Not all regression coefficients have positive expected signs. For example, a Sports Illustrated article by Jaime Diaz reported on a study of golfing putts of various lengths on the Professional Golfers’ Association (PGA) Tour.11 The article included data on the percentage of putts made (P_i) as a function of the length of the putt in feet (L_i). Since the longer the putt, the less likely even a professional is to make it, we’d expect L_i to have a negative coefficient in an equation explaining P_i. Sure enough, if you estimate an equation on the data in the article, you obtain:

P̂_i = 83.6 - 4.1L_i

a. Carefully write out the exact meaning of the coefficient of L_i.

b. Suppose someone else took the data from the article and estimated:

P_i = 83.6 – 4.1L_i + e_i

Is this the same result as that of Equation 1.22? If so, what definition do you need to use to convert this equation back to Equation 1.22?

c. Use Equation 1.22 to determine the percent of the time you’d expect a PGA golfer to make a 10-foot putt. Does this seem realistic? How about a 1-foot putt or a 25-foot putt? Do these seem as realistic?

d. Your answer to part c should suggest that there’s a problem in applying a linear regression to these data. What is that problem?