Chapter 12 Assessment Answer Key [1202] TABLE 12-8 It is believed that GPA (grade point average, based on a four point scale) should have a positive linear relationship with ACT scores. Given below is the Excel output from regressing GPA on ACT scores using a data set of 8 randomly chosen students from a Big-Ten university: Referring to Table 12-8, what is the predicted average value of GPA when ACT = 20? The table provides the coefficients for the underlying regression equation fit to the data: For a value of X = 20, the corresponding Y estimate is: 2.61 [1203] What do we mean when we say that a simple linear regression model is “statistically” useful? The linear model, where we try to predict the valiue of Y based on some X and coefficients is a better predictor of Y than the sample mean, Ybar. Chapter 12 Assessment Page 1 Chapter 12 Assessment Answer Key [1204] TABLE 12-2 A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do this, the company randomly chooses 6 small cities and offers the candy bar at different prices. Using candy bar sales as the dependent variable, the company will conduct a simple linear regression on the data below: Referring to Table 12-2, what is the estimated slope parameter for the candy bar price and sales data? Equations 12-3 and 12-4 can provide the values. Using PhStat, linear Regression. With the ANOVA / coefficients 1204 Regression Statistics Multiple R 0.88540441 R Square 0.783940969 Adjusted R Square Standard Error 0.729926212 16.29860664 Observations 6 ANOVA df SS MS Regression 1 3855.421687 3855.421687 Residual 4 1062.578313 265.6445783 Total 5 4918 Coefficients Standard Error t Stat F Significance F 14.51345897 0.018945781 P-value Lower 95% Upper 95% Intercept 161.3855422 26.16068642 6.169010231 0.003505887 88.75183241 234.0192519 Price ($) -48.19277108 12.65017211 -3.809653392 0.018945781 -83.31527952 -13.07026265 The slope coefficient is highlighted. Chapter 12 Assessment Page 2 Chapter 12 Assessment Answer Key [1205] TABLE 12-11 A company that has the distribution rights to home video sales of previously released movies would like to use the box office gross (in millions of dollars) to estimate the number of units (in thousands of units) that it can expect to sell. Following is the output from a simple linear regression along with the residual plot and normal probability plot obtained from a data set of 30 different movie titles: Chapter 12 Assessment Page 3 Chapter 12 Assessment Answer Key Referring to Table 12-11, what are, respectively, the lower and upper limits of the 95% confidence interval estimate for the average change in video unit sales as a result of a one million dollars increase in box office? These values are provided in the table. – the lower and higher limits on the number of units sold, in thousands of videos: 3.3 to 5.36 thousand. Be very careful about units, though, as the equation requires gross box office in millions. [1206] 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 H0: ρ= 0 against the one-sided alternative H1: ρ > 0 . To test H0: ρ = 0 against the two-sided alternative H1: ρ ≠ 0 at a significance level of 0.2, the p-value is: This problem does not require a calculation. Rather, it’s a definitional concern. When you are evaluating against a statistic with a symmetric distribution, going from a one-tail test to a two tail test with a given significance will double the p-value, making the outcome more likely to have occurred by chance. (0.256)(2) Chapter 12 Assessment Page 4 Chapter 12 Assessment Answer Key [1207] TABLE 12-11 A company that has the distribution rights to home video sales of previously released movies would like to use the box office gross (in millions of dollars) to estimate the number of units (in thousands of units) that it can expect to sell. Following is the output from a simple linear regression along with the residual plot and normal probability plot obtained from a data set of 30 different movie titles: Chapter 12 Assessment Page 5 Chapter 12 Assessment Answer Key Referring to Table 12-11, what is the value of the test statistic for testing whether there is a linear relationship between box office gross and home video unit sales? The coefficient in question is the Gross value – that’s what relates Gross to home unit sales. The test statistic for that coefficient is the t-stat appearing next to it, with a value around 8.6, representing a very high confidence that the relationship exists, and that the null hypothesis of a zero slope should be dismissed. [1208] Testing for the existence of correlation is equivalent to Testing for the existence of the slope β1. Correlation, by definition, is the existence of a non-zero relationship between the independent and dependent variables, which is represented by β1. . Chapter 12 Assessment Page 6 Chapter 12 Assessment Answer Key [1209] TABLE 12-11 A company that has the distribution rights to home video sales of previously released movies would like to use the box office gross (in millions of dollars) to estimate the number of units (in thousands of units) that it can expect to sell. Following is the output from a simple linear regression along with the residual plot and normal probability plot obtained from a data set of 30 different movie titles Chapter 12 Assessment Page 7 Chapter 12 Assessment Answer Key " True or False: Referring to Table 12-11, the normality of error assumption appears to have been violated. False. The text explains in section 12.5 that we need to at the residuals, the difference between the observed and the predicted data for each value of x. Look at the Normal Probability. The apparent clustering of the observations near the zero point, gradual spreading symmetrically as the Z values (x axis) become more extreme, and a roughly equivalent number of points in the extreme values of Z, all give some sense that this data is normally distributed. More advanced statistics classes provide formal mechanisms for judging this. Chapter 12 Assessment Page 8 Chapter 12 Assessment Answer Key [1213] TABLE 12-2 A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do this, the company randomly chooses 6 small cities and offers the candy bar at different prices. Using candy bar sales as the dependent variable, the company will conduct a simple linear regression on the data below: Referring to Table 12-2, what is the standard error of the regression slope estimate, Sb1? Using PhStat, linear Regression. With the ANOVA / coefficients 1204 Regression Statistics Multiple R 0.88540441 R Square 0.783940969 Adjusted R Square Standard Error 0.729926212 16.29860664 Observations 6 ANOVA df SS MS Regression 1 3855.421687 3855.421687 Residual 4 1062.578313 265.6445783 Total 5 4918 Coefficients Standard Error t Stat F Significance F 14.51345897 0.018945781 P-value Lower 95% Upper 95% Intercept 161.3855422 26.16068642 6.169010231 0.003505887 88.75183241 234.0192519 Price ($) -48.19277108 12.65017211 -3.809653392 0.018945781 -83.31527952 -13.07026265 The slope standard error is highlighted. Chapter 12 Assessment Page 9