# Question

When potential workers apply for a job that requires extensive manual assembly of small intricate parts, they are initially given three different tests to measure their manual dexterity. The ones who are hired are then periodically given a performance rating on a 0 to 100 scale that combines their speed and accuracy in performing the required assembly operations. The file P11_49.xlsx lists the test scores and performance ratings for a randomly selected group of employees. It also lists their seniority (months with the company) at the time of the performance rating.

a. Look at a matrix of correlations. Can you say with certainty (based only on these correlations) that the R2 value for the regression will be at least 35%? Why or why not?

b. Is there any evidence (from the correlation matrix) that multicollinearity will be a problem? Why or why not?

c. Run the regression of Performance Rating versus all four explanatory variables. List the equation, the value of R2, and the value of se. Do all of the coefficients have the signs (negative or positive) you would expect? Briefly explain.

d. Referring to the equation in part c, if a worker (outside of the 80 in the sample) has 15 months of seniority and test scores of 57, 71, and 63, find a prediction and an approximate 95% prediction interval for this worker’s Performance Rating score.

e. One of the t-values for the coefficients in part c is less than 1. Explain briefly why this occurred. Does it mean that this variable is not related to Performance Rating?

f. Arguably, the three test measures provide overlapping (or redundant) information. For the sake of parsimony (explaining “the most with the least”), it might be sensible to regress Performance Rating versus only two explanatory variables, Seniority and Average Test, where Average Test is the average of the three test scores—that is, Average Test = (Test1 + Test2 + Test3)/3. Run this regression and report the same measures as in part c: the equation itself, R2, and se. Can you argue that this equation is just as good as the equation in part c? Explain briefly.

a. Look at a matrix of correlations. Can you say with certainty (based only on these correlations) that the R2 value for the regression will be at least 35%? Why or why not?

b. Is there any evidence (from the correlation matrix) that multicollinearity will be a problem? Why or why not?

c. Run the regression of Performance Rating versus all four explanatory variables. List the equation, the value of R2, and the value of se. Do all of the coefficients have the signs (negative or positive) you would expect? Briefly explain.

d. Referring to the equation in part c, if a worker (outside of the 80 in the sample) has 15 months of seniority and test scores of 57, 71, and 63, find a prediction and an approximate 95% prediction interval for this worker’s Performance Rating score.

e. One of the t-values for the coefficients in part c is less than 1. Explain briefly why this occurred. Does it mean that this variable is not related to Performance Rating?

f. Arguably, the three test measures provide overlapping (or redundant) information. For the sake of parsimony (explaining “the most with the least”), it might be sensible to regress Performance Rating versus only two explanatory variables, Seniority and Average Test, where Average Test is the average of the three test scores—that is, Average Test = (Test1 + Test2 + Test3)/3. Run this regression and report the same measures as in part c: the equation itself, R2, and se. Can you argue that this equation is just as good as the equation in part c? Explain briefly.

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