Exam 13: Simple Linear Regression

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 population slope?

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 Stedistics Multiple R 0.8857 RSquare 0.7845 Adjusted R Square 0.7801 Standard Error 1.3704 $\text { Observations } \quad 51$ $ANOVA$ df SS MS F Significance F Regression 1 335.0472 335.0473 178.3859 Residual 1.8782 Total 50 427.0798 Coeffcients Standard 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 -Referring to Table 13-9, to test the claim that average SALARY depends positively on HOURS against the null hypothesis that average SALARY does not depend linearly on HOURS, the p-value of the test statistic 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 13-3, the least squares estimate of the slope 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 Clients 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 least squares estimate of the Y-intercept 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 90% confidence interval for the average change in SALARY (in thousands of dollars) as a result of spending an extra hour per day studying is

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 value of the test statistic 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, set up a scatter diagram.

TABLE 13-7 An investment specialist claims that if one holds a portfolio that moves in the opposite direction to the market index like the S&P 500, then it is possible to reduce the variability of the portfolio's return. In other words, one can create a portfolio with positive returns but less exposure to risk. A sample of 26 years of S&P 500 index and a portfolio consisting of stocks of private prisons, which are believed to be negatively related to the S&P 500 index, is collected. A regression analysis was performed by regressing the returns of the prison stocks portfolio (Y) on the returns of S&P 500 index (X) to prove that the prison stocks portfolio is negatively related to the S&P 500 index at a 5% level of significance. The results are given in the following EXCEL output. Coefficients Standard Error T Stat p -value Intercept 4.866004258 0.35743609 13.61363441 8.7932-13 S\& P -0.502513506 0.071597152 -7.01862425 294942-07 -Referring to Table 13-7, to test whether the prison stocks portfolio is negatively related to the S&P 500 index, the appropriate null and alternative hypotheses are, respectively,

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 least squares estimate of the slope is _____.

TABLE 13-10 The management of a chainelectronicstore would like to develop a model for predicting the weekly sales (in thousandof dollars) for individual stores based on the number of customers whomade purchases. A random sample of 12 stores yields the followingresults: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Regression analysis is used for prediction, while correlation analysis is used to measure the strength of the association between two numerical variables.

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) =?₀ + ?₁X The results of the simple linear regression are provided below. $\hat{Y}=-2,700+20 X, S_{Y X}=65$ , two-tailed $p$ value $=0.034\left(\right.$ for testing $\left.\beta_{1}\right)$ -Referring to Table 13-1, a 95% confidence interval for ?₁ is (15, 30). Interpret the interval.

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 3.0. The value of the test statistic is_______ .

TABLE 13-10 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousand of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Referring to Table 13-3, the director of cooperative education wanted to test the hypothesis that the true slope was equal to 3.0. For a test with a level of significance of 0.05, the null hypothesis should be rejected if the value of the test statistic is __________.

TABLE 13-2 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: City Price (\ ) Sales River Falls 1.30 100 Hudson 1.60 90 Ellsworth 1.80 90 Prescott 2.00 40 Rock Elm 2.40 38 Stillwater 2.90 32 -Referring to Table 13-2, what is the coefficient of correlation for these data?

TABLE 13-10 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousand of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Customers Sales (Thousands of Dollars) 907 11.20 926 11.05 713 8.21 741 9.21 780 9.42 898 10.08 510 6.73 529 7.02 460 6.12 872 9.52 650 7.53 603 7.25 -Referring to Table 13-10, generate the scatter plot.

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 13-5, the partner wants to test for autocorrelation using the Durbin-Watson statistic. Using a level of significance of 0.05, the critical values of the test are d_L = _____ , and d_U = ______.

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 error sum of squares (SSE) of the above regression is

In performing a regression analysis involving two numerical variables, we are assuming

$\text { TABLE 13-10 }$ The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousand of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results: Cugtomera 5aleg (Thougardaof Dollara) 11.20 11.05 713 741 780 510 529 7.02 460 7.53 7.25 -Referring to Table 13-10, the residual plot indicates possible violation of which assumptions?

TABLE 13-7 An investment specialist claims that if one holds a portfolio that moves in the opposite direction to the market index like the S&P 500, then it is possible to reduce the variability of the portfolio's return. In other words, one can create a portfolio with positive returns but less exposure to risk. A sample of 26 years of S&P 500 index and a portfolio consisting of stocks of private prisons, which are believed to be negatively related to the S&P 500 index, is collected. A regression analysis was performed by regressing the returns of the prison stocks portfolio (Y) on the returns of S&P 500 index (X) to prove that the prison stocks portfolio is negatively related to the S&P 500 index at a 5% level of significance. The results are given in the following EXCEL output. Coefficients Standard Error T Stat p-value Intercept 4.866004258 0.35743609 13.61363441 8.7932-13 S\& P -0.502513506 0.071597152 -7.01862425 294942-07 -Referring to Table 13-7, to test whether the prison stocks portfolio is negatively related to the S&P 500 index, the measured value of the test statistic is