To analyze the genetics of the Venezuelan poodle moth population, we can use the Hardy-Weinberg equilibrium equation:
p^2 + 2pq + q^2 = 1
where p is the frequency of the dominant allele (white coat) and q is the frequency of the recessive allele (brown coat). The equation states that the frequencies of the two alleles in the population must add up to 1, and that the frequencies of the three possible genotypes (white-white, white-brown, and brown-brown) must also add up to 1.
We can use the observed frequencies of the two coat colors to estimate the frequencies of the two alleles. Let's assume that the brown coat color is caused by a recessive allele, so q^2 represents the frequency of the homozygous brown-brown genotype. Then, we can set up the following equations:
q^2 = 138/500 = 0.276
p^2 + 2pq + q^2 = 1
p^2 + 2p(1-p) + 0.276 = 1
p^2 - 2p + 0.276 = 0
We can solve this quadratic equation for p using the quadratic formula:
p = [2 ± sqrt(4 - 4(0.276))]/2 = [2 ± 0.874]/2 = 1.437 or 0.563
Since p represents the frequency of the dominant allele (white coat), it must be less than 1. Therefore, we can discard the root p = 1.437 and conclude that:
p = 0.563
q = sqrt(0.276) = 0.526
These allele frequencies can be used to calculate the expected frequencies of the three genotypes:
white-white: p^2 = (0.563)^2 = 0.317
white-brown: 2pq = 2(0.563)(0.526) = 0.593
brown-brown: q^2 = (0.526)^2 = 0.276
We can compare these expected frequencies to the observed frequencies to test whether the population is in Hardy-Weinberg equilibrium. Let's use a chi-squared test with one degree of freedom:
χ^2 = Σ[(observed - expected)^2 / expected]
= [(362 - 317)^2 / 317] + [(138 - 276)^2 / 276]
= 5.70
The critical value of the chi-squared distribution with one degree of freedom and a significance level of 0.05 is 3.84. Since our calculated value of 5.70 is greater than the critical value of 3.84, we can reject the null hypothesis that the population is in Hardy-Weinberg equilibrium. This suggests that there may be some evolutionary forces at work, such as genetic drift, migration, mutation, or selection, that are causing the allele frequencies to deviate from the expected values.