Exam 13: Simple Linear Regression

TABLE 13-4 The managers of a brokerage firm are interested in finding out if the number of new clients a broker brings into the firm affects the sales generated by the broker. They sample 12 brokers and determine the number of new clients they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows. Broker Cliente Sales 1 27 52 2 11 37 3 42 64 4 33 55 5 15 29 6 15 34 7 25 58 8 36 59 9 28 44 10 30 48 11 17 31 12 22 38 -Referring to Table 13-4, the managers of the brokerage firm wanted to test the hypothesis that the true slope was equal to 0. The value of the test statistic is _____.

TABLE 13- 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: Regression Statistics Multiple R 0.8531 RSquare 0.7278 Adjusted R Square 0.7180 Standard Error 47.8668 Observations 30 ANOVA d f SS MS F Significance F Regression 1 171499.78 171499.78 74.8505 2.1259E-09 Residual 28 64154.42 2291.23 Total 29 235654.20 Coefficients Standard Error t Stat p -value Lower 95\% Upper 95\% Intercept 76.5351 11.8318 6.4686 5.24-07 52.2987 100.7716 Gross 4.3331 0.5008 8.6516 2.13-09 3.3072 5.3590 -Referring to Table 13-11, the Durbin-Watson statistic is inappropriate for this data set.

TABLE 13-4 The managers of a brokerage firm are interested in finding out if the number of new clients a broker brings into the firm affects the sales generated by the broker. They sample 12 brokers and determine the number of new clients they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows. Broker Cliente Sales 1 27 52 2 11 37 3 42 64 4 33 55 5 15 29 6 15 34 7 25 58 8 36 59 9 28 44 10 30 48 11 17 31 12 22 38 -Referring to Table 13-4, suppose the managers of the brokerage firm want to obtain a 99% confidence interval estimate for the mean sales made by brokers who have brought into the firm 24 new clients. The t critical value they would use is______ .

TABLE 13-4 The managers of a brokerage firm are interested in finding out if the number of new clients a broker brings into the firm affects the sales generated by the broker. They sample 12 brokers and determine the number of new clients they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows. Broker Cliente Sales 1 27 52 2 11 37 3 42 64 4 33 55 5 15 29 6 15 34 7 25 58 8 36 59 9 28 44 10 30 48 11 17 31 12 22 38 -Referring to Table 13-4, ______% of the total variation in sales generated can be explained by the number of new clients brought in.

TABLE 13- 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: Regression Statistics Multiple R 0.8531 RSquare 0.7278 Adjusted R Square 0.7180 Standard Error 47.8668 Observations 30 ANOVA d f SS MS F Significance F Regression 1 171499.78 171499.78 74.8505 2.1259E-09 Residual 28 64154.42 2291.23 Total 29 235654.20 Coefficients Standard Error t Stat p -value Lower 95\% Upper 95\% Intercept 76.5351 11.8318 6.4686 5.24-07 52.2987 100.7716 Gross 4.3331 0.5008 8.6516 2.13-09 3.3072 5.3590 -Referring to Table 13-11, the normality of error assumption appears to have been violated.

TABLE 13- 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: Regression Statistics Multiple R 0.8531 RSquare 0.7278 Adjusted R Square 0.7180 Standard Error 47.8668 Observations 30 ANOVA d f SS MS F Significance F Regression 1 171499.78 171499.78 74.8505 2.1259E-09 Residual 28 64154.42 2291.23 Total 29 235654.20 Coefficients Standard Error t Stat p -value Lower 95\% Upper 95\% Intercept 76.5351 11.8318 6.4686 5.24-07 52.2987 100.7716 Gross 4.3331 0.5008 8.6516 2.13-09 3.3072 5.3590 -Referring to Table 13-11, the homoscedasticity of error assumption appears to have been violated.

TABLE 13-8 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. \text {Regressing GPA on \mathrm { ACT }} Regressing GPA on Regression Statistics Multiple R 0.7598 R Square 0.5774 Adjusted R Square 0.5069 Standard E rror 0.2691 Observations 8 ANOVA d f SS MS F Significance F Regression 1 0.5940 0.5940 8.1986 0.0286 Residual 6 0.4347 0.0724 Total 7 1.0287 Coefficients Standard Error T Stat p -value Lower 95\% Upper 95\% Intercept 0.5681 0.9284 0.6119 0.5630 -1.7036 2.8398 ACT 0.1021 0.0356 2.8633 0.0286 0.0148 0.1895 -Referring to Table 13-8, the interpretation of the coefficient of determination in this regression is

The Regression Sum of Squares (SSR) can never be greater than the Total Sum of Squares (SST).

When r = - 1, it indicates a perfect relationship between X and Y.

TABLE 13-8 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. Regressing GPA on Regression Statistics Multiple R 0.7598 R Square 0.5774 Adjusted R Square 0.5069 Standard E rror 0.2691 Observations 8 ANOVA d f SS MS F Significance F Regression 1 0.5940 0.5940 8.1986 0.0286 Residual 6 0.4347 0.0724 Total 7 1.0287 Coefficients Standard Error T Stat p -value Lower 95\% Upper 95\% Intercept 0.5681 0.9284 0.6119 0.5630 -1.7036 2.8398 ACT 0.1021 0.0356 2.8633 0.0286 0.0148 0.1895 -Referring to Table 13-8, what are the decision and conclusion on testing whether there is any linear relationship at 1% level of significance between GPA and ACT scores?

TABLE 13-12 The manager of the purchasing department of a large banking organization would like to develop a model to predict the amount of time (measured in hours) it takes to process invoices. Data are collected from a sample of 30 days, and the number of invoices processed and completion time in hours is recorded. Below is the regression output: Regression Statistics Multiple R 0.9947 R Square 0.8924 Adjusted R Square 0.8886 Standard Error 0.3342 ations 30 d f SS MS F Significance F Regression 1 25.9438 25.9438 232.2200 4.3946-15 Residual 28 3.1282 0.1117 Total 29 29.072 Coefficients Standard Error t Stat p -value Lower 95\% Upper 95\% Invoices 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Processed 0.0126 0.0008 15.2388 4.3946-15 0.0109 0.0143 -Referring to Table 13-12, the estimated average amount of time it takes to process one additional invoice is

TABLE 13-12 The manager of the purchasing department of a large banking organization would like to develop a model to predict the amount of time (measured in hours) it takes to process invoices. Data are collected from a sample of 30 days, and the number of invoices processed and completion time in hours is recorded. Below is the regression output: Regression Statistics Multiple R 0.9947 R Square 0.8924 Adjusted R Square 0.8886 Standard Error 0.3342 ations 30 d f SS MS F Significance F Regression 1 25.9438 25.9438 232.2200 4.3946-15 Residual 28 3.1282 0.1117 Total 29 29.072 Coefficients Standard Error t Stat p -value Lower 95\% Upper 95\% Invoices 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Processed 0.0126 0.0008 15.2388 4.3946-15 0.0109 0.0143 -Referring to Table 13-12, the p-value of the measured F-test statistic to test whether the number of invoices processed affects the amount of time is

TABLE 13-4 The managers of a brokerage firm are interested in finding out if the number of new clients a broker brings into the firm affects the sales generated by the broker. They sample 12 brokers and determine the number of new clients they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows. Broker Cliente Sales 1 27 52 2 11 37 3 42 64 4 33 55 5 15 29 6 15 34 7 25 58 8 36 59 9 28 44 10 30 48 11 17 31 12 22 38 -Referring to Table 13-4, suppose the managers of the brokerage firm want to obtain a 99% prediction interval for the sales made by a broker who has brought into the firm 18 new clients. The t critical value they would use is ____.

TABLE 13-9 It is believed that, the average numbers of hours spent studying per day (HOURS) during undergraduate education should have a positive linear relationship with the starting salary (SALARY, measured in thousands of dollars per month) after graduation. Given below is the Excel output from regressing starting salary on number of hours spent studying per day for a sample of 51 students. NOTE: Some of the numbers in the output are purposely erased. Regression Statistics Multiple R 0.8857 R Square 0.7845 Adjusted R Square 0.7801 Standard Error 1.3704 Observations 51 $\text { ANOVA }$ df SS MS F Significance F Regression 1 335.0472 335.0473 178.3859 Residual 1.8782 Total 50 427.0798 -Referring to Table 13-9, the estimated average change in salary (in thousands of dollars) as a result of spending an extra hour per day studying is

TABLE 13-5 }\] The managing partner of an advertising agency believes that his company's sales are related to the industry sales. He uses Microsoft Excel's Data Analysis tool to analyze the last 4 years of quarterly data with the following results: Regression Statistics Multiple R 0.802 R Square 0.643 Adjusted R Square 0.618 Standard Error SYX 0.9224 $\text { Observations } \quad 16$ $\text { ANOVA }$ df SS MS F Sig.F Regression 1 21.497 21.497 25.27 0.000 Error 14 11.912 0.851 Total 15 33.409 Predictor Coefficients Standard Error t Stat p-value Intercept 3.962 1.440 2.75 0.016 Industry 0.040451 0.008048 5.03 0.000 Durbin- Watson Statistic 1.59 -Referring to Table 13-5, the value of the quantity that the least squares regression line minimizes is_____ .

TABLE 13-9 It is believed that, the average numbers of hours spent studying per day (HOURS) during undergraduate education should have a positive linear relationship with the starting salary (SALARY, measured in thousands of dollars per month) after graduation. Given below is the Excel output from regressing starting salary on number of hours spent studying per day for a sample of 51 students. NOTE: Some of the numbers in the output are purposely erased. Regression Statistics Multiple R 0.8857 R Square 0.7845 Adjusted R Square 0.7801 Standard Error 1.3704 Observations 51 $\text { ANOVA }$ df SS MS F Significance F Regression 1 335.0472 335.0473 178.3859 Residual 1.8782 Total 50 427.0798 -Referring to Table 13-9, the degrees of freedom for the F test on whether HOURS affects SALARY are Coefficients Standaad Error t Stat p -value Lower 95\% Upper 95\% Intercept -1.8940 0.4018 -4.7134 2.051-05 -2.7015 -1.0865 Hours 0.9795 0.0733 13.3561 5.944-18 0.8321 1.1269

TABLE 13- 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: Regression Statistics Multiple R 0.8531 RSquare 0.7278 Adjusted R Square 0.7180 Standard Error 47.8668 Observations 30 ANOVA d f SS MS F Significance F Regression 1 171499.78 171499.78 74.8505 2.1259E-09 Residual 28 64154.42 2291.23 Total 29 235654.20 Coefficients Standard Error t Stat p -value Lower 95\% Upper 95\% Intercept 76.5351 11.8318 6.4686 5.24-07 52.2987 100.7716 Gross 4.3331 0.5008 8.6516 2.13-09 3.3072 5.3590 -Referring to Table 13-11, 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?

TABLE 13-3 The director of cooperative education at a state college wants to examine the effect of cooperative education job experience on marketability in the work place. She takes a random sample of 4 students. For these 4, she finds out how many times each had a cooperative education job and how many job offers they received upon graduation. These data are presented in the table below. Student CoopJobs JobOffer 1 1 4 2 2 6 3 1 3 4 0 1 -Referring to Table 13-3, the director of cooperative education wanted to test the hypothesis that the true slope was equal to 0. The p-value of the test is between _________ and_________.

Data that exhibit an autocorrelation effect violate the regression assumption of independence.

TABLE 13-01 A large national bank charges local companies for using their services. A bank official reported the results of a regression analysis designed to predict the bank's charges (Y) -- measured in dollars per month -- for services rendered to local companies. One independent variable used to predict service charge to a company is the company's sales revenue (X) -- measured in millions of dollars. Data for 21 companies who use the bank's services were used to fit the model: $E(Y)=\beta_{0}+\beta_{1} X$ The results of the simple linear regression are provided below. $\hat{Y}=-2,700+20 X, \mathrm{~S}_{Y X}=65, \text { two-tailed } p \text { value }=0.034\left(\text { for testing } \beta_{1}\right)$ -Referring to Table 13-1, interpret the estimate of þ0, the Y-intercept of the line.