Exam 12: Simple Linear Regression

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 12-5 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 with the following results: Regression Statistics Multiple R 0.802 R Square 0.643 Adjusted R Square 0.618 Standard Error Syv 0.9234 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 Predictor Coef StdError t Stat P-value Intercept 3.962 1.440 2.75 0.016 Industry 0.040451 0.008048 5.03 0.000 $\text { Durbin-Watson Statistic } \quad 1.59$ -Referring to Scenario 12-5, the value of the quantity that the least squares regression line minimizes is _.

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 least squares estimate of the slope 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, what are the values of the estimated intercept and slope?

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 : $\rho$ = 0 against the one-sided alternative H1 : $\rho$ $\gt$ 0 .To testH0 : $\rho$ = 0 against the two-sided alternative H1 : $\rho$$\neq$ 0 at a significance level of 0.1, the p-value is

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 director of cooperative education wanted to test the hypothesis that the population slope was equal to 3.0.The value of the test statistic is .

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. Regression Statistics Multiple R 0.3122 R Square 0.0975 Adjusted R 0.0701 Square Standard 26.9122 Error Observations 35 ANOVA df SS MS F Regression 1 2581.5759 Residual 724.2674 Total 34 26482.4000 $\begin{array}{l}\begin{array} { l c c c c } \hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & P \text {-value } \\\hline \text { Intercept } & 595.540251 & 22.115176 & & \\\text { Spending per } & & & & \\\text { Student(\$) } & 0.007996 & 0.004235 & \\\hline\end{array}\end{array}$ -Referring to Scenario 12-13, the critical value at 5% level of significance of the F test on whether spending per student affects composite score is .

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).

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 standered Error tStat 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 value of the measured t-test statistic to test whether meanSALARY depends linearly on HOURS is

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 director of cooperative education wanted to test the hypothesis that the population slope was equal to 3.0.The p-value of the test is between and_.

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 Error 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 test statistic to test whether there is any linear relationship between GPA and ACT is

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. Regression Statistics Multiple R 0.3122 R Square 0.0975 Adjusted R 0.0701 Square Standard 26.9122 Error Observations 35 ANOVA df SS MS F Regression 1 2581.5759 Residual 724.2674 Total 34 26482.4000 $\begin{array}{l}\begin{array} { l c c c c } \hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & P \text {-value } \\\hline \text { Intercept } & 595.540251 & 22.115176 & & \\\text { Spending per } & & & & \\\text { Student(\$) } & 0.007996 & 0.004235 & \\\hline\end{array}\end{array}$ -Referring to Scenario 12-13, the value of the measured t-test statistic to test whether composite score depends linearly on spending per student is _.

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, suppose the director of cooperative education wants to construct a95% prediction interval for the number of job offers received by a student who has had exactly two cooperative education jobs.The prediction 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, 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?

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 coefficient of determination?

The coefficient of determination (r2) tells you

If the correlation coefficient (r) = 1.00, then

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 Error 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, what is the predicted value of GPA when ACT = 20?

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 residual plot indicates possible violation of which assumptions?

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, construct a 95% prediction interval for the weekly sales of a store that has 600 purchasing customers.