III. (19 points, 6-2-6-5) It is believed by sports analysts that the mean batting average in 2001 is greater than the mean batting average in 1988. The batting average in 1988 is normally distributed with unknown mean x and unknown variance (:2. The batting average in 2001 is normally distributed with unknown mean y and unknown variance (:2. Among the eight players sampled from 1988, the mean batting average was 0.251 with standard deviation 0.093. Among the nine players sampled from 2001, the mean batting average was 0.293 with standard deviation 0.069. (1) Construct a 95% lower condence interval for pg; pay. (2) What is the probability that the condence interval in (a) contains the actual value of pg; y? (3) Test Ho : HX = pay versus Ho : #X 7E y with a = 0.05. State your conclusion. (4) Give the p-value for the test performed in part (3)