Exam 13: Simple Linear Regression

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.

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 t-test statistic to test whether total milk production depends linearly on the number of milking cows is ________.

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

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 value of the measured t-test statistic to test whether total milk production depends linearly on the number of milking cows is ________.

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, construct the residual plot for the 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, suppose the managers of the brokerage firm want to construct a 99% confidence interval estimate for the mean sales made by brokers who have brought into the firm 24 new clients. The t critical value they would use is ________.

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

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, the p-value of the measured t-test statistic to test whether total milk production depends linearly on the number of milking cows is ________.

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, the conclusion on the test of whether total milk production depends linearly on the number of milking cows using a 1% level of significance 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 p-value of the t test and F test should be the same when testing whether the number of customers who make purchases is a good predictor for weekly sales.

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 value of the F test statistic on whether the number of milking cows is a good predictor for the total milk production is _________.

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 percentage of the total variation in candy bar sales is explained by prices?

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, 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 13-3 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 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, there is a 95% probability that the mean amount of time needed to record one additional loan application is somewhere between 0.0109 and 0.0143 hours.

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 least squares estimate of the slope is __________.

The coefficient of determination (r2) tells you

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, which is the correct null hypothesis for testing whether the number of customers who make a purchase affects weekly sales? a) $H _ { 0 } : \beta _ { 0 } = 0$ b) $H _ { 0 } : \beta _ { 1 } = 0$ c) $H _ { 0 } : \mu = 0$ d) $H _ { 0 } : \pi = 0$

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 standard error of the regression slope estimate, Sb1 ?

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 regression mean square (MSR) of the above regression is _____.