Exam 13: Simple Linear Regression

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 p-value of the measured F-test statistic to test whether the number of milking cows is a good predictor for the total milk production 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, what percentage of the variation in total milk production can be explained by the variation in the number of milking cows?

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, what is the predicted value of GPA when ACT = 20?

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 coefficient of correlation is __________.

If the residuals in a regression analysis of time-ordered data are not correlated, the value of the Durbin-Watson D statistic should be near __________.

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 p-value of the measured t-test statistic to test whether the number of loan applications recorded affects the amount of time is _____.

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 test statistic to test whether there is any linear relationship between GPA and ACT 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, to test the claim that the mean amount of time depends positively on the number of loan applications recorded against the null hypothesis that the mean amount of time does not depend linearly on the number of invoices processed, the p-value of the test statistic is ____.

If the Durbin-Watson statistic has a value close to 4, which assumption is violated?

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 estimated mean change in the sales of the candy bar if price goes up by $1.00?

If the Durbin-Watson statistic has a value close to 0, which assumption is violated?

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 value of the quantity that the least squares regression line minimizes is ________.

The coefficient of determination represents the ratio of SSR to SST.

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. The p-value of the test is between ________ and ________.

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, if the price of the candy bar is set at $2, the estimated mean sales will be

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 denominator of the test statistic is sb1 . The value of sb1 in this sample 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 residual plot indicates possible violation of which assumptions?

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, 93.98% of the total variation in weekly sales can be explained by the variation in the number of customers who make purchases.

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 value of the measured t-test statistic to test whether composite score depends linearly on spending per student is ________.

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 p-value of the measured F-test statistic to test whether HOURS affects SALARY is _____.