Chapter 11.4, Problem 8P

In Problems 7-12, parts (a) and (b) relate to testing $\rho$. Part (c) requests the value of $S_e$. Parts (d) and (c) relate to confidence intervals for prediction. Part (f) relates to testing $\beta$.

Baseball: Batting Average and Strikeouts Let x be a random variable that represents the batting average of a professional baseball player. Let y be a random variable that represents the percentage of strikeouts of a professional baseball player. A random sample of n = 6 professional baseball players gave the following information (Reference: The Baseball Encyclopedia, Macmillan):

x	0.328	0.290	0.340	0.248	0.367	0.269
y	3.2	7.6	4.0	8.6	3.1	11.1

(a) Verify that $\begin{array}{l}{\displaystyle \sum x}=1.842,{\displaystyle \sum y}=37.6,{\displaystyle \sum x^2}=0.575838,{\displaystyle \sum y^2}=290.78,\hfill \\ {\displaystyle \sum xy}=10.87,\text{ and }r\approx -\mathrm{0.891.}\hfill \end{array}$

(b) Use a 5% level of significance to test the claim that $\rho \ne 0$.

(c) Verify that $S_e\approx 1.6838,a\approx 26.247,\text{ and }b\approx -\mathrm{65.081.}$

(d) Find the predicted percentage of strikeouts for a player with an x = 0.300 batting average.

(e) Find an 80% confidence interval for y when x = 0.300.

(f) Use a 5% level of significance to test the claim that $\beta \ne 0$.