Question

A plant breeder observed that for a certain leaf trait of maize that shows two phenotypes (phenotype 1 and phenotype 2), the \(\mathrm{F}_{1}\) generation exhibits 200 plants with phenotype 1 and 160 with phenotype 2. Using two different null hypotheses and chi-square analysis, compute if the data fits (a) a 3: 1 ratio, and (b) a 1: 1 ratio.

Short answer

Explain your answer. Answer: The data better fits the 1:1 ratio, though the fit is not strict. The calculated chi-square value of 4.44 is slightly greater than the critical chi-square value of 3.84, indicating a stronger support for the 1:1 ratio even though it does not strictly fit the data. Meanwhile, the data does not fit the 3:1 ratio, since the calculated chi-square value (72.59) is far greater than the critical chi-square value (3.84).

Step-by-step solution

Collect given data

We have 200 plants with phenotype 1 and 160 plants with phenotype 2 in the \(\mathrm{F}_{1}\) generation. We will first perform a chi-square analysis for a 3:1 ratio and then for a 1:1 ratio.

Calculate the expected numbers for the 3:1 ratio hypothesis

A 3:1 ratio means that there will be 3 times more plants with phenotype 1 than with phenotype 2. To calculate the expected number of plants in each phenotype, we can first calculate the total number of plants and divide that by the ratio denominator (3+1) = 4, and then multiply each ratio part by the quotient. Total number of plants = 200 + 160 = 360 For phenotype 1 (3 parts of ratio): Expected number = (360 / 4) * 3 = 270 For phenotype 2 (1 part of ratio): Expected number = (360 / 4) * 1 = 90

Perform Chi-square test for the 3:1 ratio hypothesis

The chi-square formula is: \(\chi^2 = \sum\frac{(O-E)^2}{E}\), where \(O\) is the observed value and \(E\) is the expected value. For phenotype 1: \(\chi^2_1 = \frac{(200-270)^2}{270} = \frac{(-70)^2}{270} = \frac{4900}{270} = 18.15\) For phenotype 2: \(\chi^2_2 = \frac{(160-90)^2}{90} = \frac{70^2}{90} = \frac{4900}{90} = 54.44\) Total chi-square value: \(\chi^2 = 18.15 + 54.44 = 72.59\)

Calculate the expected numbers for the 1:1 ratio hypothesis

Now, let's test if the plant population fits a 1:1 ratio. Each phenotype will have half the total number of plants. Total number of plants = 360 For phenotype 1 (1 part of ratio): Expected number = (360 / 2) = 180 For phenotype 2 (1 part of ratio): Expected number = (360 / 2) = 180

Perform Chi-square test for the 1:1 ratio hypothesis

We will use the same chi-square formula from step 3 for the 1:1 ratio hypothesis. For phenotype 1: \(\chi^2_1 = \frac{(200-180)^2}{180} = \frac{20^2}{180} = \frac{400}{180} = 2.22\) For phenotype 2: \(\chi^2_2 = \frac{(160-180)^2}{180} = \frac{(-20)^2}{180} = \frac{400}{180} = 2.22\) Total chi-square value: \(\chi^2 = 2.22 + 2.22 = 4.44\)

Determine if data fits any of the ratios based on chi-square values

We have performed chi-square tests for both the 3:1 and 1:1 ratio hypotheses and obtained chi-square values of 72.59 and 4.44, respectively. To determine if the data fits either of these ratios, we will look for the critical chi-square value for a 5% level of significance and 1 degree of freedom (since there are 2 phenotypes). We find that the critical chi-square value is 3.84. Comparing the calculated chi-square values to the critical chi-square: - For the 3:1 ratio, the calculated chi-square (72.59) is far greater than the critical chi-square (3.84), suggesting that the observed data does not fit the 3:1 ratio. - For the 1:1 ratio, the calculated chi-square (4.44) is slightly greater than the critical chi-square (3.84), suggesting that the observed data does not strictly fit the 1:1 ratio either. However, the difference is minor, and it is relatively closer to the 1:1 ratio than the 3:1 ratio. In conclusion, the data does not fit the 3:1 ratio but provides stronger support for the 1:1 ratio even though the difference between calculated and critical chi-squares is minimal.