Exam 13: Simple Linear Regression

SCENARIO 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 \bullet Standard Error \bullet T Stat \bullet P- Coefficients Standard Error 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 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, a) $H _ { 0 } : \rho \geq 0$ vs. $H _ { 1 } : \rho < 0$ b) $H _ { 0 } : \rho \leq 0$ vs. $H _ { 1 } : \rho > 0$ c) $H _ { 0 } : r \geq 0$ vs. $H _ { 1 } : r < 0$ d) $H _ { 0 } : r \leq 0$ vs. $H _ { 1 } : r > 0$

SCENARIO 13-14-B You are the CEO of a dairy company. You are planning to expand production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters) and the number of milking cows. The data are shown below and also available in the Excel file Scenario13-14-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 85478 91902 110374 125364 159401 102883 113151 133297 140073 145434 152513 128275 138040 161276 208079 224119 231071 122114 132222 155092 177273 196399 205329 89564 94838 94920 97577 102163 103754 239585 239773 241293 249157 294388 318813 66462 100444 103846 154118 155460 COWS 21 20 22 17 16 20 21 19 21 19 22 24 26 26 27 22 27 24 27 24 29 21 25 26 33 31 33 23 27 29 27 28 32 22 26 24 26 28 26 39 44 42 41 42 47 18 21 22 27 27 You believe that the number of milking cows is the best predictor for total milk production on any given farm. -Referring to Scenario 13-14-B, if you purchase 40 cows for the new farm, the lower and upper limit of the 95% prediction interval for the total milk production are _____ liters and _____ liters, respectively.

SCENARIO 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 with the following results: -Referring to Scenario 13-5, the standard error of the estimated slope coefficient is ________.

SCENARIO 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 Scenario 13-3, set up a scatter plot.

Testing for the existence of correlation is equivalent to a) testing for the existence of the slope $\left( \beta _ { 1 } \right)$ . b) testing for the existence of the $Y$ -intercept $\left( \beta _ { 0 } \right)$ . c) the confidence interval estimate for predicting $Y$ . d) None of the above.

EXPLANATION: The t-test statistic is $t = \frac { \left( b _ { 1 } - \beta _ { 1 } \right) } { S _ { b _ { 1 } } } = \frac { ( 2.5 - 3 ) } { 0.3536 } = - 1.4142$ KEYWORDS: t test on slope, p-value, slope SCENARIO 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 Scenario 13-4, the managers of the brokerage firm wanted to test the hypothesis that the population slope was equal to 0. The p-value of the test is ________.

SCENARIO 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 Scenario 13-3, suppose the director of cooperative education wants to construct a 95% 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 ________.

EXPLANATION: The t-test statistic is $t = \frac { \left( b _ { 1 } - \beta _ { 1 } \right) } { S _ { b _ { 1 } } } = \frac { ( 2.5 - 3 ) } { 0.3536 } = - 1.4142$ KEYWORDS: t test on slope, p-value, slope SCENARIO 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 Scenario 13-4, 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 firm 24 new clients. The confidence interval is from ________ to ________.

SCENARIO 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 $\text { ANOVA }$ 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 -Referring to Scenario 13-12, the value of the measured t-test statistic to test whether the amount of time depends linearly on the number of loan applications recorded is

SCENARIO 13-14-B You are the CEO of a dairy company. You are planning to expand production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters) and the number of milking cows. The data are shown below and also available in the Excel file Scenario13-14-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 85478 91902 110374 125364 159401 102883 113151 133297 140073 145434 152513 128275 138040 161276 208079 224119 231071 122114 132222 155092 177273 196399 205329 89564 94838 94920 97577 102163 103754 239585 239773 241293 249157 294388 318813 66462 100444 103846 154118 155460 COWS 21 20 22 17 16 20 21 19 21 19 22 24 26 26 27 22 27 24 27 24 29 21 25 26 33 31 33 23 27 29 27 28 32 22 26 24 26 28 26 39 44 42 41 42 47 18 21 22 27 27 You believe that the number of milking cows is the best predictor for total milk production on any given farm. -Referring to Scenario 13-14-B, the degrees of freedom for the F test on whether the number of milking cows is a good predictor for the total milk production are _________.

EXPLANATION: The t-test statistic is $t = \frac { \left( b _ { 1 } - \beta _ { 1 } \right) } { S _ { b _ { 1 } } } = \frac { ( 2.5 - 3 ) } { 0.3536 } = - 1.4142$ KEYWORDS: t test on slope, p-value, slope SCENARIO 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 Scenario 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. At a level of significance of 0.01, the decision that should be made implies that the number of new clients brought in _____ (had or did not have) a positive impact on the amount of sales generated.

SCENARIO 13-14-B You are the CEO of a dairy company. You are planning to expand production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters) and the number of milking cows. The data are shown below and also available in the Excel file Scenario13-14-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 85478 91902 110374 125364 159401 102883 113151 133297 140073 145434 152513 128275 138040 161276 208079 224119 231071 122114 132222 155092 177273 196399 205329 89564 94838 94920 97577 102163 103754 239585 239773 241293 249157 294388 318813 66462 100444 103846 154118 155460 COWS 21 20 22 17 16 20 21 19 21 19 22 24 26 26 27 22 27 24 27 24 29 21 25 26 33 31 33 23 27 29 27 28 32 22 26 24 26 28 26 39 44 42 41 42 47 18 21 22 27 27 You believe that the number of milking cows is the best predictor for total milk production on any given farm. -Referring to Scenario 13-14-B, if you purchase 40 cows for the new farm, its predicted total milk production is _________ liters.

SCENARIO 13-11 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: -Referring to Scenario 13-11, what is the standard error of estimate?

SCENARIO 13-11 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: -Referring to Scenario 13-11, what is the p-value for testing whether there is a linear relationship between revenue and the number of downloads at a 5% level of significance?

SCENARIO 13-1 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: $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \varepsilon _ { i }$ The results of the simple linear regression are provided below. $\hat { Y } = - 2,700 + 20 X , S _ { X X } = 65 , \text { two-tail } p \text { value } = 0.034 \left( \text { for testing } \beta _ { 1 } \right)$ -Referring to Scenario 13-1, interpret the estimate of ?, the standard deviation of the random error term (standard error of the estimate) in the model.

SCENARIO 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 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 Observations 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 13-8, the value of the measured (observed) test statistic of the F-test for $H _ { 0 } : \beta _ { 1 } = 0 \text { vs. } H _ { 1 } : \beta _ { 1 } \neq 0$

SCENARIO 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 $\text { ANOVA }$ 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 -Referring to Scenario 13-12, predict the amount of time it would take to process 150 invoices.

SCENARIO 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 Scenario 13-3, the standard error of estimate is __________.

SCENARIO 13-13 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 $\text { ANOVA }$ -Referring to Scenario 13-13, if the state decides to spend 1,000 dollar more per student, the estimated change in mean composite score is _________.

SCENARIO 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: $\begin{array}{ccc}\underline{\text { City }} &\underline{ \text { Price (\$)}} &\underline{\text { Sales }}\\\text { River Falls } & 1.30 & 100 \\\text { Hudson } & 1.60 & 90 \\\text { Ellsworth } & 1.80 & 90 \\\text { Prescott } & 2.00 & 40 \\\text { Rock Elm } & 2.40 & 38 \\\text { Stillwater } & 2.90 & 32\end{array}$ -Referring to Scenario 13-2, what is the coefficient of correlation for these data?