Example

See - Hanson, W.D. 1959. Minimum family size for the planning of genetic experiments. Agron J.51:711-715.

Let H(a) have the ratio a₂:a₁ and H(b) have the ratio b₂:b₁

a₂ is the expected proportion of the dominant class under H(a). b₂ is the expected proportion of the dominant class under H(b). For the binomial distribution the expected mean number of recessive progeny is u = nQ and the variance is nQ(1-Q). Q is the probability of a recessive genotype, 1-Q is the probability of a dominant genotype, and n is the F₂ family size. We will solve for n which is the minimum F₂ family size to distinguish between the hypothesized ratios under H(a) and H(b). We will approximate the binomial distribution with the normal distribution, because npq is expected to be greater than 25. We can use the 't' value test and a = 0.05. The the 't' value required to detect a significant difference between two proportions is t = 1.96. The variance of Q = Q(1-Q)/n and the standard error of Q is equal to the square root of [Q(1-Q)/n].

The expected mean number of a progeny is a n and the expected number of b₁ progeny is b₁n. This is because Q is a₁ under H(a) and Q is b₁ under H(b).

H(a) ratio = a2:a1	H(b) ratio=b2:b1
mean # of recessives = a1	mean # of recessives = b1n
Var(a) =  a1a2                  n	Var(b) =  b1b2                 n

=

= 1