## To determine if the population is in Hardy-Weinberg equilibrium, we need to compare the observed genetic distribution with the expected distribution based on the Hardy-Weinberg equilibrium equation.

Step 1: Assign variables to the different genetic frequencies:

Let p represent the frequency of one allele.

Let q represent the frequency of the other allele.

Step 2: Use the given data to calculate the observed genetic frequencies:

From the information given, we know that:

- 56 samples had a single band of 6.3kb, representing the frequency of q^2.

- 6 samples had a single band of 4.1kb, representing the frequency of p^2.

- 38 samples showed both the 6.3kb and 4.1kb bands, representing the frequency of 2pq.

So, the observed genetic frequencies are:

q^2 = 56/100 = 0.56

p^2 = 6/100 = 0.06

2pq = 38/100 = 0.38

Step 3: Apply the Hardy-Weinberg equilibrium equation:

According to the Hardy-Weinberg equilibrium, p + q = 1, and p^2 + 2pq + q^2 = 1.

Since p + q = 1, we can solve for p:

p = 1 - q

Now substitute each term in the equation p^2 + 2pq + q^2 = 1 with the assigned variable:

(1 - q)^2 + 2(1 - q)q + q^2 = 1

Simplifying the equation gives us:

1 - 2q + q^2 + 2q - 2q^2 + q^2 = 1

-q^2 + q^2 + 1 - 2q + 2q = 1

0 = 0

Step 4: Interpret the results:

Since the equation simplifies to 0 = 0, we can conclude that the observed frequencies match the expected frequencies based on the Hardy-Weinberg equilibrium equation. Therefore, we can assume that the population is in Hardy-Weinberg equilibrium.

Step 5: Calculate the chi-square statistic:

To formally test if the observed frequencies are significantly different from the expected frequencies, we can use the chi-square test. The chi-square test calculates the chi-square statistic, which will tell us if the differences between the observed and expected frequencies are statistically significant. The formula for the chi-square statistic is:

Ï‡Â² = âˆ‘ [(observed - expected)Â² / expected]

Observation: The observed frequencies are:

q^2 = 0.56

p^2 = 0.06

2pq = 0.38

Expected frequencies can be calculated based on the Hardy-Weinberg equilibrium equation:

Expected q^2 frequency = p^2 * total population = 0.06 * 100 = 6

Expected 2pq frequency = 2pq * total population = 0.38 * 100 = 38

Expected p^2 frequency = q^2 * total population = 0.56 * 100 = 56

Now substitute these values into the chi-square formula:

Ï‡Â² = [(56 - 6)Â² / 6] + [(6 - 6)Â² / 6] + [(38 - 38)Â² / 38] = ...

Calculate the chi-square value and compare it to the critical chi-square value for the desired level of significance and degrees of freedom to determine if the observed frequency distribution significantly deviates from the expected distribution.

Note: The degrees of freedom for the chi-square test in this case is (number of phenotypes - 1), which is (3 - 1) = 2.

I hope this explanation helps you work through the problem. Good luck with your analysis!