 √ 51. f (x) = 3 6x − 2 53. f (x) = √ 6x − 2 x2 − 36 55. s(t) = t t − 8 57. b(θ) = 59. α(y 61. T (t) = √ t − 8 5 − t 40. f (x) = 3x x2 + x − 2 42. f (x) = 44. f (x) = 46. f (x) = 48. f (x) = 2x x2 − 3 x − 2 x − 2 √ 2x + 5 √ 7 − x x2 + 1 |
50. f (x) = √ 6 6x − 2 6 √ 6x − 2 52. f (x) = 54. f (x) = 56. Q(r) = 58. A(x) = 60. g(v) = 62. u(w) = 4 − √ 3 6x − 2 x2 + 36 √ r r − 8 √ 1 4 − 1 v2 66 Relations and Functions 63. The area A enclosed by a square, in square inches, is a function of the length of one of its sides x, when measured in inches. This relation is expressed by the formula A(x) = x2 for x > 0. Find A(3) and solve A(x) = 36. Interpret your answers to each. Why is x restricted to x > 0? 64. The area A enclosed by a circle, in square meters, is a function of its radius r, when measured in meters. This relation is expressed by the formula A(r) = πr2 for r > 0. Find A(2) and solve A(r) = 16π. Interpret your answers to each. Why is r restricted to r > 0? 65. The volume V enclosed by a cube, in cubic centimeters, is a function of the length of one of its sides x, when measured in centimeters. This relation is expressed by the formula V (x) = x3 for x > 0. Find V (5) and solve V (x) = 27. Interpret your answers to each. Why is x restricted to x > 0? 66. The volume V enclosed by a sphere, in cubic feet, is a function of the radius of the sphere r, 3 r3 for r > 0. Find when measured in feet. This relation is expressed by the formula V (r) = 4π V (3) and solve V (r) = 32π 3. Interpret your answers to each. Why is r restricted to r > 0? 67. The height of an object dropped from the roof of an eight story building is modeled by: h(t) = −16t2 + 64, 0 ≤ t ≤ 2. Here, h is the height of the object off the ground, in feet, t seconds after the object is dropped. Find h(0) and solve h(t) = 0. Interpret your answers to each. Why is t restricted to 0 ≤ t ≤ 2? |
68. The temperature T in degrees Fahrenheit t hours after 6 AM is given by T (t) = − 1 2 t2 + 8t + 3 for 0 ≤ t ≤ 12. Find and interpret T (0), T (6) and T (12). 69. The function C(x) = x2 − 10x + 27 models the cost, in hundreds of dollars, to produce x thousand pens. Find and interpret C(0), C(2) and C(5). 70. Using data from the Bureau of Transportation Statistics, the average fuel economy F in miles per gallon for passenger cars in the US can be modeled by F (t) = −0.0076t2 + 0.45t + 16, 0 ≤ t ≤ 28, where t is the number of years since 1980. Use your calculator to find F (0), F (14) and F (28). Round your answers to two decimal places and interpret your answers to each. 71. The population of Sasquatch in Portage County can be modeled by the function P (t) = 150t t+15, where t represents the number of years since 1803. Find and interpret P (0) and P (205). Discuss with your classmates what the applied domain and range of P should be. 72. For n copies of the book Me and my Sasquatch, a print on-demand company charges C(n) dollars, where C(n) is determined by the formula C(n) = 15n 13.50n 12n if if if 1 ≤ n ≤ 25 25 < n ≤ 50 n > 50 (a) Find and interpret C(20). 1.4 Function Notation 67 (b) How much does it cost to order 50 copies of the book? What about 51 copies? (c) Your answer to 72b should get you thinking. Suppose a bookstore estimates it will sell 50 copies of the book. How many books can, in fact, be ordered for the same price as those 50 copies? (Round your answer to a whole number of books.) 73. An on-line comic book retailer charges shipping costs according to the following formula S(n) = 1.5n + 2.5 0 if if 1 ≤ n ≤ 14 n ≥ 15 where n is the number of comic books purchased and S(n) is the shipping cost in dollars. (a) What |
is the cost to ship 10 comic books? (b) What is the significance of the formula S(n) = 0 for n ≥ 15? 74. The cost C (in dollars) to talk m minutes a month on a mobile phone plan is modeled by C(m) = 25 25 + 0.1(m − 1000) 0 ≤ m ≤ 1000 if if m > 1000 (a) How much does it cost to talk 750 minutes per month with this plan? (b) How much does it cost to talk 20 hours a month with this plan? (c) Explain the terms of the plan verbally. 75. In Section 1.1.1 we defined the set of integers as Z = {..., −3, −2, −1, 0, 1, 2, 3,...}.14 The greatest integer of x, denoted by x, is defined to be the largest integer k with k ≤ x. (a) Find 0.785, 117, −2.001, and π + 6 (b) Discuss with your classmates how x may be described as a piecewise defined function. HINT: There are infinitely many pieces! (c) Is a + b = a + b always true? What if a or b is an integer? Test some values, make a conjecture, and explain your result. 76. We have through our examples tried to convince you that, in general, f (a + b) = f (a) + f (b). It has been our experience that students refuse to believe us so we’ll try again with a different approach. With the help of your classmates, find a function f for which the following properties are always true. (a) f (0) = f (−1 + 1) = f (−1) + f (1) 14The use of the letter Z for the integers is ostensibly because the German word zahlen means ‘to count.’ 68 Relations and Functions (b) f (5) = f (2 + 3) = f (2) + f (3) (c) f (−6) = f (0 − 6) = f (0) − f (6) (d) f (a + b) = f (a) + f (b) regardless of what two numbers |
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