Chap 012

Chapter 12Simple Regression True / False Questions 1. A scatter plot is used to visualize the association (or lack of association) between two quantitative variables. True False 2. The correlation coefficient r measures the strength of the linear relationship between two variables. True False 3. Pearson's correlation coefficient (r) requires that both variables be interval or ratio data. True False 4. If r = .55 and n = 16, then the correlation is significant at α = .05 in a two-tailed test. True False 5. A sample correlation r = .40 indicates a stronger linear relationship than r = -.60. True False 6. A common source of spurious correlation between X and Y is when a third unspecified variable Z affects both X and Y. True False 7. The correlation coefficient r always has the same sign as b1 in Y = b0 + b1X. True False 8. The fitted intercept in a regression has little meaning if no data values near X = 0 have been observed. True False 9. The least squares regression line is obtained when the sum of the squared residuals is minimized. True False 10 In a simple regression, if the coefficient for X is positive and . significantly different from zero, then an increase in X is associated with an increase in the mean (i.e., the expected value) of Y. True False 11 In least-squares regression, the residuals e1, e2, . . . , en will always . have a zero mean. True False 12 When using the least squares method, the column of residuals always . sums to zero. True False 13 In the model Sales = 268 + 7.37 Ads, an additional $1 spent on ads . will increase sales by 7.37 percent. True False 14 If R2 = .36 in the model Sales = 268 + 7.37 Ads with n = 50, the two. tailed test for correlation at α = .05 would say that there is a significant correlation between Sales and Ads. True False 15 If R2 = .36 in the model Sales = 268 + 7.37 Ads, then Ads explains 36 . percent of the variation in Sales. True False 16 The ordinary least squares regression line always passes through the . point . True False 17 The least squares regression line gives unbiased estimates of β0 and . β1. True False 18 In a simple regression, the correlation coefficient r is the square root of . R2. True False 19 If SSR is 1800 and SSE is 200, then R2 is .90. . True False 20 The width of a prediction interval for an individual value of Y is less . than standard error se. True False 21 If SSE is near zero in a regression, the statistician will conclude that the . proposed model probably has too poor a fit to be useful. True False 22 For a regression with 200 observations, we expect that about 10 . residuals will exceed two standard errors. True False True False 29 In linear regression between two variables. be normally distributed. estimated slope and intercept. True False 30 The larger the absolute value of the t statistic of the slope in a simple . True False . are very small. causal system. a significant relationship . running the regression twice and seeing whether Y = β0 + β1X or X = β1 + β0Y has the larger R2.23 Confidence intervals for predicted Y are less precise when the residuals . exists when the p-value of the t test statistic for the slope is greater than α. True False 24 Cause-and-effect direction between X and Y may be determined by . negative p-value for r. True False 28 A negative correlation between two variables X and Y usually yields a . True False 25 The ordinary least squares method of estimation minimizes the . True False 27 If you have a strong outlier in the residuals. it may represent a different . True False 26 Using the ordinary least squares method ensures that the residuals will . linear regression. the stronger the linear relationship exists between X and Y. estimated from sums of squares in the ANOVA table. outlier). the model (large residual). interval for Y become wider. True False 32 In simple linear regression. True False 38 The studentized residuals permit us to detect cases where the . True False 37 "High leverage" would refer to a data point that is poorly predicted by . regression predicts poorly. True False 34 A prediction interval for Y is narrower than the corresponding . the p-value of the F statistic. True False . squares (SSR). the coefficient of determination (R2) is . confidence interval for the mean of Y. True False 36 The total sum of squares (SST) will never exceed the regression sum of . the p-value of the slope will always equal .31 In simple linear regression. True False 33 An observation with high leverage will have a large residual (usually an . True False 35 When X is farther from its mean. the prediction interval and confidence . g..567)..291 to 18. True False 40 Ill-conditioned refers to a variable whose units are too large or too . True False 45 The regression line must pass through the origin. True False 46 Outliers can be detected by examining the standardized residuals. True False 41 A simple decimal transformation (e. misspecification. leverage. significantly from zero in a two-tailed test will also be significantly greater than zero or less than zero in a one-tailed test at the same α. significant in a two-tailed test at the same level of significance α. $2. True False . from 18. . . True False 44 Omission of a relevant predictor is a common source of model .291) often .39 A poor prediction (large residual) indicates an observation with high . small (e. improves data conditioning. True False 42 Two-tailed t-tests are often used because any predictor that differs .434.g. True False 43 A predictor that is significant in a one-tailed t-test will also be . with the error sum of squares (SSE). a sample correlation . MSR to the MSE. True False 50 A different confidence interval exists for the mean value of Y for each . . True False 55 High leverage for an observation indicates that X is far from its mean. neither X nor Y is designated as the .2 degrees of freedom associated . True False .42 with n = 25 is significantly different than zero. there are n . True False 48 In a simple regression. coefficient r = 0. . True False 52 In a two-tailed test for correlation at α = . independent variable. True False 54 A negative value for the correlation coefficient (r) implies a negative .05. the F statistic is calculated by taking the ratio of . True False 53 In correlation analysis. in the response variable Y that is explained by the predictor X. value for the slope (b1).47 In a simple regression. True False 51 A prediction interval for Y is widest when X is near its mean. True False 49 The coefficient of determination is the percentage of the total variation . different value of X. independent variable. . True False 58 When the errors in a regression model are not independent. A. C. . response variable. B. relationship between X and Y. Fcalc = tcalc2. True False 59 In a simple bivariate regression. True False 57 There are usually several possible regression lines that will minimize . using cross-sectional data. dependent variable. True False Multiple Choice Questions 61 The variable used to predict another variable is called the: .56 Autocorrelated errors are not usually a concern for regression models . the sum of squared errors. regression variable. True False 60 Correlation analysis primarily measures the degree of the linear . D. regression model is said to have autocorrelation. the . is in squared units of the dependent variable. (days) of its shipments as a function of the distance traveled (miles).02 4.15 64 A local trucking company fitted a regression to relate the travel time . A. 2. The fitted regression is Time = -7. A. D. 2. may be cut in half to get an approximate 95 percent prediction interval.62 The standard error of the regression: .0053.126 + 0. C. 63 A local trucking company fitted a regression to relate the travel time .0214 Distance. Find the value of tcalc to test for zero slope. is based on squared deviations from the regression line.04 3.101 2. based on a sample of 20 shipments. The estimated standard error of the slope is 0. C.05. C.734 .46 5. may assume negative values if b1 < 0.0053. D. The fitted regression is Time = -7. A.126 + . B. using α = .552 1. B.0214 Distance.960 1. The estimated standard error of the slope is 0. based on a sample of 20 shipments. B. D. (days) of its shipments as a function of the distance traveled (miles). Find the critical value for a right-tailed test to see if the slope is positive. then the computed test statistic would be: A. test whether the population correlation is zero. The test statistic can never be negative. 3. C. It is a test for overall fit of the model. 2. B. 67 Which of the following is not a characteristic of the F-test in a simple . 1.071.960.75 Temperature. B. D.65 If the attendance at a baseball game is to be predicted by the equation . regression? A.597.750 9. C.500 .750 12. D. B. Attendance = 16. The F-test gives a different p-value than the t-test. If the sample consists of 25 observations and the correlation coefficient is 0. . C. 6. 1.645. It requires a table with numerator and denominator degrees of freedom. what would be the predicted attendance if Temperature is 90 degrees? A.60. 020 66 A hypothesis test is conducted at the 5 percent level of significance to .250 10. D. B. C. participation rate among females) to try to predict Cancer (death rate per 100. states.000 population due to cancer) in the 50 U. . 05. The standard error is too high for this model to be of any predictive use. The 95 percent confidence interval for the coefficient of Femlab is -4.05. The two-tailed p-value for Femlab will be less than .29 to -0. Significant correlation exists between Femlab and Cancer at α = . D.S.68 A researcher's Excel results are shown below using Femlab (labor force .28. Which of the following statements is not true? A. the cancer rate in your state will decline.000 population due to cancer) in the 50 U. . B.05 level of significance. C. participation rate among females) to try to predict Cancer (death rate per 100. A rise in female labor participation rate will cause the cancer rate to decrease within a state. Which statement is valid regarding the relationship between Femlab and Cancer? A. This model explains about 10 percent of the variation in state cancer rates. there isn't enough evidence to say the two variables are related. If your sister starts working. D. At the . states.S.69 A researcher's results are shown below using Femlab (labor force . the intercept is irrelevant since zero median income is impossible in a large city. wealthy individuals tend to commit more crimes. We can conclude that: A.S. D. . B. You may conclude that: A.0982 . B. 72 William used a sample of 68 large U. His estimated regression equation was Crime = 428 + 0. on average. causation is in serious doubt. people should have more children so they can get better jobs.000 persons) and Income (median annual income per capita.S.8395 . What is the R2 for this regression? A. C.050 Income. cities to estimate the .1605 71 A news network stated that a study had found a positive correlation .9018 . D. C.70 A researcher's results are shown below using Femlab (labor force . in dollars). D. statisticians have small families. participation rate among females) to try to predict Cancer (death rate per 100. the data are erroneous because the correlation should be negative. the slope is small so Income has no effect on Crime. . states. crime seems to create additional income in a city.000 population due to cancer) in the 50 U. between the number of children a worker has and his or her earnings last year. C. relationship between Crime (annual property crimes per 100. B. decrease by 50.4296. cities to estimate the relationship . 9 percent of the variation in Days is explained by Size. the relationship between Days and Size is significant. 75 Prediction intervals for Y are narrowest when: . 1. correlation is: A.73 Mary used a sample of 68 large U. in dollars). From this information we can conclude that: A. If Income decreases by 1000. D. 2. C. B. D. D. . the mean of X is small. 74 Amelia used a random sample of 100 accounts receivable to estimate . the relationship between Days (number of days from billing to receipt of payment) and Size (size of balance due in dollars). 76 If n = 15 and r = . the mean of X differs greatly from the mean of Y. we would expect that Crime will: A. remain unchanged.000 persons) and Income (median annual income per capita.300. C. A.S. 7. increase by 428.0047 Size with a correlation coefficient of .048. C. B. Her estimated regression equation was Days = 22 + 0. B.050 Income. between Crime (annual property crimes per 100. the mean of X is near the mean of Y. D.862. the value of X is near the mean of X. Her estimated regression equation was Crime = 428 + 0. autocorrelation is likely to be a problem.715. impossible to determine without α. larger accounts usually take less time to pay. the corresponding t-statistic to test for zero . increase by 500. C. B. 2004. between X and Y? A. B. D. a likely data entry error. we would reject the .77 Using a two-tailed test at α = .205 indicates: . an extreme outlier in the residuals.2992. A. neither the slope nor the intercept. C. C. 80 In a simple regression. C. Either a negative F statistic or a negative p-value . B. C. only the slope. . A negative p-value for the correlation coefficient C. 79 A standardized residual ei = -2. . an observation with high leverage. a rather poor prediction. . A negative F statistic B. A negative correlation coefficient D. only the intercept. Large p-value for the estimated slope Large t statistic for the slope Large p-value for the F statistic Small t-statistic for the slope 81 Which is indicative of an inverse relationship between X and Y? . hypothesis of zero correlation if the absolute value of r exceeds: A. D. B. D.0250. . A.05 for n = 30. D. A.3609. which would suggest a significant relationship . B. both the slope and intercept. 78 The ordinary least squares (OLS) method of estimation will minimize: . . It is also sometimes called Pearson's r. D. A. A.00. C. B.82 Which is not correct regarding the estimated slope of the OLS . B. D. B. no generalization is possible about their comparative width. there are only two independent variables.g. the data are presented in a simple and clear way. C. a given regression analysis: A. C. It is the square of the coefficient of correlation. It may be regarded as zero if its p-value is less than α. B. It is divided by its standard error to obtain its t statistic. It shows the change in Y for a unit change in X. we have only one explanatory variable. SSE. It is tested for significance using a t-test. C. A. we have only a few observations. B. SSR. It reports the percent of the variation in Y explained by X. D. It is calculated using sums of squares (e. It is chosen so as to minimize the sum of squared errors. 86 Which is not true of the coefficient of determination? . It is negative when there is an inverse relationship between X and Y. It assumes that Y is the dependent variable. D. D. 85 When comparing the 90 percent prediction and confidence intervals for . SST). regression line? A. the prediction interval is wider than the confidence interval. there is no difference between the size of the prediction and confidence intervals. 83 Simple regression analysis means that: . . 84 The sample coefficient of correlation does not have which property? . C.00 up to +1. It can range from -1. the prediction interval is narrower than the confidence interval. n = 25). will yield a Durbin-Watson statistic near 2.S.87 If the fitted regression is Y = 3. X = size of the police force in a city (thousands of police). we expect that a plot of the residuals . cities in 2008. city (thousands of robberies). B. incorrect to conclude that: A.5. Autocorrelated residuals (because this is time-series data) B.25. Heteroscedastic residuals (because we are using totals uncorrected for city size) C. we would be least likely to see which problem? A. D. will show no pattern at all. C.5 + 2. Nonnormal residuals (because a few larger cities may skew the residuals) D. C. the estimated regression line crosses the Y axis at 3. the sample correlation coefficient must be positive. 88 In a simple regression Y = b0 + b1X where Y = number of robberies in a . B. crosses the centerline too many times.50. Y increases 2. D. will form approximately a straight line. and n = 45 randomly chosen large U. .1X (R2 = . the value of the sample correlation coefficient is 0. it is .1 percent for a 1 percent increase in X. High leverage for some observations (because some cities may be huge) 89 When homoscedasticity exists. versus the fitted Y: A. C. banks ($millions). autocorrelated residuals due to time-series data. a negative slope because banks hold less currency when they are robbed. Autocorrelation is mainly a concern if we are using time-series data. D. A. A non-standardized residual whose value is ei = 4. X = annual value of currency held by all U. . reported bank robbery losses in all U. Spurious correlation can often be reduced by expressing X and Y in per capita terms. Non-normal residuals imply biased coefficient estimates. A leverage statistic of 0. nonnormal residuals due to skewed data as bank size increases over time.S. B. B. 92 A regression was estimated using these variables: Y = annual value of . banks ($millions).90 Which statement is not correct? . heteroscedastic residuals due to the wide variation in data magnitudes.22 would be considered an outlier. D. is most nearly correct? A. which statement . D. B.16 or more would indicate high leverage. Standardized residuals make it easy to identify outliers or instances of poor fit. 91 In a simple bivariate regression with 25 observations. Heteroscedastic residuals will have roughly the same variance for any value of X. n = 100 years (1912 through 2011). C. a major problem. C.S. We would not anticipate: A. Standardizing the residuals will eliminate any heteroscedasticity. B. A. C. = 20 + 7 Study. Normal X values Non-autocorrelated errors Homoscedastic errors Normal errors 95 In a simple bivariate regression with 60 observations there will be _____ . Bob studied 9 hours. Hardtack's class showed Score . The regression yielded R2 = 0. The quick 95 percent prediction interval for Bob's grade is approximately: A. B. 69 to 97. 67 to 99. D. D. C. B. D. (R2)? A.SST/SSE . A. SSR/SSE SSR/SST 1 . 94 Which is not an assumption of least squares regression? . C.93 A fitted regression for an exam in Prof. 60 59 58 57 96 Which is correct to find the value of the coefficient of determination . C. 76 to 90. where Score is the student's exam score and Study is the student's study hours. 75 to 91.50 and SE = 8. residuals. B. C.01. B.497 . B. D. 100 In a sample of n = 23.01. α = .960 98 In a sample of size n = 23. the critical value of the correlation coefficient .01 but not α = . = .05 nor α = .412 ±. the Student's t test statistic for a correlation of r . ±. neither α = .500 would be: A. D.05 and α = .05 is: A.500 ±. C. B. C. C.05.725 ±2.559.97 The critical value for a two-tailed test of H0: β1 = 0 at α = . simple regression with 22 observations is: A. 2. a sample correlation of r = . for a two-tailed test at α = . can't say without knowing α. sufficient evidence to conclude that the population correlation coefficient exceeds zero in a right-tailed test at: A.400 provides . 2.05 but not α = . D.01. both α = . B. ±1.086 ±2. 2. D.05 in a .646. 99 In a sample of n = 23.528 ±1.819.524 ±. α = . 444 ±.05 is: A. r = .05. can't say without knowing if it's a two-tailed or one-tailed test. B.101 In a sample of n = 23. C. D. C. α = .05.852 D.819 ±2. neither α = .229 ±2.025 but not α = . B. D.05 but not α = .412 ±.646 ±2. C. ±2.080 102 In a sample of n = 40.025 nor α = .400 provides . 1. both α = .497 . 103 In a sample of n = 20. the critical value of Student's t for a two-tailed .025 and α = .110 B. 104 In a sample of n = 20. test of significance for a simple bivariate regression at α = . D.05. the critical value of the correlation coefficient . a sample correlation of r = . 1. α = . for a two-tailed test at α = . sufficient evidence to conclude that the population correlation coefficient exceeds zero in a right-tailed test at: A. the Student's t test statistic for a correlation of .025.645 C.05 is: A. B.400 would be: A.587 ±. 2. ±. D. neither α = .074 106 In a sample of size n = 36. C. -2. B.938. for a two-tailed test at α = . D. ±2. B. the Student's t test statistic for a correlation of . D.450 provides . C. test of significance for a simple bivariate regression at α = .030.05. -2.05.01 α = .052 ±2. 108 In a sample of n = 36. ±. the critical value of the correlation coefficient .105 In a sample of n = 27.05 is: A.01 and α = . 107 In a sample of n = 36.450 would be: A. can't say without knowing α. -2. α = . B.05 is: A.05 both α = . the critical value of Student's t for a two-tailed . D.01 nor α = .329 ±.898 ±2. a sample correlation of r = -.387 ±. r = -.060 ±2.110. C. C.423 ±.497 . B. sufficient evidence to conclude that the population correlation coefficient differs significantly from zero in a two-tailed test at: A. 126 + 0.13 days 2.07 days 7.73 days .05 is: A.074 110 A local trucking company fitted a regression to relate the travel time .724 2. C. 2. (days) of its shipments as a function of the distance traveled (miles).109 In a sample of n = 36. B. 1. C. The fitted regression is Time = -7. D.14 days 1. the critical value of Student's t for a two-tailed .032 2.0214 Distance. B. D. If Distance increases by 50 miles.938 2. the expected Time would increase by: A. test of significance of the slope for a simple regression at α = . 111 A local trucking company fitted a regression to relate the cost of its . $301. Based on this estimated relationship. shipments as a function of the distance traveled. B. when distance increases by 50 miles. D. $104. C. . The Excel fitted regression is shown. the coefficient of determination is . C. D. the SST would be smaller than SSR. A.81. then: . the standard error would be large. $143. 112 If SSR is 2592 and SSE is 608. $286. B. the expected shipping cost would increase by: A. the slope is likely to be insignificant. D. A.833 3.8911 .294 0. 1. C. C.113 Find the sample correlation coefficient for the following data. . .762 -2. . B. A.228 = b0 + b1x.9124 .9822 .9556 114 Find the slope of the simple regression . B. D. 115 Find the sample correlation coefficient for the following data. B.109 -2. D. . C.8736 .221 1. C. B. D.7291 .9118 .9563 116 Find the slope of the simple regression .884 = b0 + b1x. A. .595 1. 2. . A. 2.333. 5.349.998.284]. [1.217]. The 95 percent confidence interval for the slope is: A.398].268.284]. D. [ -0.282. [1. C. observations. -1. B. [1.064]. D. . C.026].449].117 A researcher's results are shown below using n = 25 observations.118.118. 2.998]. B. 2. -0. +0. 2. [ -3. [1. 118 A researcher's regression results are shown below using n = 8 . . [ -4. [1.602. The 95 percent confidence interval for the slope is: A. C.119 Bob thinks there is something wrong with Excel's fitted regression. The estimated equation is obviously incorrect. Bob needs to increase his sample size to decide. The relationship is linear. What do you say? A. . The R2 looks a little high but otherwise it looks OK. D. Short Answer Questions . so the equation is credible. B. so he performed a . His results are shown below. using what you have learned in this chapter.120 Pedro became interested in vehicle fuel efficiency. Write a brief analysis of these results. Is the intercept meaningful in this regression? Make a prediction of CityMPG when Weight = 3000. simple regression using 93 cars to estimate the model CityMPG = β0 + β1 Weight where Weight is the weight of the vehicle in pounds. what change would you predict in its CityMPG? . and also when Weight = 4000. Do these predictions seem believable? If you could make a car 1000 pounds lighter. 121 Mary noticed that old coins are smoother and more worn. and also when Age = 20. and then performed a simple regression to estimate the model Weight = β0 + β1 Age where weight is the weight of the coin in grams and Age is the age of the coin in years. weighed 31 nickels and recorded their age. using what you have learned in this chapter. What does this tell you? Is the intercept meaningful in this regression? . Make a prediction of Weight when Age = 10. Write a brief analysis of these results. She . Her results are shown below. Chapter 12 Simple Regression Answer Key True / False Questions 1. TRUE The scatter plot shows association between two quantitative variables. TRUE A correlation coefficient measures linearity between two variables. A scatter plot is used to visualize the association (or lack of association) between two quantitative variables. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Visual Displays and Correlation Analysis . The correlation coefficient r measures the strength of the linear relationship between two variables. Topic: Visual Displays and Correlation Analysis 2. AACSB: Analytic Blooms: Understand Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. then the correlation is significant at α = .2)/(1 .f.60.552)]1/2 = 2.2 = 14.025 = 2. FALSE The sign only indicates the direction. Topic: Visual Displays and Correlation Analysis 5. If r = . not the strength. = 16 . A sample correlation r = .145 for d.. Topic: Visual Displays and Correlation Analysis .3. Pearson's correlation coefficient (r) requires that both variables be interval or ratio data. TRUE tcalc = r[(n . AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Visual Displays and Correlation Analysis 4. TRUE Correlation assumes quantitative data with at least interval measurements.55 and n = 16. of the linear relationship.r2)]1/2 = (.05 in a two-tailed test.40 indicates a stronger linear relationship than r = -.464 > t.55)[(16 .2)/(1 . Topic: Regression Terminology 8. TRUE The t-test for the slope in simple regression gives the same result as the t-test for r. AACSB: Analytic Blooms: Understand Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Simple Regression . Topic: Visual Displays and Correlation Analysis 7. TRUE Predicting Y for X = 0 makes little sense if the observed data have no values near X = 0. AACSB: Analytic Blooms: Understand Difficulty: 1 Easy Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. The correlation coefficient r always has the same sign as b1 in Y = b0 + b1X.6. AACSB: Analytic Blooms: Understand Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. The fitted intercept in a regression has little meaning if no data values near X = 0 have been observed. TRUE Both X and Y could be influenced by Z. A common source of spurious correlation between X and Y is when a third unspecified variable Z affects both X and Y. Topic: Ordinary Least Squares Formulas 10. AACSB: Analytic Blooms: Understand Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. .e. then an increase in X is associated with an increase in the mean (i. the expected value) of Y. Topic: Simple Regression 11. the residuals e1. Topic: Ordinary Least Squares Formulas . so their mean is zero. en will always have a zero mean. TRUE The conditional mean of Y depends on X (unless the slope is effectively zero). if the coefficient for X is positive and significantly different from zero.. The least squares regression line is obtained when the sum of the squared residuals is minimized. In least-squares regression. . In a simple regression. . TRUE The OLS method minimizes the sum of squared residuals. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. TRUE The residuals must sum to zero if the OLS method is used. . e2.9. 37 Ads. the twotailed test for correlation at α = . In the model Sales = 268 + 7. Topic: Ordinary Least Squares Formulas 13. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. not percent.36 in the model Sales = 268 + 7.05 would say that there is a significant correlation between Sales and Ads. TRUE tcalc = r[(n .37 percent.196 > t.2)/(1 . If R2 = .36)]1/2 = 5. Topic: Simple Regression 14. AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. FALSE The slope coefficient is in the same units as Y (dollars.12. the column of residuals always sums to zero. an additional $1 spent on ads will increase sales by 7.2 = 48. in this case).011 for d. When using the least squares method.f.r2)]1/2 = (. = 50 .. TRUE The residuals must sum to zero if the OLS method is used.2)/(1 . AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-02 Interpret the slope and intercept of a regression equation.60)[(50 . Topic: Visual Displays and Correlation Analysis .025 = 2.37 Ads with n = 50. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot.15. TRUE The OLS formulas require the line to pass through this point.37 Ads. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. ANOVA table. The ordinary least squares regression line always passes through the point . Topic: Ordinary Least Squares Formulas 16. and F test. Topic: Regression Terminology 17. If R2 = . The least squares regression line gives unbiased estimates of β0 and β1.36 in the model Sales = 268 + 7. R2. Topic: Ordinary Least Squares Formulas . AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-08 Interpret the standard error. TRUE We can interpret R2 as the fraction of variation in Y explained by X (expressed as a percent). then Ads explains 36 percent of the variation in Sales. TRUE The expected values of the OLS estimators b0 and b1 are the true parameters β0 and β1. R2. Topic: Tests for Significance 20. TRUE In fact. Topic: Confidence and Prediction Intervals for Y . we could use the notation r2 instead of R2 when talking about simple regression. and F test. In a simple regression. R2. and F test. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. TRUE R2 = SSR/SST = SSR/(SSR + SSE) = 1800/(1800 + 200) = . If SSR is 1800 and SSE is 200. ANOVA table. FALSE The formula for the interval width multiplies the standard error by an expression > 1. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y.90. the correlation coefficient r is the square root of R2. ANOVA table.18. then R2 is .90. The width of a prediction interval for an individual value of Y is less than standard error se. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. Topic: Ordinary Least Squares Formulas 19. and F test. If SSE is near zero in a regression. which would be smaller if the fit is good. FALSE SSE is the sum of the square residuals. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. Topic: Tests for Significance 22.21. the statistician will conclude that the proposed model probably has too poor a fit to be useful. Confidence intervals for predicted Y are less precise when the residuals are very small. Topic: Unusual Observations 23. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. TRUE If the residuals are normal. R2. ANOVA table. 95.44 percent (190 of 200) will lie within ±2se (so 10 outside). Topic: Confidence and Prediction Intervals for Y . AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. we expect that about 10 residuals will exceed two standard errors. For a regression with 200 observations. FALSE Small residuals imply a small standard error and thus a narrower prediction interval. Topic: Simple Regression 25. Topic: Residual Tests . FALSE OLS produces unbiased estimates but cannot ensure normality of the residuals. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-10 Test residuals for violations of regression assumptions. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. Using the ordinary least squares method ensures that the residuals will be normally distributed. FALSE Cause and effect cannot be determined in the context of simple regression models. Topic: Ordinary Least Squares Formulas 26. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-02 Interpret the slope and intercept of a regression equation.24. Cause-and-effect direction between X and Y may be determined by running the regression twice and seeing whether Y = β0 + β1X or X = β1 + β0Y has the larger R2. FALSE OLS minimizes the sum of squared residuals. The ordinary least squares method of estimation minimizes the estimated slope and intercept. FALSE The p-value cannot be negative. AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. A negative correlation between two variables X and Y usually yields a negative p-value for r. it may represent a different causal system. If you have a strong outlier in the residuals.27. Topic: Visual Displays and Correlation Analysis 29. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. In linear regression between two variables. Topic: Other Regression Problems (Optional) 28. TRUE Outliers might come from a different population or causal system. a significant relationship exists when the p-value of the t test statistic for the slope is greater than α. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. Topic: Tests for Significance . FALSE Reject β1 = 0 if the p-value is less than α. In simple linear regression. Topic: Ordinary Least Squares Formulas 32. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. and F test. ANOVA table. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error.30. and F test. In simple linear regression. R2. the coefficient of determination (R2) is estimated from sums of squares in the ANOVA table. The larger the absolute value of the t statistic of the slope in a simple linear regression. R2. regardless of its sign (+ or -). Topic: Analysis of Variance: Overall Fit . TRUE This is true only if there is one predictor (but is no longer true in multiple regression). TRUE R2 = SSR/SST or R2 = 1 . ANOVA table. Topic: Tests for Significance 31. TRUE The correlation coefficient measures linearity.SSE/SST. the p-value of the slope will always equal the p-value of the F statistic. the stronger the linear relationship exists between X and Y. AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. Topic: Confidence and Prediction Intervals for Y . A prediction interval for Y is narrower than the corresponding confidence interval for the mean of Y. FALSE Predicting an individual case requires a wider confidence interval than predicting the mean. the prediction interval and confidence interval for Y become wider. TRUE The width increases when X differs from its mean (review the formula). AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. Topic: Confidence and Prediction Intervals for Y 35. An observation with high leverage will have a large residual (usually an outlier). Topic: Unusual Observations 34. When X is farther from its mean. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. FALSE The concepts are distinct (a high-leverage point could have a good fit).33. A large studentized tvalue (e. TRUE Studentized residuals resemble a t-distribution. and F test.00 or t > + 2. FALSE A high-leverage observation may have a good fit (only its X value determines its leverage).g.. Topic: Analysis of Variance: Overall Fit 37. ANOVA table.00) would implies a poor fit. Topic: Unusual Observations . FALSE The identity is SSR + SSE = SST. R2. The studentized residuals permit us to detect cases where the regression predicts poorly. Topic: Unusual Observations 38. t < -2. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. "High leverage" would refer to a data point that is poorly predicted by the model (large residual). The total sum of squares (SST) will never exceed the regression sum of squares (SSR).36. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-08 Interpret the standard error. 4.434. 4. Only its X value determines its leverage.3E + 06).g. Topic: Unusual Observations 40.291) often improves data conditioning.39.g. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-07 Perform regression analysis with Excel or other software..3E + 06).g. from 18.. Ill-conditioned refers to a variable whose units are too large or too small (e. TRUE In Excel. a symptom of poor data conditioning is exponential notation (e. A simple decimal transformation (e. A poor prediction (large residual) indicates an observation with high leverage.. Topic: Other Regression Problems (Optional) 41. TRUE Keeping data magnitudes similar helps avoid exponential notation (e. FALSE High leverage indicates an unusually large or small X value (not a poor prediction). AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-07 Perform regression analysis with Excel or other software.291 to 18.g. A high-leverage observation may have a good fit or a poor fit.. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. Topic: Other Regression Problems (Optional) .567). $2. Two-tailed t-tests are often used because any predictor that differs significantly from zero in a two-tailed test will also be significantly greater than zero or less than zero in a one-tailed test at the same α. TRUE True because the critical t is larger in the two-tailed test (the default in most software). simple regression may be inadequate. TRUE In a multivariate world. A predictor that is significant in a one-tailed t-test will also be significant in a two-tailed test at the same level of significance α. FALSE False because the critical t would be larger in a two-tailed test.42. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-07 Perform regression analysis with Excel or other software. Topic: Tests for Significance 44. Topic: Other Regression Problems (Optional) . Topic: Tests for Significance 43. Omission of a relevant predictor is a common source of model misspecification. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. but the fitted intercept is rarely zero. larger than ±3 would be an outlier). ANOVA table.g. in general. The regression line must pass through the origin. Topic: Analysis of Variance: Overall Fit .2 degrees of freedom associated with the error sum of squares (SSE).. Topic: Ordinary Least Squares Formulas 46. Topic: Unusual Observations 47. there are n . AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-08 Interpret the standard error. R2. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot.45. TRUE This is true in simple regression because we estimate two parameters (β0 and β1). We might be unable to reject a zero intercept if a t-test. Outliers can be detected by examining the standardized residuals. equal zero. FALSE The OLS intercept estimate does not. In a simple regression. and F test. TRUE A poor fit implies a large t-value (e. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. and F test. In a simple regression. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. TRUE R2 = SSR/SST or R2 = 1 . and F test. Fcalc = MSR/MSE (obtained from the ANOVA table). R2. ANOVA table.48. Topic: Ordinary Least Squares Formulas 50. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. TRUE By definition.SSE/SST lies between 0 and 1 and often is expressed as a percent. Topic: Analysis of Variance: Overall Fit 49. R2. A different confidence interval exists for the mean value of Y for each different value of X. TRUE Both the interval width and also E(Y|X) =β0 + β1 X depend on the value of X. the F statistic is calculated by taking the ratio of MSR to the MSE. ANOVA table. The coefficient of determination is the percentage of the total variation in the response variable Y that is explained by the predictor X. Topic: Confidence and Prediction Intervals for Y . Topic: Visual Displays and Correlation Analysis 53. neither X nor Y is designated as the independent variable. A prediction interval for Y is widest when X is near its mean.)2 in the numerator. TRUE tcalc = r[(n . TRUE In correlation analysis.2)/(1 . The minimum would be when xi = .2)/(1 . Review the formula.42 with n = 25 is significantly different than zero. = 25 .069 for d.025 = 2. Topic: Confidence and Prediction Intervals for Y 52.422)]1/2 = 2. In correlation analysis.51.f.42)[(25 .05. a sample correlation coefficient r = 0. Topic: Visual Displays and Correlation Analysis .2 = 23. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. FALSE The prediction interval is narrowest when X is near its mean." AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance.219 > t. X and Y covary without designating either as "independent. In a two-tailed test for correlation at α = .r2)]1/2 = (.. which has a term (xi . A negative value for the correlation coefficient (r) implies a negative value for the slope (b1). Autocorrelated errors are not usually a concern for regression models using cross-sectional data. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. TRUE The sign of r must be the same as the sign of the slope estimate b1. High leverage for an observation indicates that X is far from its mean. Topic: Unusual Observations 56. TRUE We more often expect autocorrelated residuals in time series data. observations have higher leverage when X is far from its mean. Topic: Ordinary Least Squares Formulas 55. TRUE By definition. Topic: Residual Tests . AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-10 Test residuals for violations of regression assumptions.54. 57. There are usually several possible regression lines that will minimize the sum of squared errors. FALSE The OLS solution for the estimators b0 and b1 is unique. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. Topic: Ordinary Least Squares Formulas 58. When the errors in a regression model are not independent, the regression model is said to have autocorrelation. TRUE For example, in first-order autocorrelation εt depends on εt-1. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-10 Test residuals for violations of regression assumptions. Topic: Residual Tests 59. In a simple bivariate regression, Fcalc = tcalc2. TRUE This statement is true only in a simple regression (one predictor). AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error; R2; ANOVA table; and F test. Topic: Analysis of Variance: Overall Fit 60. Correlation analysis primarily measures the degree of the linear relationship between X and Y. TRUE The sign of r indicates the direction and its magnitude indicates the degree of linearity. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Visual Displays and Correlation Analysis Multiple Choice Questions 61. The variable used to predict another variable is called the: A. B. C. D. response variable. regression variable. independent variable. dependent variable. We might also call the independent variable a predictor of Y. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. Topic: Simple Regression 62. The standard error of the regression: A. is based on squared deviations from the regression line. B. may assume negative values if b1 < 0. C. is in squared units of the dependent variable. D. may be cut in half to get an approximate 95 percent prediction interval. In a simple regression, the standard error is the square root of the sum of the squared residuals divided by (n - 2). AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error; R2; ANOVA table; and F test. Topic: Tests for Significance 63. A local trucking company fitted a regression to relate the travel time (days) of its shipments as a function of the distance traveled (miles). The fitted regression is Time = -7.126 + 0.0214 Distance, based on a sample of 20 shipments. The estimated standard error of the slope is 0.0053. Find the value of tcalc to test for zero slope. A. B. C. D. tcalc = 2.46 5.02 4.04 3.15 = (0.0214)/(0.0053) = 4.038. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. Topic: Tests for Significance 6.05 = 1.75(90) = 9. Topic: Tests for Significance 65. using α = . Appendix D gives t.250 10.126 + . = n . Find the critical value for a right-tailed test to see if the slope is positive.734 For d. 020 The predicted Attendance is 16. 2.2 = 20 .552 1.64.2 = 18. C.734. Topic: Simple Regression .960 1.750 9.101 2. C.500 . The estimated standard error of the slope is 0. B. what would be the predicted attendance if Temperature is 90 degrees? A.500 .0053. If the attendance at a baseball game is to be predicted by the equation Attendance = 16.0214 Distance.f.750. D. based on a sample of 20 shipments. A. AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. D. The fitted regression is Time = -7. B. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests.05. A local trucking company fitted a regression to relate the travel time (days) of its shipments as a function of the distance traveled (miles).75 Temperature.750 12. 66. A hypothesis test is conducted at the 5 percent level of significance to test whether the population correlation is zero. If the sample consists of 25 observations and the correlation coefficient is 0.60, then the computed test statistic would be: A. B. C. D. 2.071. 1.960. 3.597. 1.645. tcalc = r[(n - 2)/(1 - r2)]1/2 = (.60)[(25 - 2)/(1 - .602)]1/2 = 3.597. Comment: Requires formula handout or memorizing the formula. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Visual Displays and Correlation Analysis 67. Which of the following is not a characteristic of the F-test in a simple regression? A. It is a test for overall fit of the model. B. The test statistic can never be negative. C. It requires a table with numerator and denominator degrees of freedom. D. The F-test gives a different p-value than the t-test. Fcalc is the ratio of two variances (mean squares) that measures overall fit. The test statistic cannot be negative because the variances are non-negative. In a simple regression, the F-test always agrees with the t-test. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error; R2; ANOVA table; and F test. Topic: Analysis of Variance: Overall Fit 68. A researcher's Excel results are shown below using Femlab (labor force participation rate among females) to try to predict Cancer (death rate per 100,000 population due to cancer) in the 50 U.S. states. Which of the following statements is not true? A. The standard error is too high for this model to be of any predictive use. B. The 95 percent confidence interval for the coefficient of Femlab is -4.29 to -0.28. C. Significant correlation exists between Femlab and Cancer at α = . 05. D. The two-tailed p-value for Femlab will be less than .05. The magnitude of se depends on Y (and, in this case, the tcalc indicates significance). AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. Topic: Tests for Significance 69. A researcher's results are shown below using Femlab (labor force participation rate among females) to try to predict Cancer (death rate per 100,000 population due to cancer) in the 50 U.S. states. Which statement is valid regarding the relationship between Femlab and Cancer? A. A rise in female labor participation rate will cause the cancer rate to decrease within a state. B. This model explains about 10 percent of the variation in state cancer rates. C. At the .05 level of significance, there isn't enough evidence to say the two variables are related. D. If your sister starts working, the cancer rate in your state will decline. It is customary to express the R2 as a percent (here, the tcalc indicates significance). AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error; R2; ANOVA table; and F test. Topic: Ordinary Least Squares Formulas Topic: Visual Displays and Correlation Analysis . B. and F test. What is the R2 for this regression? A. ANOVA table. A researcher's results are shown below using Femlab (labor force participation rate among females) to try to predict Cancer (death rate per 100.745. D. . states. A news network stated that a study had found a positive correlation between the number of children a worker has and his or her earnings last year.0982.8395 .70.9018 . AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Ordinary Least Squares Formulas 71.S.836)/(54. causation is in serious doubt. There is no a priori basis for expecting causation. R2.000 population due to cancer) in the 50 U.0982 . B. C. C. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. D. the data are erroneous because the correlation should be negative.377.1605 R2 = SSR/SST = (5. people should have more children so they can get better jobs. statisticians have small families. You may conclude that: A.225) = . C. in dollars). Mary used a sample of 68 large U. crime seems to create additional income in a city. D.000 persons) and Income (median annual income per capita. decrease by 50. we would expect that Crime will: A. . wealthy individuals tend to commit more crimes. William used a sample of 68 large U.050 ΔIncome = 0. remain unchanged. D.S.050 Income. cities to estimate the relationship between Crime (annual property crimes per 100.000 persons) and Income (median annual income per capita. His estimated regression equation was Crime = 428 + 0. increase by 428. If Income decreases by 1000. cities to estimate the relationship between Crime (annual property crimes per 100. We can conclude that: A. AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. Topic: Simple Regression 73. Zero median income makes no sense (significance cannot be assessed from given facts).050 Income. increase by 500.72. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests.050(1000) = -50. B. on average. C. the slope is small so Income has no effect on Crime.S. B. in dollars). Her estimated regression equation was Crime = 428 + 0. the intercept is irrelevant since zero median income is impossible in a large city. The constant has no effect so ΔCrime = 0. Topic: Ordinary Least Squares Formulas 75. 9 percent of the variation in Days is explained by Size. B. the value of X is near the mean of X. D.09. Prediction intervals for Y are narrowest when: A. We cannot judge significance without more information. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. C. AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-08 Interpret the standard error.0047 Size with a correlation coefficient of . The minimum would be when xi = .302 = . Topic: Confidence and Prediction Intervals for Y . Review the formula. the mean of X is near the mean of Y. Her estimated regression equation was Days = 22 + 0. R2 = . B. R2. C. which has (xi . ANOVA table. These are not time-series data. D.300. the mean of X differs greatly from the mean of Y. Amelia used a random sample of 100 accounts receivable to estimate the relationship between Days (number of days from billing to receipt of payment) and Size (size of balance due in dollars). the mean of X is small. autocorrelation is likely to be a problem. so there is no reason to expect autocorrelation. the relationship between Days and Size is significant.)2 in the numerator.Topic: Simple Regression 74. From this information we can conclude that: A. larger accounts usually take less time to pay. and F test. B.3609. tcalc = r[(n . 1. D.3609 for d.2 = 28. impossible to determine without α. Topic: Visual Displays and Correlation Analysis . Use rcrit = t. .048)/(2. C.2)/(1 .4296.0250.715.048. 7.2)1/2 = . AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. D.4296)[(15 . C. the corresponding t-statistic to test for zero correlation is: A.715.2992.05 for n = 30. we would reject the hypothesis of zero correlation if the absolute value of r exceeds: A. 2.. If n = 15 and r = . .0482 + 30 .2)/(1 . .r2)]1/2 = (. Using a two-tailed test at α = . AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance.0252 + n .2)1/2 = (2. . B.2004. = 30 . Topic: Visual Displays and Correlation Analysis 77.f.76.025/(t.42962)]1/2 = 1.862. a likely data entry error. Topic: Ordinary Least Squares Formulas 79. a rather poor prediction. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. only the slope. C. only the intercept.78. D. D. both the slope and intercept. A standardized residual ei = -2. C. an extreme outlier in the residuals. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. Topic: Residual Tests .205 indicates: A. OLS method minimizes the sum of squared residuals. This residual is beyond ±2se but is not an outlier (and without xi we cannot assess leverage). B. neither the slope nor the intercept. B. an observation with high leverage. The ordinary least squares (OLS) method of estimation will minimize: A. Topic: Analysis of Variance: Overall Fit . C. which would suggest a significant relationship between X and Y? A. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-08 Interpret the standard error. A negative F statistic B. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. Topic: Tests for Significance 81. In a simple regression. and F test. D. ANOVA table. A negative p-value for the correlation coefficient C. B. Which is indicative of an inverse relationship between X and Y? A. Large p-value for the estimated slope Large t statistic for the slope Large p-value for the F statistic Small t-statistic for the slope The larger the tcalc the more we feel like rejecting H0: β1 = 0. R2. Either a negative F statistic or a negative p-value Fcalc and the p-value cannot be negative. A negative correlation coefficient D.80. C. Which is not correct regarding the estimated slope of the OLS regression line? A. Topic: Simple Regression . AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. Simple regression analysis means that: A. B. It is divided by its standard error to obtain its t statistic. D. It may be regarded as zero if its p-value is less than α. C. D. It is chosen so as to minimize the sum of squared errors. the data are presented in a simple and clear way. Topic: Tests for Significance 83.82. It shows the change in Y for a unit change in X. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. we have only one explanatory variable. B. we have only a few observations. there are only two independent variables. We would reject H0: β1 = 0 if its p-value is less than the level of significance. Multiple regression has more than one independent variable (predictor). C.84. Individual values of Y vary more than the mean of Y. there is no difference between the size of the prediction and confidence intervals. Correlation analysis makes no assumption of causation or dependence. It is tested for significance using a t-test. It assumes that Y is the dependent variable. the prediction interval is narrower than the confidence interval. no generalization is possible about their comparative width. Topic: Confidence and Prediction Intervals for Y . B. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. It can range from -1. AACSB: Analytic Blooms: Remember Difficulty: 1 Easy Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. It is also sometimes called Pearson's r.00 up to +1.00. When comparing the 90 percent prediction and confidence intervals for a given regression analysis: A. Topic: Visual Displays and Correlation Analysis 85. C. D. B. the prediction interval is wider than the confidence interval. The sample coefficient of correlation does not have which property? A. D. Topic: Simple Regression .g. B. and F test. the estimated regression line crosses the Y axis at 3. n = 25). D. SSE. It is negative when there is an inverse relationship between X and Y. R2 cannot be negative. Units are not percent unless Y is already a percent. It is calculated using sums of squares (e. It is the square of the coefficient of correlation.1X (R2 = . it is incorrect to conclude that: A. B. SST). R2. It reports the percent of the variation in Y explained by X. ANOVA table. SSR.1 percent for a 1 percent increase in X.5 + 2.25. the value of the sample correlation coefficient is 0.50. Topic: Ordinary Least Squares Formulas 87. Which is not true of the coefficient of determination? A.86.5. Y increases 2. D. If the fitted regression is Y = 3.. C. the sample correlation coefficient must be positive. AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. C. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. and unusual leverage. X = size of the police force in a city (thousands of police). D. High leverage for some observations (because some cities may be huge) It is not a time series. cities in 2008. we expect that a plot of the residuals versus the fitted Y: A. we would be least likely to see which problem? A.88. Topic: Residual Tests 89. AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-10 Test residuals for violations of regression assumptions.S. will yield a Durbin-Watson statistic near 2. Nonnormal residuals (because a few larger cities may skew the residuals) D. When homoscedasticity exists. Autocorrelated residuals (because this is time-series data) B. In a simple regression Y = b0 + b1X where Y = number of robberies in a city (thousands of robberies). so autocorrelation would not be expected. Heteroscedastic residuals (because we are using totals uncorrected for city size) C. Homoscedastic residuals exhibit no pattern (equal variance for all Y). nonnormality. but the "size effect" is likely to produce heteroscedasticity. crosses the centerline too many times. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-10 Test residuals for violations of regression assumptions. will show no pattern at all. will form approximately a straight line. B. and n = 45 randomly chosen large U. Topic: Residual Tests . C. For simple regression. Standardizing is only a scale shift so does not reduce heteroscedasticity. Which statement is not correct? A. We cannot judge a residual's magnitude without knowing the standard error se. AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-11 Identify unusual residuals and high-leverage observations. which statement is most nearly correct? A. a major problem. D. C. B. Heteroscedastic residuals will have roughly the same variance for any value of X. Standardized residuals make it easy to identify outliers or instances of poor fit. A non-standardized residual whose value is ei = 4.16 or more would indicate high leverage.16. Autocorrelation is mainly a concern if we are using time-series data. Non-normal errors do not bias the OLS estimates. the "high leverage criterion" is hi > 4/n = 4/25 = . A leverage statistic of 0.22 would be considered an outlier. C. Heteroscedastic residuals exhibit different variance for different X or Y values. Spurious correlation can often be reduced by expressing X and Y in per capita terms. In a simple bivariate regression with 25 observations. Standardizing the residuals will eliminate any heteroscedasticity. Topic: Unusual Observations .90. Non-normal residuals imply biased coefficient estimates. B. AACSB: Analytic Blooms: Understand Difficulty: 2 Medium Learning Objective: 12-10 Test residuals for violations of regression assumptions. D. Topic: Residual Tests 91. X = annual value of currency held by all U.50 and SE = 8. 69 75 67 76 to to to to 97. 90.S. The quick 95 percent prediction interval for Bob's grade is approximately: A. B. C. a negative slope because banks hold less currency when they are robbed. B. 91.S. where Score is the student's exam score and Study is the student's study hours. The quick interval is ypredicted ±2se or 83 ± (2)(8) or 83 ± 16. The regression yielded R2 = 0. . A fitted regression for an exam in Prof. nonnormal residuals due to skewed data as bank size increases over time. Hardtack's class showed Score = 20 + 7 Study. A regression was estimated using these variables: Y = annual value of reported bank robbery losses in all U. D. Topic: Residual Tests 93. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-09 Distinguish between confidence and prediction intervals for Y. Bob studied 9 hours. We would not anticipate: A. AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-10 Test residuals for violations of regression assumptions. n = 100 years (1912 through 2011).92. heteroscedastic residuals due to the wide variation in data magnitudes. C. so autocorrelation would be expected. autocorrelated residuals due to time-series data. 99. banks ($millions). banks ($millions). and the "size effect" is likely to produce heteroscedasticity and nonnormality. D. It is a time series. but growth in both X and Y would yield a positive slope. Normal X values Non-autocorrelated errors Homoscedastic errors Normal errors The predictor X is not assumed to be a random variable at all. B. A. AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-03 Make a prediction for a given x value using a regression equation. C. Topic: Ordinary Least Squares Formulas 95. B. Topic: Regression Terminology . C. 60 59 58 57 There is one residual for every observation. D. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot.Topic: Confidence and Prediction Intervals for Y 94. D. In a simple bivariate regression with 60 observations there will be _____ residuals. Which is not an assumption of least squares regression? A. 2 = 20. B. The critical value for a two-tailed test of H0: β1 = 0 at α = .SST/SSE We use the ANOVA sums of squares to calculate R2. and F test. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. Topic: Ordinary Least Squares Formulas 97.96.086 for d.086 ±2.960 From Appendix D.528 ±1. C. ANOVA table.2 = 22 .05 in a simple regression with 22 observations is: A. SSR/SSE SSR/SST 1 . tcrit = ±2. Which is correct to find the value of the coefficient of determination (R2)? A. B. = n . AACSB: Analytic Blooms: Remember Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error.725 ±2. D.f. R2. C. Topic: Tests for Significance . ±1. 2.502)]1/2 = 2. neither α = .40)[(23 . both α = .f. = 23 .500 would be: A.. tcalc = r[(n .559. B.2)/(1 . can't say without knowing α.400 provides sufficient evidence to conclude that the population correlation coefficient exceeds zero in a right-tailed test at: A.2)/(1 .2)/(1 .518.05 but not α = . tcalc = r[(n .50)[(23 . C. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Visual Displays and Correlation Analysis .2 = 21.01 but not α = . However. In a sample of size n = 23.2)/(1 .r2)]1/2 = (..402)]1/2 = 2. In a sample of n = 23.721 for d. α = .646.05 nor α = . AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. a sample correlation of r = .01.01 = 2. D.819.r2)]1/2 = (.98.05 = 1. α = . Topic: Visual Displays and Correlation Analysis 99. the test would not be significant for t. 2.000 > t.05. C. 2. D. the Student's t test statistic for a correlation of r = .01.01. B.05 and α = .646. B.f.05 is: A.0252 + n . D.2 = 21.2 = 21.646 ±2.0692 + 23 .025 = ±2. C. In a sample of n = 23.2 = 23 . C.819 ±2.4115 for d.069)/(2. = 23 .524 ±.500 ±. ±2. B. Topic: Visual Displays and Correlation Analysis 101. In a sample of n = 23. Topic: Tests for Significance .080 for d. = n . D. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. ±.f.497 Use rcrit = t.229 ±2.025/(t. t. the critical value of the correlation coefficient for a two-tailed test at α = .05 is: A.412 ±.2)1/2 = .100.2)1/2 = (2. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. the critical value of Student's t for a two-tailed test of significance for a simple bivariate regression at α = .080 From Appendix D. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance..025 and α = .r2)]1/2 = (. tcalc = r[(n . the Student's t test statistic for a correlation of r = .686.690 > t. tcalc = r[(n . α = . 2.024 for d.05 but not α = .025 but not α = .2)/(1 .400 provides sufficient evidence to conclude that the population correlation coefficient exceeds zero in a right-tailed test at: A. neither α = . AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-01 Calculate and test a correlation coefficient for significance.402)]1/2 = 2. can't say without knowing if it's a two-tailed or one-tailed test. D.025.402)]1/2 = 1. = 40 . Topic: Visual Displays and Correlation Analysis 103. a sample correlation of r = . both α = .400 would be: A. In a sample of n = 20.05. 1.025 nor α = .05.40)[(20 .025 = 2.2)/(1 .852. 1.2)/(1 . The test would also be significant a fortiori if we used t. In a sample of n = 40.102.f. α = .110 B..645 C.40)[(40 . C.r2)]1/2 = (.05 = 1.2)/(1 .05.2 = 38. B. Topic: Visual Displays and Correlation Analysis .852 D. 587 ±. the critical value of the correlation coefficient for a two-tailed test at α = . In a sample of n = 20.025 = ±2.4437 for d. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. In a sample of n = 27.104. C.2)1/2 = (2.060 for d.412 ±.060 ±2.05 is: A.1012 + 20 . = 20 .025/(t. Topic: Tests for Significance .898 ±2.f.497 Use rcrit = t. ±2. the critical value of Student's t for a two-tailed test of significance for a simple bivariate regression at α = .0252 + n .074 From Appendix D.f.101)/(2.052 ±2.05 is: A. C. B. ±.2 = 27 .2 = 25. D.444 ±. = n .2 = 18. D. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests. t.2)1/2 = . B. Topic: Visual Displays and Correlation Analysis 105. 728 for d.450 would be: A.(-. The test would also be significant a fortiori if we used t. α = .40)2)]1/2 = -2. a sample correlation of r = -. C. D.110. tcalc = r[(n . the Student's t test statistic for a correlation of r = -.938.005 = -2.450 provides sufficient evidence to conclude that the population correlation coefficient differs significantly from zero in a two-tailed test at: A. B.f. Topic: Visual Displays and Correlation Analysis .(-.45)[(36 . C.r2)]1/2 = (-.2)/(1 .01 and α = .938.05 both α = .05.938 < t. = 34. -2. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance.030. B. -2.01 nor α = . can't say without knowing α. In a sample of size n = 36.2)/(1 .032 AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-01 Calculate and test a correlation coefficient for significance.r2)]1/2 = (-.05.40)2)]1/2 = -2. D. -2. tcalc = r[(n .2)/(1 .2)/(1 . neither α = .01 α = . In a sample of n = 36.45)[(36 . Topic: Visual Displays and Correlation Analysis 107.025 = -2.106. 2)1/2 = .3191 for d.724 2.025 = ±2. D.2 = 34.387 ±. t. B. In a sample of n = 36. 2.05 is: A.032)/(2. ±.05 is: A.032 2. B.2 = 34.423 ±. the critical value of Student's t for a two-tailed test of significance of the slope for a simple regression at α = .497 Use rcrit = t.074 From Appendix D. Topic: Tests for Significance .025/(t.329 ±.2)1/2 = (2. = n . the critical value of the correlation coefficient for a two-tailed test at α = .032 for d. D. C. Topic: Visual Displays and Correlation Analysis 109.f.938 2. = 36 . C.f. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. In a sample of n = 36.108.0252 + n .2 = 36 . AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests.0322 + 36 . the expected Time would increase by: A.73 days days days days 50(0. B. AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation.0214 Distance. D.126 + 0. A local trucking company fitted a regression to relate the travel time (days) of its shipments as a function of the distance traveled (miles). C.110. 1. The fitted regression is Time = -7. Topic: Simple Regression .07 7. If Distance increases by 50 miles.0214) = 1.07.13 2.14 1. $104. B. A local trucking company fitted a regression to relate the cost of its shipments as a function of the distance traveled.8666(50) = $143. the expected shipping cost would increase by: A. Based on this estimated relationship.111. $143. Topic: Simple Regression .33. C. The Excel fitted regression is shown. 2. D. when distance increases by 50 miles. $301. $286. AACSB: Analytic Blooms: Apply Difficulty: 1 Easy Learning Objective: 12-02 Interpret the slope and intercept of a regression equation. then: A. the slope is likely to be insignificant. Topic: Ordinary Least Squares Formulas . C.81. the standard error would be large. the coefficient of determination is . R2 = SSR/SST = SSR/(SSR + SSE) = 2592/(2592 + 608) = . the SST would be smaller than SSR. and F test. R2.81. ANOVA table.112. D. SST cannot be smaller than SSR because SST = SSR + SSE. B. If SSR is 2592 and SSE is 608. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-08 Interpret the standard error. The significance and standard error cannot be judged without more information. 113. B. Find the sample correlation coefficient for the following data. YData) to verify your calculation using the formula for r. .8911 .9556 Use Excel =CORREL(XData. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance. Topic: Visual Displays and Correlation Analysis .9124 . D. C.9822 . A. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. B.762 -2. = b0 + b1x.228 Use Excel to verify your calculations using the formulas for b0 and b1. 1.833 3. C. Find the slope of the simple regression A. Topic: Ordinary Least Squares Formulas .114. D.294 0. YData) to verify your calculation using the formula for r. B.8736 . D. A. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-01 Calculate and test a correlation coefficient for significance.115. C. .7291 .9118 . Find the sample correlation coefficient for the following data.9563 Use Excel =CORREL(XData. Topic: Visual Displays and Correlation Analysis . D. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot.884 Use Excel to verify your calculations using the formulas for b0 and b1.109 -2. Topic: Ordinary Least Squares Formulas . B. = b0 + b1x.221 1.595 1. C. 2.116. Find the slope of the simple regression A. A researcher's results are shown below using n = 25 observations.284].118.998]. -0. For d.99855). [ -3. -1. 5.117.2 = 23.2 = 25 . D.217].025 = 2.f.069) (0.069.2834 ± (2. +0.282.998. so -2. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-05 Calculate and interpret confidence intervals for regression coefficients. [ -4. C. = n . The 95 percent confidence interval for the slope is: A.349.026]. [ -0. t. Topic: Tests for Significance . [1. B. 2. For d.268. so 1.f.333. [1. [1. [1.284]. t.449]. = n .447. Topic: Tests for Significance .447) (0. A researcher's regression results are shown below using n = 8 observations. The 95 percent confidence interval for the slope is: A.064].118. D. 2. B. 2. 2. C.2 = 8 .602. [1.118.398].8333 ± (2.2307).025 = 2. AACSB: Analytic Blooms: Apply Difficulty: 2 Medium Learning Objective: 12-05 Calculate and interpret confidence intervals for regression coefficients.2 = 6. 0) = 2. C. so the indicated slope less than 1 must be wrong. AACSB: Analytic Blooms: Apply Difficulty: 3 Hard Learning Objective: 12-04 Fit a simple regression on an Excel scatter plot. What do you say? A. plus the visual intercept is 100 (not 154. Bob thinks there is something wrong with Excel's fitted regression. Bob needs to increase his sample size to decide.625. so the equation is credible.119. D. B. Topic: Ordinary Least Squares Formulas Short Answer Questions .2284. A visual estimate of the slope is Δy/Δx = (625 . The R2 looks a little high but otherwise it looks OK. The relationship is linear.100)/(200 . The estimated equation is obviously incorrect.61) and the fit seems better than R2 = . ..0001) and the F statistic is huge. If Weight = 3000. we predict MPG = 47..711).0484 . Is the intercept meaningful in this regression? Make a prediction of CityMPG when Weight = 3000. Pedro became interested in vehicle fuel efficiency. so he performed a simple regression using 93 cars to estimate the model CityMPG = β0 + β1 Weight where Weight is the weight of the vehicle in pounds.0484 .0484 ..05 mpg. The intercept is not meaningful since no vehicle has zero weight or a weight close to zero.0080(4000) = 15.0080(3000) = 23. If Weight = 4000. Write a brief analysis of these results. The highly significant predictor Weight is consistent with the high coefficient of determination (R2 = .0080 Weight = 47. which says that well over half the variation in MPG is explained by Weight. His results are shown below.0080 Weight = 47.05 mpg. we predict MPG = 47. Do these predictions seem believable? If you could make a car 1000 pounds lighter. and also when Weight = 4000.0484 . We expect a negative slope (heavier vehicles would get lower MPG). The coefficient of Weight differs from zero at any common value of α (the p-value is less than .120. using what you have learned in this chapter. The confidence interval for the coefficient of the predictor Weight does not include zero. what change would you predict in its CityMPG? It is reasonable that a causal relationship might exist between a vehicle's weight and its MPG. . which says that well over half the variation in MPG is explained by Weight. When Weight = 4000. The intercept is not meaningful since no vehicle has zero weight or any weight close to zero. we predict MPG = 47.Feedback: It is reasonable to postulate that a causal relationship might exist between a vehicle's weight and its MPG. The coefficient of Weight differs from zero at any common value of α (the p-value is less than .711). as anticipated a priori.0080(4000) = 15. The slope's sign is negative. Our a priori expectation would be that the slope should be negative since we would expect that heavier vehicles would get lower MPG. we would predict MPG = 47. Topic: Tests for Significance ...0080 Weight = 47.05 mpg. If Weight = 3000.0484 .0484 .0080 Weight = 47.05 mpg.0484 .0001) and the F statistic is huge.0484 . AACSB: Reflective Thinking Blooms: Evaluate Difficulty: 3 Hard Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests..0080(3000) = 23. The highly significant predictor Weight is consistent with the high coefficient of determination (R2 = . The confidence interval for the coefficient of the predictor Weight does not include zero. . 0210 .. The intercept is meaningful if Age = 0 was in the sample data set (or at least some Age value near zero). The confidence interval for the coefficient of Age does not include zero. Mary noticed that old coins are smoother and more worn. Make a prediction of Weight when Age = 10.121.. If Age = 10. the coefficient of determination (R2 = . we predict Weight = 5. Her results are shown below.0040(10) = 4. we predict Weight = 5.0040(20) = 4. Write a brief analysis of these results.0001) and the F test statistic is large.0210 . and its sign is negative. What does this tell you? Is the intercept meaningful in this regression? It is reasonable to postulate a causal relationship between a coin's age and its weight (negative slope..0210 . .0210 . She weighed 31 nickels and recorded their age. The coefficient of Age differs from zero at any common α (the p-value is less than . and then performed a simple regression to estimate the model Weight = β0 + β1 Age where weight is the weight of the coin in grams and Age is the age of the coin in years. using what you have learned in this chapter.0040 Age = 5. since we would expect that coins will wear down with usage). If Age = 20..941 gm. and also when Age = 20..e. as anticipated a priori. The intercept is logically meaningful because Age = 0 is something we might observe (i.0040 Age = 5.981 gm. a newly minted nickel). Despite the significant predictor Age.442) shows that less than half the variation in nickel weights is explained by Age. .0040(20) = 4. Topic: Tests for Significance . The coefficient of Age differs from zero at any common value of α (the p-value is less than .Feedback: It is reasonable to postulate that a causal relationship might exist between a coin's age and its weight.941 gm. we would predict Weight = 5. and its sign is negative. If Age = 20.981 gm.0210 .0001) and the F test statistic is quite large.442) shows that less than half the variation in nickel weights is explained by Age. Our a priori expectation would be that the slope should be negative since we would expect that coins will wear down with usage.. The intercept is meaningful..0040 Age = 5.0210 ..e. The confidence interval for the coefficient of Age does not include zero. a newly minted nickel).0210 .. Despite the highly significant predictor Age. the coefficient of determination (R2 = .0210 .0040(10) = 4. The intercept is logically meaningful a priori because Age = 0 is something we might easily observe (i. Our predictions: If Age = 10. as anticipated a priori. assuming that Age = 0 years was included in the sample data set (or at least some Age value near zero).0040 Age = 5. we would predict Weight = 5. AACSB: Reflective Thinking Blooms: Evaluate Difficulty: 3 Hard Learning Objective: 12-06 Test hypotheses about the slope and intercept by using t tests.