Fruit flies which were heterozygous for both autosomal dominant traits aristapedia (Ar) and
star eyes (S) were mated. Since aristapedia and star eyes are both lethal when homozygous
and unlinked, one would predict ratio of 4:2:2:1 for Ar-S to Ar to S to wild type (+). Among
the offspring flies, there were 113 Ar-S, 55 Ar, 73 S, and 38 +. Test the hypothesis of a
4:2:2:1 phenotypic ratio at the 5% level of significance.
To test the hypothesis of a 4:2:2:1 phenotypic ratio, we will use a chi-square test. This statistical test compares the observed frequencies of the different phenotypes to the expected frequencies based on the given ratio.
Here are the steps to conduct the chi-square test:
Step 1: Set up the null and alternative hypotheses:
- Null hypothesis (H0): The observed phenotypic ratios follow the expected 4:2:2:1 ratio.
- Alternative hypothesis (Ha): The observed phenotypic ratios do not follow the expected 4:2:2:1 ratio.
Step 2: Determine the degrees of freedom:
To determine the degrees of freedom for this test, we subtract 1 from the number of categories. In this case, there are 4 categories (Ar-S, Ar, S, +), so the degrees of freedom (df) will be 4 - 1 = 3.
Step 3: Set the significance level:
The given significance level is 5% or 0.05. This represents the probability of rejecting the null hypothesis when it is true.
Step 4: Calculate the expected frequencies:
The expected frequencies are calculated by multiplying the total number of offspring by the expected ratio for each phenotype. In this case, if the expected ratio is 4:2:2:1 and the total number of offspring is 279 (113 + 55 + 73 + 38), then we can calculate the expected frequencies as follows:
- Ar-S: (4/9) * 279 ≈ 124.00
- Ar: (2/9) * 279 ≈ 62.00
- S: (2/9) * 279 ≈ 62.00
- +: (1/9) * 279 ≈ 31.00
Step 5: Calculate the chi-square statistic:
The chi-square statistic is calculated using the formula:
χ^2 = Σ [(Observed frequency - Expected frequency)^2 / Expected frequency]
Using the observed and expected frequencies, we can calculate the chi-square statistic as follows:
χ^2 = [(113 - 124)^2 / 124] + [(55 - 62)^2 / 62] + [(73 - 62)^2 / 62] + [(38 - 31)^2 / 31]
Step 6: Look up the critical value:
Using the degrees of freedom (3) and the chosen significance level (0.05), you can consult a chi-square distribution table or use statistical software to find the critical value. For a significance level of 0.05 and 3 degrees of freedom, the critical value is approximately 7.815.
Step 7: Compare the chi-square statistic to the critical value:
If the chi-square statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 8: Interpret the results:
If the null hypothesis is rejected, it implies that the observed phenotypic ratios do not follow the expected 4:2:2:1 ratio. If the null hypothesis is not rejected, it suggests that the observed ratios are in line with the expected ratio.
That's how you can test the hypothesis of a 4:2:2:1 phenotypic ratio at the 5% level of significance using a chi-square test.