Questions: PHENOTYPES, OBSERVE, EXPECTED, O-E, (O-E)^2 Dark Brown eyes, 125, , , Light brown cyes, 78, , , Light brown eyes, 87, , , White cyes, 25, , , TOTAL, , , ,

Solution Steps

The expected frequencies for each phenotype, assuming equal distribution, are calculated as follows:

$$E = \frac{N}{k} = \frac{315}{4} = 78.75$$

where $N$ is the total number of observations (315) and $k$ is the number of phenotypes (4). Thus, the expected frequencies are:

$$E = [78.75, 78.75, 78.75, 78.75]$$

The Chi-Square test statistic ($\chi^2$) is calculated using the formula:

$$\chi^2 = \sum_i \frac{(O_i - E_i)^2}{E_i}$$

Substituting the observed ($O$) and expected ($E$) values:

$$\chi^2 = \frac{(125 - 78.75)^2}{78.75} + \frac{(78 - 78.75)^2}{78.75} + \frac{(87 - 78.75)^2}{78.75} + \frac{(25 - 78.75)^2}{78.75}$$

Calculating this gives:

$$\chi^2 = 64.7206$$

The degrees of freedom ($df$) for this test is given by:

$$df = k - 1 = 4 - 1 = 3$$

Using a significance level of $\alpha = 0.05$, the critical value for the Chi-Square distribution with 3 degrees of freedom is:

$$\chi^2(0.95, 3) = 7.8147$$

The p-value associated with the calculated Chi-Square statistic is:

$$P = P(\chi^2 > 64.7206) = 0.0$$

Final Answer