The variance-to-mean ratio is a useful measurement of how clumped or dispersed events are in space or time relative to the random expectation (see Section 8.6). A high ratio indicates that events are clumped, whereas a low ratio indicates “overdispersion” of events. Davis et al. (2009) used this approach to examine the dispersion of “compensatory mutations” affecting protein sequences. Most mutations to genes are harmful, but compensatory mutations occasionally occur that counteract some of the damage caused by other mutations. The authors gathered data on 77 harmful mutations of varying lengths and a total of 328 compensatory mutations that have been discovered in the same genes. These data come from organisms ranging from viruses to fruit flies. The authors recorded the number of compensatory mutations at each amino acid position in the genes. They calculated the mean and variance in the number of compensatory mutations per amino acid position and calculated variance/mean=2.64. They used simulation to test the null hypothesis that mutations were randomly and independently located in the genes. Compensatory mutations were placed at random and independently. After each simulation, the computer calculated the resulting variance-to-mean ratio in the number of compensatory mutations per amino acid position. The results of 10,000 such simulations are plotted in the accompanying histogram.

a. What does this frequency distribution estimate?

b. Using the results shown in the frequency distribution and the observed variance/mean ratio of 2.64, test the null hypothesis that the true variance/mean ratio is as expected from the random placement of compensatory mutations.