In assessing data that fell into two phenotypic classes, a geneticist observed values of 250:150. She decided to perform a $\chi$² analysis by using the following two different null hypotheses:
(a) the data fit a 3:1 ratio, and
(b) the data fit a 1:1 ratio.
Calculate the $\chi$ ² values for each hypothesis. What can be concluded about each hypothesis?

Verified step by step guidance

1. Step 1: Identify the observed values and total number of observations. Here, the observed counts are 250 and 150, so the total is 250 + 150 = 400.
2. Step 2: For hypothesis (a), where the expected ratio is 3:1, calculate the expected counts by dividing the total according to this ratio. The expected count for the first class is \(\frac{3}{4} \times 400\), and for the second class is \(\frac{1}{4} \times 400\).
3. Step 3: For hypothesis (b), where the expected ratio is 1:1, calculate the expected counts by dividing the total equally. Each class is expected to have \(\frac{1}{2} \times 400\) individuals.
4. Step 4: Use the chi-square formula to calculate the chi-square value for each hypothesis: \(\chi^{2} = \sum \frac{(O - E)^2}{E}\) where \(O\) is the observed count and \(E\) is the expected count for each phenotypic class. Calculate this sum separately for each hypothesis using their respective expected counts.
5. Step 5: Compare the calculated chi-square values to the critical chi-square value from the chi-square distribution table at the appropriate degrees of freedom (df = number of classes - 1) and significance level (commonly 0.05). If the calculated value is less than the critical value, the hypothesis fits the data well; if greater, the hypothesis is rejected.

Key Concepts

Chi-Square (χ²) Test

The chi-square test is a statistical method used to compare observed data with expected data based on a specific hypothesis. It calculates a χ² value that measures the difference between observed and expected frequencies. A higher χ² value indicates a greater deviation from the expected ratio, helping to determine if the null hypothesis can be rejected.

Null Hypothesis in Genetic Ratios

In genetics, the null hypothesis often states that observed phenotypic ratios fit a predicted Mendelian ratio, such as 3:1 or 1:1. This hypothesis assumes no significant difference between observed and expected data. Testing different null hypotheses helps identify which genetic model best explains the observed data.

Interpreting Chi-Square Results

After calculating the χ² value, it is compared to a critical value from the chi-square distribution table based on degrees of freedom and significance level. If χ² is less than the critical value, the null hypothesis is accepted; if greater, it is rejected. This interpretation determines whether observed deviations are due to chance or indicate a different genetic pattern.