1. A soybean breeder is interested in assessing whether an experimental family of soybean recombinant inbred lines has an increased amount of allelic diversity. Previous work suggests that the population mean allelic diversity of soybean is 71% of that of Asian soybean landraces. Thus, this researcher has gathered data to determine if there is enough evidence to conclude that the population mean allelic diversity has increased. A total of n =381 plants from this family had a sample mean allelic diversity of compared to Asian soybean landraces and sample standard deviation of . Assume that she is about to conduct a hypothesis test at , and that the sample size is sufficiently large for s to estimate .

a) Calculate the probability of a Type II error (i.e., ) if the actual population mean allelic diversity is 74%. As part of this draw the density function for the distribution under the null hypothesis and then also draw the density function assuming is 74%. On these density functions show the rejection region and shade in the area that corresponds to the probability of a Type II error.