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.