A random sample of size 64 is to be used to test the null hypothesis that for a certain age group the mean score on an achievement test (the mean of a normal population with ?2 = 256) is less than or equal to 40.0 against the alternative that it is greater than 40.0. If the null hypothesis is to be rejected if and only if the mean of the random sample exceeds 43.5, find
a) The probabilities of type I error when ? = 37.0, 38.0, 39.0, and 40.0
b) The probabilities of type II error when ? = 41.0, 42.0, 43.0, 44.0, 45.0, 46.0, 47.0, and 48.0.