Question

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?

Short answer

Answer: Based on the calculated \(\chi^{2}\) values (\(33.333\) for hypothesis (a) and \(25\) for hypothesis (b)), hypothesis (b), the 1:1 ratio, better fits the given data. However, a formal conclusion should be drawn by using \(\chi^{2}\) critical values or p-values, along with the level of significance and degrees of freedom.

Step-by-step solution

Calculate the Expected values for each hypothesis

For hypothesis (a), the expected ratio is 3:1 and for hypothesis (b), the expected ratio is 1:1. We need to calculate the expected values for each phenotype class according to these ratios. First, we find the total number of individuals: \(250 + 150 = 400\). For hypothesis (a), the expected ratio is 3:1, so we have: - Expected value for the first class: \(\frac{3}{(3+1)} \times 400 = 300\) - Expected value for the second class: \(\frac{1}{(3+1)} \times 400 = 100\) For hypothesis (b), the expected ratio is 1:1, so we have: - Expected value for the first class: \(\frac{1}{(1+1)} \times 400 = 200\) - Expected value for the second class: \(\frac{1}{(1+1)} \times 400 = 200\)

Calculate the \(\chi^{2}\) values for each hypothesis

For both hypotheses, we will use the \(\chi^{2}\) formula: \(\chi^{2} = \sum \frac{(O - E)^2}{E}\) Where \(O\) is the observed value, and \(E\) is the expected value. For hypothesis (a): - \(\chi^{2} = \frac{(250-300)^{2}}{300} + \frac{(150-100)^{2}}{100}\) - \(\chi^{2} = \frac{(-50)^{2}}{300} + \frac{50^{2}}{100}\) - \(\chi^{2} = \frac{2500}{300} + \frac{2500}{100} = 8.333 + 25 = 33.333\) For hypothesis (b): - \(\chi^{2} = \frac{(250-200)^{2}}{200} + \frac{(150-200)^{2}}{200}\) - \(\chi^{2} = \frac{50^{2}}{200} + \frac{(-50)^{2}}{200}\) - \(\chi^{2} = \frac{2500}{200} + \frac{2500}{200} = 12.5+12.5 = 25\)

Interpret the results

The calculated \(\chi^{2}\) values are \(33.333\) for hypothesis (a) and \(25\) for hypothesis (b). The lower the \(\chi^2\) value, the better the fit between the observed data and the expected values under a given null hypothesis. Therefore, based on our calculations, hypothesis (b) (i.e., the data fits a 1:1 ratio) is more likely to be true. However, to formally conclude which hypothesis best fits or clearly reject one of them, the analysis should be completed using \(\chi^2\) critical values or p-values, which requires the level of significance (such as \(0.05\) or \(0.01\)), and a degree of freedom (in this case, \(1\)) in order to draw any definite conclusions.