Exam 12: Simple Linear Regression

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 correlation coefficient 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 estimate for the number of job offers received by students who have had exactly one cooperative education job.The prediction interval is from to .

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 population slope was equal to 0.For a test with a level of significance of 0.01, the null hypothesis should be rejected if the 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 standard error of the estimated slope coefficient is _.

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, to test the claim that SALARY depends positively on HOURS against the null hypothesis that SALARY does not depend linearly on HOURS, the p-value of the test statistic 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, suppose the managers of the brokerage firm want to construct a 99% prediction interval for the sales made by a broker who has brought into the firm 18 new clients.The prediction interval is from to _.

In performing a regression analysis involving two numerical variables, you are assuming

The value of r is always positive.

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 conclusion on the test of whether composite score depends linearly on spending per student using a 10% level of significance is

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 12-3 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 0.The value of the test statistic is _.

Data that exhibit an autocorrelation effect violate the regression assumption of independence.

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 regression mean square (MSR) of the above regression is_.

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 $\text { ANOVA }$ df SS MS F Significance F Regression 1 171062.9193 171062.9193 86.4759 0.0000 Residual 28 55388.4309 1978.1582 Total 29 226451.3503 Simple Linear Regression 12-41 -Referring to Scenario 12-11, what is the standard deviation around the regression line?

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 $\text { ANOVA }$ df SS MS F Significance F Regression 1 171062.9193 171062.9193 86.4759 0.0000 Residual 28 55388.4309 1978.1582 Total 29 226451.3503 Simple Linear Regression 12-41 -Referring to Scenario 12-11, predict the revenue when the number of downloads is 30 thousand.

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 $\text { ANOVA }$ df SS MS F Significance F Regression 1 171062.9193 171062.9193 86.4759 0.0000 Residual 28 55388.4309 1978.1582 Total 29 226451.3503 Simple Linear Regression 12-41 -Referring to Scenario 12-11, the homoscedasticity of error assumption appears to have been violated.

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: $\begin{array}{llr}\text { City }&\text { Price(\$)}&\text { Sales }\\\text { River Falls } & 1.30 & 100 \\\text { Hudson } & 1.60 & 90 \\\text { Ellsworth } & 1.80 & 90 \\\text { Prescott } & 2.00 & 40 \\\text { Rock Elm } & 2.40 & 38 \\\text { Stillwater } & 2.90 & 32\end{array}$ -Referring to Scenario 12-2, what is the standard error of the estimate, SYX, for the data?

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, which is the correct null hypothesis for testing whether the number of customers who make a purchase affects weekly sales?

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 error sum of squares (SSE) of the above regression is .

You give a pre-employment examination to your applicants.The test is scored from 1 to 100.You have data on their sales at the end of one year measured in dollars.You want to know if there is any linear relationship between pre-employment examination score and sales.An appropriate test to use is the t test of the population correlation coefficient.