Exam 13: Simple Linear Regression

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. The p-value of the test is ________.

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, which of the following is the correct alternative hypothesis for testing whether there is a linear relationship between revenue and the number of downloads? a) $H _ { 1 } : b _ { 1 } = 0$ b) $H _ { 1 } : b _ { 1 } \neq 0$ c) $H _ { 1 } : \beta _ { 1 } = 0$ d) $H _ { 1 } : \beta _ { 1 } \neq 0$

SCENARIO 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 ANOVA Note: $2.051 E - 05 = 2.051 * 10 ^ { - 05 }$ and $5.944 E - 18 = 5.944 * 10 ^ { - 18 }$ . . -Referring to Scenario 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

SCENARIO 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 Scenario 13-10, generate the scatter plot.

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 both a 99% 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 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 ANOVA Note: $2.051 E - 05 = 2.051 * 10 ^ { - 05 }$ and $5.944 E - 18 = 5.944 * 10 ^ { - 18 }$ . . -Referring to Scenario 13-9, the estimated change in mean salary (in thousands of dollars) as a result of spending an extra hour per day studying is

SCENARIO 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 Scenario 13-10, what is the value of the F test statistic when testing whether the number of customers who make purchases is a good predictor for weekly sales?

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 error or residual sum of squares (SSE) is __________.

SCENARIO 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 Scenario 13-10, 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.

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 coefficient of determination is __________.

When r = - 1, it indicates a perfect relationship between X and Y.

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 arethe lower and upper limits of the 95% confidence interval estimate for population slope?

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 prediction for the amount of sales (in $1,000s) for a person who brings 25 new clients into the firm 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, the p-value of the measured t-test statistic to test whether composite score depends linearly on spending per student is ________.

Which of the following assumptions concerning the probability distribution of the random error term is stated incorrectly?

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. At a level of significance of 0.01, the null hypothesis should be _______ (rejected or not rejected).

Based on the residual plot below, you will conclude that there might be a violation of which of the following assumptions.

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, what percentage of the variation in the amount of time needed can be explained by the variation in the number of invoices processed?

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 model appears to be adequate based on the residual analyses.

The Regression Sum of Squares (SSR) can never be greater than the Total Sum of Squares (SST).