Exam 13: Simple Linear Regression

The Durbin-Watson D statistic is used to check the assumption of normality.

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 mean weekly sales will increase by an estimated $10 for each additional purchasing customer.

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, the Durbin-Watson statistic is inappropriate for this data set.

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, the null hypothesis that there is no linear relationship between revenue and the number of downloads should be rejected at a 5% level of significance.

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 director of cooperative education wanted to test the hypothesis that the population slope was equal to 0. For a test with a level of significance of 0.05, the null hypothesis should be rejected if the value of the test statistic 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 interpretation for the slope coefficient?

SCENARIO 13-14-A 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-DataA.XLSX. MILK 84729 101969 103314 70574 76144 86695 87600 105272 120422 68749 85541 91910 110465 125369 159470 102930 113214 133328 140086 145514 152589 128304 138093 161368 208136 224205 231090 122204 132311 155154 177268 196449 205324 89562 94840 94997 97625 102226 103832 239673 239861 241298 249196 294455 318871 66483 100459 103847 154141 155516 COWS 18 24 22 19 21 19 22 19 24 18 20 25 26 27 29 24 25 28 26 28 27 21 24 27 33 32 32 22 25 26 29 29 28 25 21 23 24 28 25 39 40 40 39 42 47 19 20 21 25 31 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-A, what is the standard deviation of total milk production around the regression line?

The slope (b1) represents

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 interpretation for the coefficient of determination?

Assuming a linear relationship between X and Y, if the coefficient of correlation (r) equals - 0.30, a) there is no correlation. b) the slope $\left( b _ { 1 } \right)$ is negative. c) variable $X$ is larger than variable $Y$ . d) the variance of $X$ is negative.

SCENARIO 13-6 The following Excel tables are obtained when "Score received on an exam (measured in percentage points)" (Y) is regressed on "percentage attendance" (X) for 22 students in a Statistics for Business and Economics course. Regression Statistics Multiple R 0.142620229 R Square 0.02034053 Standard Error 20.25979924 Observations 22 Coefficients Standard Error T Stat P -value Intercept 39.39027309 37.24347659 1.057642216 0.302826622 Attendance 0.340583573 0.52852452 0.644404489 0.526635689 -Referring to Scenario 13-6, which of the following statements is true?

SCENARIO 13-14-A 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-DataA.XLSX. MILK 84729 101969 103314 70574 76144 86695 87600 105272 120422 68749 85541 91910 110465 125369 159470 102930 113214 133328 140086 145514 152589 128304 138093 161368 208136 224205 231090 122204 132311 155154 177268 196449 205324 89562 94840 94997 97625 102226 103832 239673 239861 241298 249196 294455 318871 66483 100459 103847 154141 155516 COWS 18 24 22 19 21 19 22 19 24 18 20 25 26 27 29 24 25 28 26 28 27 21 24 27 33 32 32 22 25 26 29 29 28 25 21 23 24 28 25 39 40 40 39 42 47 19 20 21 25 31 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-A, if you purchase one additional milking cows, the estimated change in mean total milk production is _________ liters.

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 value of the test statistic is _______.

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 error sum of squares (SSE) of the above regression 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 error sum of squares (SSE) of the above regression 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 decision on the test of whether spending per student affects composite score using a 5% level of significance is to ________ (reject or not reject) H0 .

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 partner wants to test for autocorrelation using the Durbin-Watson statistic. Using a level of significance of 0.05, the critical values of the test are dL = ________, and dU = ________.

The residual represents the discrepancy between the observed dependent variable and its _______ value.

The Y-intercept (b0) represents the

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