Exam 13: Simple Linear Regression

TABLE 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. 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 Table 13-4, the managers of the brokerage firm wanted to test the hypothesis that the true 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 ___________.

TABLE 13-12 The manager of the purchasing department of a large banking organization would like to develop a model to predict the amount of time (measured in hours) it takes to process invoices. Data are collected from a sample of 30 days, and the number of invoices processed and completion time in hours is recorded. Below is the regression output: Regression Statistics Multiple R 0.9947 R Square 0.8924 Adjusted R Square 0.8886 Standard Error 0.3342 Observations 30 d f 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\% Invoices 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Processed 0.0126 0.0008 15.2388 4.3946-15 0.0109 0.0143 -Referring to Table 13-12, what are the critical values of the Durbin-Watson test statistic using the 5% level of significance to test for evidence of positive autocorrelation?

TABLE 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 from regressing GPA on ACT scores using a data set of 8 randomly chosen students from a Big-Ten university. Regressing GPA on Regression Statistics Multiple R 0.7598 R Square 0.5774 Adjusted R Square 0.5069 Standard E rror 0.2691 Observations 8 ANOVA d f 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 Table 13-8, the value of the measured (observed) test statistic of the F-test for H0 : ?₁ = 0 versus H₁ : ?₁ ? 0

TABLE 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 Table 13-10, construct a 95% confidence interval for the change in average weekly sales when the number of customers who make purchases increases by one.

TABLE 13-4 The managers of a brokerage firm are interested in finding out if the number of new clients a broker brings intothe firm affects the sales generated by the broker. Theysample 12 brokersand determine the numberof new clients they have enrolled in the last year and their sales amountsin thousandsof 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 Table 13-4, the managers of the brokerage firm wanted to test the hypothesis that the true slope was equal to 0. At a level of significance of 0.01, the decision that should be made implies that ______(there is or there is no) linear dependent relation between the independent and dependent variables.

TABLE 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 from regressing GPA on ACT scores using a data set of 8 randomly chosen students from a Big-Ten university. \text {Regressing GPA on \mathrm { ACT }} Regressing GPA on Regression Statistics Multiple R 0.7598 R Square 0.5774 Adjusted R Square 0.5069 Standard E rror 0.2691 Observations 8 ANOVA d f 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 Table 13-8, what is the predicted average value of GPA when ACT = 20?

TABLE 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 Table 13-3, the error or residual sum of squares (SSE) is_____ .

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 : ? = 0 against the two- sided alternative H₀: ?? 0. To test H₀ :? = 0 against the one- sided alternative H0 : ? > 0 at a significance level of 0.2, the p- value is

The width of the prediction interval for the predicted value of Y is dependent on

TABLE 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 Table 13-3, suppose the director of cooperative education wants to obtain both a 95% confidence interval estimate and a 95% prediction interval for X = 2. The confidence interval estimate would be the wider of the two intervals.

TABLE 13-12 The manager of the purchasing department of a large banking organization would like to develop a model to predict the amount of time (measured in hours) it takes to process invoices. Data are collected from a sample of 30 days, and the number of invoices processed and completion time in hours is recorded. Below is the regression output: Regression Statistics Multiple R 0.9947 R Square 0.8924 Adjusted R Square 0.8886 Standard Error 0.3342 ations 30 d f 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\% Invoices 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Processed 0.0126 0.0008 15.2388 4.3946-15 0.0109 0.0143 -Referring to Table 13-12, the p-value of the measured t-test statistic to test whether the number of invoices processed affects the amount of time is

TABLE 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's Data Analysis tool to analyze the last 4 years of quarterly data with the following results: Regression Sttuistics Multiple R 0.802 R Square 0.643 Adjusted RSquare 0.618 Standard Error SYX 0.9224 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 Coeffcients Standard Error tStat p-value Intercept 3.962 1.440 2.75 0.016 Industry 0.040451 0.008048 5.03 0.000 Durbin- Watson Statistic 1.59 -Referring to Table 13-5, the standard error of the estimated slope coefficient is________ .

The slope (b₁) represents

TABLE 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's Data Analysis tool 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 SYX 0.9224 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 Coefficients Standard Error t Stat p -value Intercept 3.962 1.440 2.75 0.016 Industry 0.040451 0.008048 5.03 0.000 Durbin- Watson Statistic 1.59 -Referring to Table 13-5, the partner wants to test for autocorrelation using the Durbin-Watson statistic. Using a level of significance of 0.05, the decision he should make is

TABLE 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 Table 13-10, it is inappropriate to compute the Durbin-Watson statistic and test for autocorrelation in this case.

TABLE 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 Table 13-10, construct a 95% prediction interval for the weekly sales of a store that has 600 purchasing customers.

TABLE 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 from regressing GPA on ACT scores using a data set of 8 randomly chosen students from a Big-Ten university. Regressing GPA on $A C T$ 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 Table 13-8, the value of the measured test statistic to test whether there is any linear relationship between GPA and ACT is

TABLE 13-12 The manager of the purchasing department of a large banking organization would like to develop a model to predict the amount of time (measured in hours) it takes to process invoices. Data are collected from a sample of 30 days, and the number of invoices processed and completion time in hours is recorded. Below is the regression output: Regression Statistics Multiple R 0.9947 R Square 0.8924 Adjusted R Square 0.8886 Standard Error 0.3342 ations 30 d f 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\% Invoices 0.4024 0.1236 3.2559 0.0030 0.1492 0.6555 Processed 0.0126 0.0008 15.2388 4.3946-15 0.0109 0.0143 -Referring to Table 13-12, there is a 95% probability that the average amount of time needed to process one additional invoice is somewhere between 0.0109 and 0.0143 hours.

TABLE 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: City Price (\ ) Sales River Falls 1.30 100 Hudson 1.60 90 Ellsworth 1.80 90 Prescott 2.00 40 Rock Elm 2.40 38 Stillwater 2.90 32 -Referring to Table 13-2, what is $?(X -\bar X )^2$ for these data?

The value of r is always positive.