Exam 13: Simple Linear Regression

TABLE 13-10 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands 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, what is the value of the standard error of the estimate?

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. 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 number of new clients brought in had a positive impact on the amount of sales generated. 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. 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 population slope was equal to 0. The denominator of the test statistic is $S_{b_{1}}$ . The value of $S_{b_{1}}$ in this sample is ________.

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 4.8660 0.3574 13.6136 8.7932-13 S \&P -0.5025 0.0716 -7.0186 2.94942-07 Note: 2.94942E-07 = 2.94942*10⁻⁷ -Referring to Table 13-7, which of the following will be a correct conclusion?

TABLE 13-12 The manager of the purchasing department of a large saving and loan organization would like to develop a model to predict the amount of time (measured in hours) it takes to record a loan application. Data are collected from a sample of 30 days, and the number of applications recorded and completion time in hours is recorded. Below is the regression output: Regression Statistics Multiple R 0.9447 R Square 0.8924 Adjusted R 0.8886 Square Standard 0.3342 Error Observations 30 $\text { ANOVA }$ df 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\% Intercept 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Applications Recorded 0.0126 0.0008 15.2388 4.3946-15 0.0109 0.0143 Note: 4.3946E-15 is 4.3946*10^-15 -Referring to Table 13-12, the p-value of the measured F test statistic to test whether the number of loan applications recorded affects the amount of time 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 for predicting starting salary (Y) using number of hours spent studying per day (X) for a sample of 51 students. NOTE: Only partial output is shown. 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 Standard Coefficients 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 Note: 2.051E - 05 = 2.051*10⁻⁰⁵ and 5.944E - 18 = 5.944*10⁻¹⁸. -Referring to Table 13-9, the estimated change in mean salary (in thousands of dollars)as a result of spending an extra hour per day studying is

The Y-intercept (b₀)represents the

TABLE 13-10 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands 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, what are the degrees of freedom of the F test statistic when testing whether the number of customers who make purchases is a good predictor for weekly sales?

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 for predicting starting salary (Y) using number of hours spent studying per day (X) for a sample of 51 students. NOTE: Only partial output is shown. 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 Standard Coefficients 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 Note: 2.051E - 05 = 2.051*10⁻⁰⁵ and 5.944E - 18 = 5.944*10⁻¹⁸. -Referring to Table 13-9, the p-value of the measured F-test statistic to test whether HOURS affects SALARY is

TABLE 13-10 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands 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, the null hypothesis for testing whether the number of customers who make a purchase affects weekly sales cannot be rejected if a 1% probability of committing a type I error is desired.

If you wanted to find out if alcohol consumption (measured in fluid oz.)and grade point average on a 4-point scale are linearly related, you would perform a

TABLE 13-6 The following Excel tables are obtained when "Score received on an exam (measured in percentage points)" (Y) is regressed on "percentage attendance" (X) for 22 students in a Statistics for Business and Economics course. Regression Statistics Multiple R 0.142620229 R Square 0.02034053 Standard Error 20.25979924 Observations 22 Coefficients Standard Error T Stat P-value Intercept 39.39027309 37.24347659 1.057642216 0.302826622 Attendance 0.340583573 0.52852452 0.644404489 0.526635689 -Referring to Table 13-6, which of the following statements is true?

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 to analyze the last 4 years of quarterly data (i.e., n = 16) with the following results: Regression Statistics Multiple R 0.802 R Square 0.643 Adjusted R Square 0.618 Standard Error SYX 0.9224 Observations 16 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 p -value Intercept 3.962 1.440 2.75 0.016 Industry 0.040451 0.008048 5.03 0.000 Durbin-Watson Statistic $1.59$ -If the Durbin-Watson statistic has a value close to 4, which assumption is violated?

The coefficient of determination represents the ratio of SSR to SST.

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 error or residual sum of squares (SSE)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. 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 regression sum of squares (SSR)is ________.

TABLE 13-10 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands 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, what is the p-value of the F test statistic when testing whether the number of customers who make purchases is a good predictor for weekly sales?

TABLE 13-10 The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands 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 residual plot.

TABLE 13-12 The manager of the purchasing department of a large saving and loan organization would like to develop a model to predict the amount of time (measured in hours) it takes to record a loan application. Data are collected from a sample of 30 days, and the number of applications recorded and completion time in hours is recorded. Below is the regression output: Regression Statistics Multiple R 0.9447 R Square 0.8924 Adjusted R 0.8886 Square Standard 0.3342 Error Observations 30 $\text { ANOVA }$ df 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\% Intercept 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Applications Recorded 0.0126 0.0008 15.2388 4.3946-15 0.0109 0.0143 Note: 4.3946E-15 is 4.3946*10^-15 -Referring to Table 13-12, the 90% confidence interval for the mean change in the amount of time needed as a result of recording one additional loan application is

TB1604_00for services rendered to local companies. One independent variable used to predict service charges 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: Yᵢ = β₀ + β₁Xi + Eᵢ The results of the simple linear regression are provided below. $Y = - 2,700 + 20 X , S Y X = 65 \text {, two-tail } p \text {-value } = 0.034 \text { (for testing } \beta _ { 1 } \text { ) }$ -Referring to Table 13-1, interpret the p-value for testing whether β₁ exceeds 0.