Exam 12: Simple Linear Regression

SCENARIO 12-12 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 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 0.0126 0.0008 15.2388 0.0000 0.0109 0.0143 Recorded 12-46 Simple Linear Regression Simple Linear Regression 12-47 -Referring to Scenario 12-12, the model appears to be adequate based on the residual analyses.

SCENARIO 12-4 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 Scenario 12-4, suppose the managers of the brokerage firm want to construct both a99% confidence interval estimate and a 99% prediction interval for X = 24.The confidence interval estimate would be the (wider or narrower) of the two intervals.

SCENARIO 12-4 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 Scenario 12-4, suppose the managers of the brokerage firm want to construct a 99%confidence interval estimate for the mean sales made by brokers who have brought into the firm24 new clients.The confidence interval is from to .

SCENARIO 12-10 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 Scenario 12-10, the value of the t test statistic and F test statistic should be the same when testing whether the number of customers who make purchases is a good predictor for weekly sales.

SCENARIO 12-12 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 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 0.0126 0.0008 15.2388 0.0000 0.0109 0.0143 Recorded 12-46 Simple Linear Regression Simple Linear Regression 12-47 -Referring to Scenario 12-12, the degrees of freedom for the t test on whether the number of loan applications recorded affects the amount of time are

SCENARIO 12-7 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 StandardError T Stat P -value Intercept 4.8660 0.3574 13.6136 0.0000 S\&P -0.5025 0.0716 -7.0186 0.0000 -Referring to Scenario 12-7, to test whether the prison stocks portfolio is negatively related to the S&P 500 index, the measured value of the test statistic is

SCENARIO 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: $\begin{array}{llr}\text { City}&\text { Price(\$)}&\text { Sales }\\\hline\text { River Falls } & 1.30 & 100 \\\text { Hudson } & 1.60 & 90 \\\text { Ellsworth } & 1.80 & 90 \\\text { Prescott } & 2.00 & 40 \\\text { Rock Elm } & 2.40 & 38 \\\text { Stillwater } & 2.90 & 32\end{array}$ -Referring to Scenario 12-2, what is the percentage of the total variation in candy bar sales explained by the regression model?

SCENARIO 12-3 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 Coop Jobs Job Offer 1 1 4 2 2 6 3 1 3 4 0 1 -Referring to Scenario 12-3, the coefficient of correlation is .

SCENARIO 12-10 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 Scenario 12-10, generate the scatter plot.

SCENARIO 12-10 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 Scenario 12-10, the mean weekly sales will increase by an estimated$0.01 for each additional purchasing customer.

SCENARIO 12-10 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 Scenario 12-10, what is the value of the standard error of the estimate?

SCENARIO 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 for predicting GPA using ACT scores based a data set of 8 randomly chosen students from a Big-Ten university. Regressing GPA on ACT Regression Statistics Multiple R 0.7598 R Square 0.5774 Adjusted R Square 0.5069 Standard Error 0.2691 Qbservations 8 ANOVA df 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 Scenario 12-8, the value of the measured (observed) test statistic of the F-test forH0 : $\beta$ 1= 0 vs.H1 : $\beta$ 1 $\neq$ 0

SCENARIO 12-13 In this era of tough economic conditions, voters increasingly ask the question: "Is the educational achievement level of students dependent on the amount of money the state in which they reside spends on education?" The partial computer output below is the result of using spending per student ($) as the independent variable and composite score which is the sum of the math, science and reading scores as the dependent variable on 35 states that participated in a study.The table includes only partial results. $\begin{array}{l}\begin{array} { l c } \hline { \text { Regression Statistics } } \\\hline \text { Multiple R } & 0.3122 \\\text { R Square } & 0.0975 \\\text { Adjusted R } & 0.0701 \\\text { Square } & \\\text { Standard } & 26.9122 \\\text { Error } & \\\text { Observations } & 35 \\\hline\end{array}\\\text { ANOVA }\\\begin{array}{lrrr}&df&SS&MS&F\\\hline\text { Regression } & 1 & 2581.5759 & \\\text { Residual } & & & 724.2674 \\\text { Total } & 34 & 26482.4000 &\\\hline\end{array}\\\\\begin{array} { l c c c c } \hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & P \text {-value } \\\hline \text { Intercept } & 595.540251 & 22.115176 & & \\\text { Spending per } & & & & \\\text { Student(\$) } & 0.007996 & 0.004235 & \\\hline\end{array}\end{array}$ -Referring to Scenario 12-13, the conclusion on the test of whether composite score depends linearly on spending per student using a 10% level of significance is

SCENARIO 12-10 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 Scenario 12-10, it is inappropriate to compute the Durbin-Watson statistic and test for autocorrelation in this case.

SCENARIO 12-11 A computer software developer would like to use the number of downloads (in thousands) for the trial version of his new shareware to predict the amount of revenue (in thousands of dollars) he can make on the full version of the new shareware.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 sharewares that he has developed: Regression Statistics Multiple R 0.8691 R Square 0.7554 Adjusted R Square 0.7467 Standard Error 44.4765 Observations 30.0000 ANOVA df SS MS F Significance F Regression 1 171062.9193 171062.9193 86.4759 0.0000 Residual 28 55386.4309 1978.1582 Total 29 226451.3503 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -95.0614 26.9183 -3.5315 0.0015 -150.2009 -39.9218 Download 3.7297 0.4011 9.2992 0.0000 2.9082 4.5513 Simple Linear Regression 12-41 -Referring to Scenario 12-11, which of the following assumptions appears to have been violated?

SCENARIO 12-9 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 ANOVA df SS MS F Significance F Regression 1 335.0472 335.0473 178.3859 Residual 1.8782 Total 50 427.0798 Coefficients Standard Error t Stat P -value Lower 95\% Upper 95\% Intercept -1.8940 0.4018 -4.7134 0.0000 -2.7015 -1.0865 Hours 0.9795 0.0733 13.3561 0.0000 0.8321 1.1269 Note: 2.051E - 05 = 2.051 *10-05 and 5.944 E - 18 =5.944 *10-18 . -Referring to Scenario 12-9, 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

SCENARIO 12-1 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 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: $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \varepsilon _ { i }$ Theresultsofthesimplelinearregressionareprovidedbelow. $\bar { I } = - 2,700 + 20 X , S _ { Z X } = 65 \text {, two-tail } p \text { value } = 0.034 \text { (for testing } \beta _ { 1 } \text { ) }$ -Referring to Scenario 12-1, interpret the p-value for testing whether $\beta$ 1 exceeds 0.

SCENARIO 12-10 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 Scenario 12-10, the p-value of the t test and F test should be the same when testing whether the number of customers who make purchases is a good predictor for weekly sales.

SCENARIO 12-11 A computer software developer would like to use the number of downloads (in thousands) for the trial version of his new shareware to predict the amount of revenue (in thousands of dollars) he can make on the full version of the new shareware.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 sharewares that he has developed: Regression Statistics Multiple R 0.8691 R Square 0.7554 Adjusted R Square 0.7467 Standard Error 44.4765 Observations 30.0000 ANOVA df SS MS F Significance F Regression 1 171062.9193 171062.9193 86.4759 0.0000 Residual 28 55386.4309 1978.1582 Total 29 226451.3503 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -95.0614 26.9183 -3.5315 0.0015 -150.2009 -39.9218 Download 3.7297 0.4011 9.2992 0.0000 2.9082 4.5513 Simple Linear Regression 12-41 -Referring to Scenario 12-11, what is the critical value for testing whether there is a linear relationship between revenue and the number of downloads at a 5% level of significance?

SCENARIO 12-4 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 Scenario 12-4, 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.At a level of significance of 0.01, the null hypothesis should be (rejected or not rejected).