Using data for 1901 through 1999, annual batting averages (BAs) were tabulated for all Major League Baseball (MLB) players for each season in which the player had at least 50 times at bat. For each season, each player’s BA was converted to a standardized Z value by subtracting the average BA for all players that year and dividing by the standard deviation of BAs that year [1]. The annual Z values were grouped according to whether it was the player’s first year in the major leagues, second year, and so on. The average Z value was then calculated for each career year. For example, looking at each player’s first year in the major leagues, the average Z value is −0.337.

a. Use the data in Table 11.2 to estimate the multiple regression equation Z = α + β₁t +

β₂t².

b. Are the coefficients of t and t² statistically significant at the 5 percent level?

c. What do the signs of t and t² tell you about the shape of the fitted line?

d. Explain why a comparison of the predicted Z value for t = 1 and t = 2 does not tell us the predicted change in the Z value if all first-year players were to play a second year in the majors.

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Table 11.2: Exercise 11.34 Year t Mean Z Year t Mean Z Year t Mean Z Year t Mean Z 1 -0.337 6 0.151 11 0.112 16 0.125 2345 -0.180 7 0.095 12 0.079 17 0.131 -0.019 8 0.125 13 0.122 18 0.009 4 0.067 9 0.146 14 0.127 19 0.280 0.098 10 0.193 15 0.154 20 -0.079