Since p + q ~ 1, we can estimate the frequency of the N allele by subtraction as q = 1 - p = 1 - 0.4184 = 05816.

Using these allele frequencies allows calculation of the Hardy-Weinberg expected genotype frequency and number of individuals with each genotype, as shown in Table 2.4. In Table 2.4 we can see that the match between the observed and expected is not perfect, but we need some method to ask whether the difference is actually large enough to conclude that Hardy-Weinberg equilibrium does not hold in the sample of1066 genotypes. Remember that any allele-frequency estimate (p) could differ slightly from the true parameter (p) due to chance events as well as due to random sampling in the group of genotypes used to estimate the allele frequencies. Asking whether genotypes are in Hardy-Weinberg proportions is actually the same as asking whether a coin is "fair." With a fair coin we expect one-half heads and one-half tails if we flip it a large number of times. But even with a fair coin we can get something other than exactly

50:50 even if the sample size is large. We would consider a coin fair if in 1000 flips it produced 510 heads and 490 tails. However, the hypothesis that a coin is fair would be in doubt if we observed 250 heads and 750 tails given that we expect 500 of each.

In more general terms, the expected frequency of an event, p, times the number of trials or samples, n, gives the expected number events or np. To test the hypothesis that p is the frequency of an event in an actual population, we compare np with np. Close agreement suggests that the parameter and the estimate are the same quantity. But a large disagreement instead suggests that p and p are likely to be different probabilities. The Chi-squared (%2) distribution is a statistical test commonly used to compare np and np. The X2 test provides the probability of obtaining the difference (or more) between the observed (np) and expected (np) number of outcomes by chance alone if the null hypothesis is true. As the difference between the observed and expected grows larger it becomes less probable that the parameter and the parameter estimate are actually the same but differ in a given sample due to chance. The %2 statistic is:

(observed - expected)2 expected

The X2 formula makes intuitive sense. In the numerator there is the difference between the observed and Hardy-Weinberg expected number of individuals. This difference is squared, like a variance, since we do not care about the direction of the difference but only the magnitude of the difference. Then in the denominator we divide by the expected number of individuals to make the squared difference relative. For example, a squared difference of 4 is small if the expected number is 100 (it is 4%) but relatively

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