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Department
Administrative Studies
Course
ADMS 3330
Professor
Michael Gadsden
Semester
Fall

Description
CHAPTER 16 SIMPLE LINEAR REGRESSION AND CORRELATION SECTIONS 1 - 2 MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer. 1. The regression line= 3 + 2x has been fitted to the data points (4, 8), (2, 5), and (1, 2). The sum of the squared residuals will be: a. 7 b. 15 c. 8 d. 22 ANSWER: d 2. If an estimated regression line has a y-intercept of 10 and a slope of 4, then when x = 2 the actual value of y is: a. 18 b. 15 c. 14 d. unknown ANSWER: d 3. Given the least squares regressio= 5 –2x: a. the relationship between x and y is positive b. the relationship between x and y is negative 89 90 Chapter Sixteen c. as x increases, so does y d. as x decreases, so does y ANSWER: b 4. A regression analysis between weight (y in pounds) and height (x in inches) resulted in the following least squares line:y = 120 + 5x. This implies that if the height is increased by 1 inch, the weight, on average, is expected to: a. increase by 1 pound b. decrease by 1 pound c. increase by 5 pounds d. increase by 24 pounds ANSWER: c 5. A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line: y = 75 +6x. This implies that if advertising is $800, then the predicted amount of sales (in dollars) is: a. $4875 b. $123,000 c. $487,500 d. $12,300 ANSWER: b 6. A regression analysis between sales (in $1000) and advertising (in $) resulted in the following least squares line:yˆ = 80,000 + 5x. This implies that an: a. increase of $1 in advertising is expected, on average, to result in an increase of $5 in sales b. increase of $5 in advertising is expected, on average, to result in an increase of $5,000 in sales c. increase of $1 in advertising is expected, on average, to result in an increase of $80,005 in sales d. increase of $1 in advertising is expected, on average, to result in an increase of $5,000 in sales ANSWER: d 7. Which of the following techniques is used to predict the value of one variable on the basis of other variables? a. Correlation analysis b. Coefficient of correlation c. Covariance d. Regression analysis ANSWER: d 8. The residual is defined as the difference between: a. the actual value of y and the estimated value of y b. the actual value of x and the estimated value of x Simple Linear Regression and Correlation 91 c. the actual value of y and the estimated value of x d. the actual value of x and the estimated value of y ANSWER: a 9. In the simple linear regression model, the y-intercept represents the: a. change in y per unit change in x b. change in x per unit change in y c. value of y when x = 0 d. value of x when y = 0 ANSWER: c 10. In the first order linear regression model, the population parameters of the y-intercept and the slope are estimated respectively, by: b a. 0 and b1 b. b 0 and β 1 β c. 0 and b 1 d. β0 and β 1 ANSWER: a 11. In the simple linear regression model, the slope represents the: a. value of y when x = 0 b. average change in y per unit change in x c. value of x when y = 0 d. average change in x per unit change in y ANSWER: b 12. In regression analysis, the residuals represent the: a. difference between the actual y values and their predicted values b. difference between the actual x values and their predicted values c. square root of the slope of the regression line d. change in y per unit change in x ANSWER: a 13. In the first-order linear regression model, the population parameters of the y-intercept and the slope are, respectively, a. b 0 and b1 b. b 0 and β 1 c. β0 and b 1 d. β0 and β 1 ANSWER: d 14. In a simple linear regression problem, the following statistics are calculated from a ( x − x )( y − y ) x y sample of 10 observations: ∑ = 2250, sx = 10, ∑ = 50, ∑ = 75. The least squares estimates of the slope and y-intercept are respectively: a. 1.5 and 0.5 92 Chapter Sixteen b. 2.5 and 1.5 c. 1.5 and 2.5 d. 2.5 and –5.0 ANSWER: d 15. If a simple linear regression model has no y-intercept, then: a. all values of x are zero b. all values of y are zero c. when y = 0 so does x d. when x = 0 so does y ANSWER: d 16. In the least squares regression lin= 3 - 2x, the predicted value of y equals: a. 1.0 when x = -1.0 b. 2.0 when x = 1.0 c. 2.0 when x = -1.0 d. 1.0 when x = 1.0 ANSWER: d 17. The least squares method for determining the best fit minimizes: a. total variation in the dependent variable b. sum of squares for error c. sum of squares for regression d. All of the above ANSWER: b 18. What do we mean when we say that a simple linear regression model is “statistically” useful? a. All the statistics computed from the sample make sense b. The model is an excellent predictor of y c. The model is “practically” useful for predicting y y d. The model is a better predictor of y than the sample ANSWER: d Simple Linear Regression and Correlation 93 TRUE / FALSE QUESTIONS 19. An inverse relationship between an independent variable x and a dependent variably y means that as x increases, y decreases, and vice versa. ANSWER: T 20. A direct relationship between an independent variable x and a dependent variably y means that the variables x and y increase or decrease together. ANSWER: T 21. Another name for the residual term in a regression equation is random error. ANSWER: T ˆ y 22. A simple linear regression equation is given by y = 5.25+ 3.8x. The point estimate of when x = 4 is 20.45. ANSWER: T 23. The vertical spread of the data points about the regression line is measured by the y- intercept. ANSWER: F 24. The method of least squares requires that the sum of the squared deviations between actual y values in the scatter diagram and y values predicted by the regression line be minimized. ANSWER: T 25. A regression analysis between sales (in $1000) and advertising (in $) resulted in the following least squares line: y = 60 + 5x. This implies that an increase of $1 in advertising is expected to result in an increase of $65 in sales. ANSWER: F 26. A regression analysis between weight ( y in pounds) and height (x in inches) resulted in the following least squares line: y = 135 + 6 . This implies that if the height is increased by 1 inch, the weight is expected to increase by an average of 6 pounds. ANSWER: T 27. The residual ri is defined as the difference between the actual valuei and the estimated value yi. ANSWER: T 28. The regression line y = 2 + 3x has been fitted to the data points (4,11), (2,7), and (1,5). The sum of squares for error will be 10.0. ANSWER: T 29. A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line:y = 77 +8x. This implies that if advertising is $600, then the predicted amount of sales (in dollars) is $125,000. 94 Chapter Sixteen ANSWER: T 30. The residuals are observations of the error variab. Consequently, the minimized sum of squared deviations is called the sum of squares for error, denoted SSE. ANSWER: T 31. Statisticians have shown that sample y -intercep0 and sample slope coefficient 1are unbiased estimators of the population regression parameterand β , respectively. 0 1 ANSWER: T 32. If cov(x, y) = 7.5075 and2= 3.5, then the sample slope coefficient is 2.145. x ANSWER: T 33. The first – order linear model is sometimes called the simple linear regression model. ANSWER: T 34. To create a deterministic model, we start with a probabilistic model that approximates the relationship we want to model. ANSWER: F 35. The residual represents the discrepancy between the observed dependent variable and its Predicted or estimated average value. ANSWER: T Simple Linear Regression and Correlation 95 STATISTICAL CONCEPTS & APPLIED QUESTIONS FOR QUESTIONS 36 AND 37, USE THE FOLLOWING NARRATIVE: Narrative: Car Speed and Gas Mileage An economist wanted to analyze the relationship between the speed of a car (x) and its gas mileage (y). As an experiment a car is operated at several different speeds and for each speed the gas mileage is measured. These data are shown below. Speed 25 35 45 50 60 65 70 Gas Mileage 40 39 37 33 30 27 25 36. {Car Speed and Gas Mileage Narrative} Determine the least squares regression line. ANSWER: y = 50.6563 – 0.3531x 37. {Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph. ANSWER: When x = 70, y = 25.9393 mpg 38. The following 10 observations of variables x and y were collected. x 1 2 3 4 5 6 7 8 9 10 y 25 22 21 19 14 15 12 10 6 2 Find the least squares regression line, and the estimated value of y when x = 3 ANSWER: y = 27.733-2.389x. When x = 3,y= 20.566 39. A scatter diagram includes the following data points: x 3 2 5 4 5 y 8 6 12 10 14 Two regression models are proposed: (1) y = 1.2 + 2.5x, and (2) y = 5.5 + 4.0x. Using the least squares method, which of these regression models provide the better fit to the data? Why? ANSWER: SSE = 4.95 and 593.25 for models 1 and 2, respectively. Therefore, model (1) fits the data better than model (2). 40. Consider the following data values of variables x and y. 96 Chapter Sixteen x 2 4 6 8 10 13 y 7 11 17 21 27 36 a. Determine the least squares regression line. b. Find the predicted value of y for x = 9. c. What does the value of the slope of the regression line tell you? ANSWER: a. y = 0.934 + 2.637x b. When x = 9, y= 24.667 c. If x increases by one unit, y on average will increase by 2.637. FOR QUESTIONS 41 THROUGH 45, USE THE FOLLOWING NARRATIVE: Narrative: Sunshine and Skin Cancer A medical statistician wanted to examine the relationship between the amount of sunshine (x) in hours, and incidence of skin cancer (y). As an experiment he found the number of skin cancers detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Skin Cancer per 100,000 7 11 9 12 15 10 7 5 41. {Sunshine and Skin Cancer Narrative} Determine the least squares regression line. ANSWER: y = -1.115 + 1.846x 42. {Sunshine and Skin Cancer Narrative} Draw a scatter diagram of the data and plot the least squares regression line on it. ANSWER: Average Daily Sunshine Line Fit Plot 16 12 Skin Cancer r n Predicted Skin Cancer C 8 i S Linear (Predicted Skin 4 Cancer) 0 0 2 4 6 8 10 Average Daily Sunshine Simple Linear Regression and Correlation 97 43. {Sunshine and Skin Cancer Narrative} Estimate the number of skin cancer per 100,000 of population for 6 hours of sunshine. ANSWER: ˆ When x = 6, y = 9.961 44. {Sunshine and Skin Cancer Narrative} What does the value of the slope of the regression line tell you? ANSWER: If the amount of sunshine x increases by one hour, the amount of skin cancer y increases by an average of 1.846 per 100,000 of population. 45. {Sunshine and Skin Cancer Narrative} Calculate the residual corresponding to the pair (x, y) = (8, 15). ANSWER: e = y -ˆ= 15 – 13.653 = 1.347 FOR QUESTIONS 46 THROUGH 49, USE THE FOLLOWING NARRATIVE: NARRATIVE: Sales and Experience The general manager of a chain of furniture stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month’s sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. Salesperson Years of Experience Sales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 6 5 14 7 12 20 8 7 17 9 20 30 10 15 25 98 Chapter Sixteen 46. {Sales and Experience Narrative} Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate. ANSWER: Scatter Diagram 35 30 25 l 20 S 15 10 5 0 0 5 10 15 20 25 Years of Experience It appears that a linear model is appropriate. 47. {Sales and Experience Narrative} Determine the least squares regression line. ANSWER: y = 8.63 + 1.0817x 48. {Sales and Experience Narrative} Interpret the value of the slope of the regression line. ANSWER: For each additional year of experience, monthly sales of a salesperson increase by an average of $1,081.7. 49. {Sales and Experience Narrative} Estimate the monthly sales for a salesperson with 16 years of experience. ANSWER: yˆ When x =16, = 25.94 FOR QUESTIONS 50 THROUGH 53, USE THE FOLLOWING NARRATIVE: Narrative: Income and Education A professor of economics wants to study the relationship between income (y in $1000s) and education (x in years). A random sample eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 Simple Linear Regression and Correlation 99 50. {Income and Education Narrative} Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate. ANSWER: Scatter Diagram 60 e50 c I40 30 6 8 10 12 14 16 18 Years of Education It appears that a linear model is appropriate. 51. {Income and Education Narrative} Determine the least squares regression line. ANSWER: y = 10.6165 + 2.9098x 52. {Income and Education Narrative} Interpret the value of the slope of the regression line. ANSWER: For each additional year of education, the income increases by an average of $2,909.80. 53. {Income and Education Narrative} Estimate the income of an individual with 15 years of education. ANSWER: When x = 15, y = 54.264 (in $1000s) or $54,264.0 FOR QUESTIONS 54 THROUGH 57, USE THE FOLLOWING NARRATIVE: Narrative: Game Winnings and Education An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief she gathers data about the last eight winners of her favorite game show. She records their winnings in dollars and the number of years of education. The results are as follows. 100 Chapter Sixteen Contestant Years of Education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 6 16 300 7 13 650 8 14 400 54. {Game Winnings and Education Narrative} Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate. ANSWER: Scatter Diagram 1000 800 s i 600 i W 400 200 8 10 12 14 16 18 Years of Education It appears that a linear model is appropriate. 55. {Game Winnings and Education Narrative} Determine the least squares regression line. ANSWER: y = 1735 – 89.1667x 56. {Game Winnings and Education Narrative} Interpret the value of the slope of the regression line. ANSWER: For each additional year of education a contestant has, his or her winnings on TV game shows decreases by an average of approximately $89.20. Simple Linear Regression and Correlation 101 57. {Game Winnings and Education Narrative} Estimate the game winnings for a contestant with 15 years of education. ANSWER: When x = 15, yˆ= $397.50 FOR QUESTIONS 58 THROUGH 61, USE THE FOLLOWING NARRATIVE: Narrative: Movie Revenues A financier whose specialty is investing in movie productions has observed that, in general, movies with “big-name” stars seem to generate more revenue than those movies whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid performers in the movie for ten recently released movies. Movie Cost of Two Highest Gross Revenue Paid Performers 1 5.3 48 2 7.2 65 3 1.3 18 4 1.8 20 5 3.5 31 6 2.6 26 7 8.0 73 8 2.4 23 9 4.5 39 10 6.7 58 58. {Movie Revenues Narrative} Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate. ANSWER: It appears that a linear model is appropriate. Scatter Diagram e 70 n 60 e 50 R 40 o 20 G 10 0 0 2 4 6 8 10 Payment to Top Two Stars 102 Chapter Sixteen 59. {Movie Revenues Narrative} Determine the least squares regression line. ANSWER: y = 4.225 + 8.285x 60. {Movie Revenues Narrative} Interpret the value of the slope of the regression line. ANSWER: For each million dollar paid to the two highest paid performers, the gross revenue of the movie increases by an average of $8.285 million. 61. {Movie Revenues Narrative} Estimate the gross revenue of a movie if the two highest paid performers received 6 million dollars. ANSWER: When x = 6,yˆ= $53.935 million FOR QUESTIONS 62 THROUGH 65, USE THE FOLLOWING NARRATIVE: NARRATIVE: Cost of Books The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyze the claim, a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected books. These data are listed below. Book Number of Pages Selling Price ($) 1 844 55 2 727 50 3 360 35 4 915 60 5 295 30 6 706 50 7 410 40 8 905 53 9 1058 65 10 865 54 11 677 42 12 912 58 62. {Cost of Books Narrative} Determine the least squares regression line. ANSWER: y = 19.387 + .0414x Simple Linear Regression and Correlation 103 63. {Cost of Books Narrative} Draw a scatter diagram of the data and plot the least squares regression line on it. ANSWER: Number of Pages Line Fit Plot 70 60 Selling Price i 50 P 40 Predicted Selling Price l 30 e 20 S 10 Linear (Predicted Selling Price) 0 0 200 400 600 800 1000 1200 Number of Pages 64. {Cost of Books Narrative} Interpret the value of the slope of the regression line. ANSWER: For every additional page, the price of a book increases by an average of about 4 cents. 65. {Cost of Books Narrative} Estimate the selling price for a 650 pages book. ANSWER: When x = 650, yˆ= $46.037 FOR QUESTIONS 66 THROUGH 68, USE THE FOLLOWING NARRATIVE: Narrative: Accidents and Precipitation A statistician investigating the relationship between the amount of precipitation (in inches) and the number of automobile accidents gathered data for 10 randomly selected days. The results Day Precipitation Number of Accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 8 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 104 Chapter Sixteen 66. {Accidents and Precipitation Narrative} Find the least squares regression line. ANSWER: y = 2.3704 + 34.864x 67. {Accidents and Precipitation Narrative} Estimate the number of accidents in a day with 0.25 inches of precipitation ANSWER: When x = 0.25, y= 11.08 ≈ 11 accidents 68. {Accidents and Precipitation Narrative} What does the slope of the least squares regression line tell you? ANSWER: For each additional inch of precipitation, the number of accidents on average increases by 34.864 (about 35 accidents). FOR QUESTIONS 69 THROUGH 73, USETHE FOLLOWING NARRATIVE: Narrative: Willie Nelson Concert At a recent Willie Nelson concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year. The following data were collected: Age 62 57 40 49 67 54 43 65 54 41 Number of Concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number of Concerts 3 2 4 5 4 5 4 2 1 3 An Excel output follows : SUMMARY OUTPUT DESCRIPTIVE STATISTICS Regression Statistics Age Concerts Multiple R 0.80203 Mean 53 Mean 3.65 R Square 0.64326 Standard Error 2.1849 Standard Error 0.3424 Adjusted R Square 0.62344 Standard Deviation 9.7711 Standard Deviation 1.5313 Standard Error 0.93965 Sample Variance 95.4737 Sample Variance 2.3447 Observations 20 Count 20 Count 20 SPEARMAN RANK CORRELATION COEFFICIENT=0.8306 ANOVA df SS MS F Significance F Regression 1 28.65711 28.65711 32.45653 2.1082E-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -3.01152 1.18802 -2.53491 0.02074 -5.50746 -0.5156 Age 0.12569 0.02206 5.69706 0.00002 0.07934 0.1720 Simple Linear Regression and Correlation 105 69. {Willie Nelson Concert Narrative} Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate to describe the relationship between the age and number of concerts attended by the respondents. ANSWER: Scatter Diagram 7 r 6 c 5 o C 4 o 3 r b 2 m 1 N 0 30 35 40 45 50 55 60 65 70 75 Age A linear model appears to be appropriate to describe the relationship between the age and number of concerts attended by the respondents. 70. {Willie Nelson Concert Narrative} Determine the least squares regression line. ANSWER: y = -3.0115 + 0.1257x 71. {Willie Nelson Concert Narrative} Plot the least squares regression line on the scatter diagram. ANSWER: Scatter Diagram with Trendline 7 r 6 e n 5 C 4 f r 3 b 2 m u 1 N 0 30 35 40 45 50 55 60 65 70 75 Age 106 Chapter Sixteen 72. {Willie Nelson Concert Narrative} Interpret the value of the slope of the regression line. ANSWER: For every additional year of age, the number of concerts attended increases on average by 0.1257. Equivalently we may say, for every additional 20 years of age, the number of concerts attended increases on average by about 2.50. 73. {Willie Nelson Concert Narrative} Estimate the number of Willie Nelson concerts attended by a 64 year old person. ANSWER: When x = 64, yˆ= 5.03 (about 5 concerts) FOR QUESTIONS 74 THROUGH 77, USE THE FOLLOWING NARRATIVE: Narrative: Oil Quality and Price Quality of oil is measured in API gravity degrees – the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel. Oil degrees API Price per barrel (in $) 27.0 12.02 28.5 12.04 30.8 12.32 31.3 12.27 31.9 12.49 34.5 12.70 34.0 12.80 34.7 13.00 37.0 13.00 41.0 13.17 41.0 13.19 38.8 13.22 39.3 13.27 A partial Minitab output follows: Descriptive Statistics Variable N Mean StDev SE Mean Degrees 13 34.60 4.613 1.280 Price 13 12.730 0.457 0.127 Covariances Degrees Price Degrees 21.281667 Price 2.026750 0.208833 Regression Analysis Simple Linear Regression and Correlation 107 Predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7% Analysis of Variance Source DF SS MS F P Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 74. {Oil Quality and Price Narrative} Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate to describe the relationship between the quality of oil and price per barrel. ANSWER: Scatter Diagram 13.4 13.2 13 12.8 i12.6 P 12.4 12.2 12 11.8 20 25 30 35 40 45 Degrees A linear model appears to be appropriate to describe the relationship between the quality of oil and price per barrel. 75. {Oil Quality and Price Narrative} Determine the least squares regression line. ANSWER: y = 9.4349 + 0.095235x 76. {Oil Quality and Price Narrative} Plot the least squares regression line on the scatter diagram. 108 Chapter Sixteen ANSWER: Scatter Diagram 13.6 13.4 13.2 13 e 12.8 r P 12.6 12.4 12.2 12 11.8 20 25 30 35 40 45 Degrees 77. {Oil Quality and Price Narrative} Interpret the value of the slope of the regression line. ANSWER: For every additional API gravity degree, the price of oil per barrel increases by an average of 9.52 cents. Simple Linear Regression and Correlation 109 SECTIONS 3 - 4 MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer. 78. In a simple linear regression problem, the following sum of squares are produced: ∑ (yi− y) = 200 , ∑ (y i y i = 50 , and ∑ (yi− y) = 150 . The percentage of the variation in y that is explained by the variation in x is: a. 25% b. 75% c. 33% d. 50% ANSWER: b 79. In simple linear regression, most often we perform a two-tail test of the population slope β 1 to determine whether there is sufficient evidence to infer that a linear relationship exists. The null hypothesis is stated as: a. H 0 β 1 0 b. H 0 β 1 b 1 c. H : β = r 0 1 d. H 0 β 1 ρ s ANSWER: a 80. Testing whether the slope of the population regression line could be zero is equivalent to testing whether the: a. sample coefficient of correlation could be zero b. standard error of estimate could be zero c. population coefficient of correlation could be zero d. sum of squares for error could be zero ANSWER: c 2 2 81. Given that s x 500, sy= 750 , cov (x, y) = 100, and n = 6, the standard error of estimate is: a. 12.247 b. 24.933 c. 30.2076 d. 11.180 ANSWER: c 82. The symbol for the population coefficient of correlation is: a. r 110 Chapter Sixteen b. ρ c. r 2 2 d. ρ ANSWER: b 83. Given that the sum of squares for error is 60 and the sum of squares for regression is 140, then the coefficient of determination is: a. 0.429 b. 0.300 c. 0.700 d. 0.837 ANSWER: c 84. A regression line using 25 observations produced SSR = 118.68 and SSE = 56.32. The standard error of estimate was: a. 2.1788 b. 1.5648 c. 1.5009 d. 2.2716 ANSWER: b 85. The symbol for the sample coefficient of correlation is: a. r ρ a. b. r2 c. ρ 2 ANSWER: a 86. Given the least squares regression line y = -2.48 + 1.63x, and a coefficient of determination of 0.81, the coefficient of correlation is: a. -0.85 b. 0.85 c. -0.90 d. 0.90 ANSWER: d 87. Which value of the coefficient of correlation r indicates a stronger correlation than 0.65? a. 0.55 b. -0.75 c. 0.60 d. -0.45 ANSWER: b 88. If the coefficient of determination is 0.975, then the slope of the regression line: a. must be positive b. must be negative Simple Linear Regression and Correlation 111 c. could be either positive or negative d. None of the above. ANSWER: c 89. In regression analysis, if the coefficient of determination is 1.0, then: a. the sum of squares for error must be 1.0 b. the sum of squares for regression must be 1.0 c. the sum of squares for error must be 0.0 d. the sum of squares for regression must be 0.0 ANSWER: c 90 The sample correlation coefficient between x and y is 0.375. It has been found out that H :ρ = 0 H :ρ < 0 the p– value is 0.744 when testing o against the one-sided alternative 1 . To test theH oρ = 0 against the two-sided alternative 1ρ ≠ 0 at a significance level of 0.193, the p – value is a. 0.372 b. 1.488 c. 0.256 d. 0.512 ANSWER: d 91. Correlation analysis is used to determine: a. the strength of the relationship between x and y b. the least squares estimates of the regression parameters c. the predicted value of y for a given value of x d. the coefficient of determination ANSWER: a 92. If the coefficient of correlation is –0.80 then, the percentage of the variation in y that is explained by the variation in x is: a. 80% b. 64% c. –80% d. –64% ANSWER: b 93. If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be: a. 1.0 b. –1.0 c. either 1.0 or –1.0 d. 0.0 ANSWER: c 94. If the coefficient of correlation is –0.60, then the coefficient of determination is: a. -0.60 b. -0.36 112 Chapter Sixteen c. 0.36 d. 0.40 ANSWER: c 95. In regression analysis, if the coefficient of correlation is –1.0, then: a. the sum of squares for error is –1.0 b. the sum of squares for regression is 1.0 c. the sum of squares for error and sum of squares for regression are equal d. the sum of squares for regression and total variation in y are equal ANSWER: d 96. If the coefficient of correlation between x and y is close to 1.0, this indicates that: a. y causes x to happen b. x causes y to happen c. both (a) and (b) d. there may or may not be any causal relationship between x and y ANSWER: d 97. For the values of the coefficient of determination listed below, which one implies the greatest value of the sum of squares for regression given that the total variation in y is 1800? a. 0.69 b. 0.96 c. 0.58 d. 0.85 ANSWER: b 98. When all the actual and predicted values of y are equal, the standard error of estimate will be: a. 1.0 b. –1.0 c. 0.0 d. 2.0 ANSWER: c 99. Which of the following statistics and procedures can be used to determine whether a linear model should be employed? a. The standard error of estimate b. The coefficient of determination c. The t-test of the slope d. All of the above ANSWER: d 100. In testing the hypotheses: H 0 β 1 0 vs. H 1 β 1 0 , the following statistics are available: n = 10, b 0 = −1.8 , b1= 2.45 , s b = 1.20, and y = 6. The value of the 1 test statistic is: Simple Linear Regression and Correlation 113 a. 2.042 b. 0.306 c. –1.50 d. -0.300 ANSWER: a 101. The standard error of estimate s ε is given by: a. SSE/(n – 2) b. SSE /( n − 2) c. SSE /( n − 2) d. SSE/ n − 2 ANSWER: c 102. If the standard error of estimate s = 20 and n = 10, then the sum of squares for error, ε SSE, is: a. 400 b. 3200 c. 4000 d. 40000 ANSWER: b 103. The smallest value that the standard error of estimatesε can assume is: a. –1 b. 0 c. 1 d. –2 ANSWER: b 2 2 104. If cov(x, y) = 1260, sx= 1600 and sy = 1225 ,then the coefficient of determination is: a. 0.7875 b. 1.0286 c. 0.8100 d. 0.7656 ANSWER: c 105. The standard error of estimate s ε is a measure of the: a. variation of y around the regression line b. variation of x around the regression line c. variation of y around the mean y d. variation of x around the mean x 114 Chapter Sixteen ANSWER: a 106. The Pearson coefficient of correlation r equals 1 when there is no: a. explained variation b. unexplained variation c. y-intercept in the model d. outliers ANSWER: b 107. In regression analysis, the coefficient of determination R 2 measures the amount of variation in y that is: a. caused by the variation in x b. explained by the variation in x c. unexplained by the variation in x d. None of the above ANSWER: b 108. If we are interested in determining whether two variables are linearly related, it is necessary to: a. perform the t-test of the slope 1 ρ b. perform the t-test of the coefficient of correlation c. either (a) or (b) since they are identical d. calculate the standard error of estimate ε ANSWER: c 109. In a regression problem the following pairs of (x,y) are given: (3,1), (3,-1), (3,0), (3,-2) and (3,2). That indicates that the: a. correlation coefficient is –1 b. correlation coefficient is 0 c. correlation coefficient is 1 d. coefficient of determination is between –1 and 1 ANSWER: b 110. In a regression problem, if the coefficient of determination is 0.95, this means that: a. 95% of the y values are positive b. 95% of the variation in y can be explained by the variation in x c. 95% of the x values are equal d. 95% of the variation in x can be explained by the variation in y ANSWER: b 111. The sample correlation coefficient between x and y is 0.375. It has been found out that the p – value is 0.256 when testing H oρ = 0 against the two-sided alternative H1:ρ ≠ 0 . To test H oρ = 0 against the one-sided alternativeH1:ρ > 0 at a significant level of 0.193, the p – value will be equal to a. 0.128 Simple Linear Regression and Correlation 115 b. 0.512 c. 0.744 d. 0.872 ANSWER: a 112. In simple linear regression, which of the following statements indicate no linear relationship between the variables x and y? a. Coefficient of determination is 1.0 b. Coefficient of correlation is 0.0 c. Sum of squares for error is 0.0 d. Sum of squares for regression is relatively large ANSWER: b 113. If the sum of squared residuals is zero, then the: a. coefficient of determination must be 1.0 b. coefficient of correlation must be 1.0 c. coefficient of determination must be 0. 0 d. coefficient of correlation must be 0.0 ANSWER: a 114. In a regression problem, if all the values of the independent variable are equal, then the coefficient of determination must be: a. 1.0 b. 0.5 c. 0.0 d. –1.0 ANSWER: c 115. The standard error of the estimate is a measure of a. total variation of the y variable b. the variation around the sample regression line c. explained variation d. the variation of the x variable ANSWER: b 116. In simple linear regression, the coefficient of correlation r and the least squares estimate b 1 of the population slope 1: a. must be equal b. must have opposite signs c. must have the same sign d. may have opposite signs or the same sign 116 Chapter Sixteen ANSWER: c 2 117. The coefficient of determination R ) tells us a. that the coefficient of correlation is larger than 1 b. whether r has any significance c. that we should not partition the total variation d. the proportion of total variation in y that is explained by x ANSWER: d 118. In performing a regression analysis involving two numerical variables, we are assuming: a. the variances of x and yare equal b. the variation around the line of regression is the same for each x value c. that x and y are independent d. All of the above ANSWER: b 119. Which of the following assumptions concerning the probability distribution of the random error term is stated incorrectly? a. The distribution is normal b. The mean of the distribution is 0 c. The variance of the distribution increases as x increases d. The errors are independent ANSWER: c 120. If the correlation coefficient (r) = 1.00, then b a. The y – intercept ( o) must equal 0 b. The explained variation equals the unexplained variation c. There is no unexplained variation d. There is no explained variation ANSWER: c 121. In a simple linear regression problem, r and1 a. may have opposite signs b. must have the same sign c. must have opposite signs d. must be equal ANSWER: b 122. The sample correlation coefficient between x and y is 0.375. It has been found out that the p – value is 0.256 when testingH oρ = 0 against a two-sided alternativeH1:ρ ≠ 0 . To test H oρ = 0 against the one-sided alternativeH1:ρ < 0 at a significance level of 0.193, the p - value will be equal to a. 0.128 b. 0.512 c. 0.744 Simple Linear Regression and Correlation 117 d. 0.872 ANSWER: d 123. Which of the following in not a required condition for the error variablin the simple linear regression model? a. The probability distribution of is normal. b. The mean of the probability distribution of is zero. σ ε c. The standard deviation ε of is a constant no matter what the value of x. d. The values of ε are auto correlated. ANSWER: d 124. Testing for existence of correlation is equivalent to a. testing for the existence of the slope1( b. testing for the existence of the Y – intercepo)( c. the confidence interval estimate for predicting Y d. None of the above ANSWER: a 125. The coefficient of determination 2 measures the amount of: R a. variation in y that is explained by variation in x b. variation in x that is explained by variation in y c. variation in y that is unexplained by variation in x d. variation in x that is unexplained by variation in y ANSWER: a 126. If the coefficient of correlation is 0.90, then the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is: a. 90% b. 81% c. 0.90% d. 0.81% ANSWER: b 127. If a researcher wanted to find out if alcohol consumptions and grade point average on a 4 – point scale are linearly related, he would perform a 2 a. χ test for the difference in two proportions b. χ 2test for independence c. a z test for the difference in two proportions d. a t test for no linear relationship between the two variables ANSWER: d 118 Chapter Sixteen TRUE / FALSE QUESTIONS 128. If the value of the sum of squares for error SSE equals zero, then the coefficient of determination must equal zero. ANSWER: F 129. When the actual values y of a dependent variable and the corresponding predicted values yare the same, the standard error of the estimate will be 1.0. ANSWER: F 130. The value of the sum of squares for regression SSR can never be smaller than 0.0. ANSWER: T 131. The value of the sum of squares for regression SSR can never be smaller than 1. ANSWER: F 132. If all the values of an independent variable x are equal, then regressing a dependent variable y on x will result in a coefficient of determination of zero. ANSWER: T 133. In a simple linear regression model, testing whether the slope β1of the population regression line could be zero is the same as testing whether or not the population coefficient of correlationequals zero. ANSWER: T 134. When the actual values y of a dependent variable and the corresponding predicted values yare the same, the standard error of estimaεwill be 0.0. ANSWER: T y 135. If there is no linear relationship between two variablesx and , the coefficient of determination must be ± 1.0. ANSWER: F 136. The value of the sum of squares for regression SSR can never be larger than the value of sum of squares for error SSE. ANSWER: F 137. When the actual values y of a dependent variable and the corresponding predicted values yare the same, the standard error of estimatwill be -1.0. ε ANSWER: F 138. In a simple linear regression problem, the least squares line is= -3.75 + 1.25 , and the coefficient of determination is 0.81. The coefficient of correlation must be –0.90. ANSWER: F 139. In simple linear regression, the divisor of the standard error of esεis n – 2. Simple Linear Regression and Correlation 119 ANSWER: T 140. In a regression problem the following pairs of (x, y) are given: (4,-2), (4,-1), (4,0), (4,1) and (4,2). That indicates that the coefficient of correlation is –1. ANSWER: F 141. The value of the sum of squares for regression SSR can never be larger than the value of total sum of squares SST. ANSWER: T 142. In regression analysis, if the coefficient of determination is 1.0, then the coefficient of correlation must be 1.0. ANSWER: F 143. Correlation analysis is used to determine the strength of the relationship between an independent variable x and dependent variable y. ANSWER: T 144. If the coefficient of correlation is –0.81, then the percentage of the variation in y that is explained by the regression line is 81%. ANSWER: F 145. If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0. ANSWER: F 146. If the standard error of estimateε= 20 and n = 8, then the sum of squares for error SSE is 2,400. ANSWER: T 147. The probability distribution of the error variable is normal, with mean E( ) = 0, and standard deviation σε=1. ANSWER: F 148. In a simple linear regression problem, if the coefficient of determination is 0.95, this means that 95% of the variation in the independent variable x can be explained by regression line. ANSWER: F 149. Given that cov(x, y) = 10,sy = 15, sx= 8, and n = 12, the value of the standard error of estimate sεis 2.75. ANSWER: F 150. If the error variable is normally distributed, the test statistic for test0ng 1 0 is Student t distributed with n – 2 degrees of freedom. 120 Chapter Sixteen ANSWER: T 151. Given that cov(x, y) = 8.5, sy = 8, and sx= 10, then the value of the coefficient of determination is 0.95. ANSWER: F 152. The coefficient of determination is the coefficient of correlation squared. That is, 2 R = r ANSWER: T 153. Given that SSE = 60 and SSR = 540, the proportion of the variation in y that is explained by the variation in x is 0.90. ANSWER: T 154. Given that SSE = 84 and SSR = 358.12, the coefficient of correlation (also called the Pearson coefficient of correlation) must be 0.90. ANSWER: F 155. Except for the values r = -1, 0, and 1, we cannot be specific in our interpretation of the coefficient of correlation r. However, when we square it we produce a more meaningful statistic. ANSWER: T 156. A zero population correlation coefficient between a pair of random variables means that there is no linear relationship between the random variables. ANSWER: T s2 s2 157. Given that cov(x, y) = 8, y = 14, x = 10, and n = 6, the value of the sum of squares for error SSE is 38. ANSWER: T 158. A store manager gives a pre-employment examination to new employees. The test is scored from 1 to 100. He has data on their sales at the end of one year measured in dollars. He wants to know if there is any linear relationship between pre-employment examination score and sales. An appropriate test to use is the t test on the population correlation coefficient. ANSWER: T Simple Linear Regression and Correlation 121 STATISTICAL CONCEPTS & APPLIED QUESTIONS FOR QUESTIONS 159 THROUGH 164, USE THE FOLLOWING NARRATIVE: Narrative: Car Speed and Gas Mileage An economist wanted to analyze the relationship between the speed of a car (x) and its gas mileage (y). As an experiment a car is operated at several different speeds and for each speed the gas mileage is measured. These data are shown below. Speed 25 35 45 50 60 65 70 Gas Mileage 40 39 37 33 30 27 25 159. {Car Speed and Gas Mileage Narrative} Calculate the standard error of estimate, and describe what this statistic tells you about the regression line. ANSWER: sε = 1.448; the model’s fit to these data is good. 160. {Car Speed and Gas Mileage Narrative} Do these data provide sufficient evidence at the 5% significance level to infer that a linear relationship exists between higher speeds and lower gas mileage? ANSWER: H : ρ = 0 vs. H : ρ ≠ 0 0 1 Rejection region: | t 0.025,10.228 Test statistic: t = -9.754 Conclusion: Reject the null hypothesis. Yes, these data provide sufficient evidence at the 5% significance level to infer that a linear relationship exists between higher speeds and lower gas mileage. 161. {Car Speed and Gas Mileage Narrative} Predict with 99% confidence the gas mileage of a car traveling 55 mph. ANSWER: 31.236 ± 6.284. Thus, LCL = 24.952, and UCL = 37.52 162. {Car Speed and Gas Mileage Narrative} Calculate the Pearson coefficient of correlation. ANSWER: r = -0.975 163. {Car Speed and Gas Mileage Narrative} What does the coefficient of correlation tell you about the direction and strength of the relationship between the two variables? ANSWER: There is a very strong negative linear relationship between car speed and gas mileage. 122 Chapter Sixteen 164. {Car Speed and Gas Mileage} Calculate the coefficient of determination and interpret its value. ANSWER: R 2= 0.95. This means that 95% of the total variation in gas mileage can be explained by the speed of the car. 165. The following 10 observations of variables x and y were collected. x 1 2 3 4 5 6 7 8 9 10 y 25 22 21 19 14 15 12 10 6 2 a. Calculate the standard error of estimate. b. Test to determine if there is enough evidence at the 5% significance level to indicate that x and y are negatively linearly related. c. Calculate the coefficient of correlation, and describe what this statistic tells you about the regression line. ANSWER: a. s = 1.322 ε b. H 0 β 1 0 vs. H 1 β 1 0 Rejection region: | t |0.05,81.86 Test statistic: t = -16.402 Conclusion: Reject the null hypothesis. Yes, there is enough evidence at the 5% significance level to indicate that x and y are negatively linearly related. c. r = -0.9854. This indicates a very strong negative linear relationship between the two variables. 166. Consider the following data values of variables x and y. x 2 4 6 8 10 13 y 7 11 17 21 27 36 a. Calculate the coefficient of determination, and describe what this statistic tells you about the relationship between the two variables. b. Calculate the Pearson coefficient of correlation. What sign does it have? Why? c. What does the coefficient of correlation calculated Tell you about the direction and strength of the relationship between the two variables? ANSWER: a. R = 0.995. This means that 99.5% of the variation in the dependent variable y is explained by the variation in the independent variable x. b. r = 0.9975. It is positive since the slope of the regression line is positive. c. There is a very strong (almost perfect) positive linear relationship between the two variables. FOR QUESTIONS 167 THROUGH 171, USE THE FOLLOWING NARRATIVE: Simple Linear Regression and Correlation 123 Narrative: Sunshine and Skin Cancer A medical statistician wanted to examine the relationship between the amount of sunshine (x) and incidence of skin cancer (y). As an experiment he found the number of skin cancers detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Skin Cancer per 100,000 7 11 9 12 15 10 7 5 167. {Sunshine and Skin Cancer Narrative} Calculate the standard error of estimate, and describe what this statistic tells you about the regression line. ANSWER: s ε= 0.9608; the model’s fit to these data is good. 168. {Sunshine and Skin Cancer Narrative} Can we conclude at the 1% significance level that there is a linear relationship between sunshine and skin cancer? ANSWER: H 0 ρ = 0 vs.H 1 ρ ≠ 0 Rejection region: | t |0.005,63.707 Test statistic: t = 8.485 Conclusion: Reject the null hypothesis. Yes, we conclude at the 1% significance level that there is a linear relationship between sunshine and skin cancer. 169. {Sunshine and Skin Cancer Narrative} Calculate the coefficient of determination and interpret it. ANSWER: R = 0.9231. This means that 92.31% of the variation in the incidence of skin cancer is explained by the variation in the amount of sunshine. 170. {Sunshine and Skin Cancer Narrative} Calculate the Pearson coefficient. What sign does it have? Why? ANSWER: b R = 0.9608. It is positive since the slope of the regression 1i= 1.846) is positive. 171. {Sunshine and Skin Cancer Narrative} What does the coefficient of correlation calculated Tell you about the direction and strength of the relationship between the two variables? ANSWER: There is a very strong (almost perfect) positive linear relationship between the two variables. FOR QUESTIONS 172 THROUGH 177, USE THE FOLLOWING NARRATIVE: Narrative: Sales and Experience 124 Chapter Sixteen The general manager of a chain of furniture stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month’s sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. Salesperson Years of Experience Sales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 6 5 14 7 12 20 8 7 17 9 20 30 10 15 25 172. {Sales and Experience Narrative} Determine the standard error of estimate and describe what this statistic tells you about the regression line. ANSWER: sε = 1.5724; the model’s fit is good. 173. (Sales and Experience Narrative} Determine the coefficient of determination and discuss what its value tells you about the two variables. ANSWER: R = 0.9536, which means that 95.36% of the variation in sales is explained by the variation in years of experience of the salesperson. 174. {Sales and Experience Narrative} Calculate the Pearson correlation coefficient. What sign does it have? Why? ANSWER: r = 0.9765. It has a positive sign since the slope of the regression line ( = 1.0817) is 1 positive. 175. {Sales and Experience Narrative} Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between years of experience and sales. Simple Linear Regression and Correlation 125 ANSWER: H 0 ρ = 0 vs. H 1 ρ ≠ 0 t = Rejection region: | t | 0.025,82.306 Test statistic: t = 12.8258 Conclusion: Reject the null hypothesis. Yes, a linear relationship exists between years of experience and sales. 176. {Sales and Experience Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between years of experience and sales. ANSWER: H 0 β1= 0 vs. H1: β1≠ 0 Rejection region: | t |0.025,82.306 Test statistic: t = 12.8258 Conclusion: Reject the null hypothesis. Yes, a linear relationship exists between years of experience and sales. 177. {Sales and Experience Narrative} Do the tests of ρ and β in the previous two questions 1 provide the same results? Explain. ANSWER: Yes; both tests have the same value of the test statistic, the same rejection region, and of course the same conclusion. This is not a coincidence; the two tests are identical. FOR QUESTIONS 178 THROUGH 183, USE THE FOLLOWING NARRATIVE: Narrative: Income and Education A professor of economics wants to study the relationship between income (y in $1000s) and education (x in years). A random sample eight individuals is taken and the results are shown below. Education 16 11 15 8 12 10 13 14 Income 58 40 55 35 43 41 52 49 178. {Income and Education Narrative} Determine the standard error of estimate and describe what this statistic tells you about the regression line. ANSWER: sε =2.436; the model’s fit to these data is good. 179. {Income and Education Narrative} Determine the coefficient of determination and discuss what its value tells you about the two variables. 126 Chapter Sixteen AN2WER: R = 0.9223, which means that 92.03% of the variation in income is explained by the variation in years of education. 180. {Income and Education Narrative} Calculate the Pearson correlation coefficient. What sign does it have? Why? ANSWER: r = 0.9604. It has a positive sign since the slope of the regression line 1 = 2.9098) is positive. 181. {Income and Education Narrative} Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a linear relationship exists between years of education and income. ANSWER: H 0 ρ = 0 vs. H 1 ρ ≠ 0 t = Rejection region: | t | 0.025,62.447 Test statistic: t = 8.439 Conclusion: Reject the null hypothesis. Yes, a linear relationship exists between years of education and income. 182. {Income and Education Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between years of education and income. ANSWER: H : β = 0 H : β ≠ 0 0 1 , 1 1 Rejection region: | t | 0.025,62.447 Test statistic: t = 8.439 Conclusion: Reject the null hypothesis. Yes, a linear relationship exists between years of education and income. ρ 183. {Income and Education Narrative} Do the tests of and β1 in the previous two provide the same results? Explain. ANSWER: Yes; both tests have the same value of the test statistic, the same rejection region, and of course the same conclusion. This is not a coincidence; the two tests are identic
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