A cross between white and yellow summer squash gave progeny of the following colors:
| Color | White | Yellow | Green |
| ------------------ | ----- | ------ | ----- |
| Number of progeny | 155 | 40 | 10 |
Are these data consistent with the 12:3:1 ratio predicted by a certain genetic model? Use the chi-square test at alpha [tex]$\alpha =0.05$[/tex].
(e) State your conclusion in plain terms.
A. We reject [tex]$H _0$[/tex]. There is not enough reason to suspect the 12:3:1 ratio.
B. We reject [tex]$H _0$[/tex]. There is enough reason to suspect the 12:3:1 ratio.
C. We do NOT reject [tex]$H _0$[/tex]. There is not enough reason to suspect the 12:3:1 ratio.
D. We do NOT reject [tex]$H _0$[/tex]. There is enough reason to suspect the 12:3:1 ratio.
Answer (2)
Calculate the expected values based on the 12:3:1 ratio: White = 153.75, Yellow = 38.4375, Green = 12.8125.
Calculate the chi-square statistic: χ 2 ≈ 0.69046 .
Determine the degrees of freedom: df = 2, and find the critical value for alpha = 0.05: 5.991.
Since the chi-square statistic is less than the critical value, we do not reject the null hypothesis: There is not enough reason to suspect the 12:3:1 ratio. The answer is: We do NOT reject H 0 . There is not enough reason to suspect the 12:3:1 ratio.
Explanation
Analyze the problem and data We are given observed data for the colors of progeny from a cross between white and yellow summer squash: White (155), Yellow (40), and Green (10). The total number of progeny is 155 + 40 + 10 = 205. We want to test if these data are consistent with a 12:3:1 ratio using a chi-square test at a significance level of alpha = 0.05.
Calculate expected values First, we calculate the expected values for each color based on the 12:3:1 ratio. The total ratio is 12 + 3 + 1 = 16. So, the expected proportions are 12/16 for White, 3/16 for Yellow, and 1/16 for Green. We multiply these proportions by the total number of progeny (205) to get the expected values:
Expected White = (12/16) * 205 = 153.75 Expected Yellow = (3/16) * 205 = 38.4375 Expected Green = (1/16) * 205 = 12.8125
Calculate the chi-square statistic Next, we calculate the chi-square statistic using the formula: χ 2 = ∑ E i ( O i − E i ) 2 , where O i is the observed value and E i is the expected value for each color.
χ 2 = 153.75 ( 155 − 153.75 ) 2 + 38.4375 ( 40 − 38.4375 ) 2 + 12.8125 ( 10 − 12.8125 ) 2
χ 2 = 153.75 ( 1.25 ) 2 + 38.4375 ( 1.5625 ) 2 + 12.8125 ( − 2.8125 ) 2
χ 2 = 153.75 1.5625 + 38.4375 2.44140625 + 12.8125 7.90140625
$\chi^2 \approx 0.01016 + 0.06352 + 0.61678 \approx 0.69046
Determine degrees of freedom and critical value The degrees of freedom (df) are calculated as the number of categories minus 1. Since there are 3 categories (White, Yellow, Green), df = 3 - 1 = 2.
We are given a significance level of alpha = 0.05. The critical value from the chi-square distribution table for alpha = 0.05 and df = 2 is 5.991.
Compare chi-square statistic with critical value We compare the calculated chi-square statistic (0.69046) with the critical value (5.991). Since 0.69046 < 5.991, we do not reject the null hypothesis.
State the conclusion Since we do not reject the null hypothesis, we conclude that there is not enough evidence to suspect that the data is inconsistent with the 12:3:1 ratio. Therefore, the data is consistent with the 12:3:1 ratio.
Examples
In genetics, the chi-square test is used to determine if observed data from experiments involving genetic crosses match expected ratios predicted by Mendelian inheritance. For example, if we cross two heterozygous plants for a single trait, we expect a 3:1 phenotypic ratio in the offspring. If we observe a different ratio, the chi-square test can help us determine if the difference is statistically significant or due to random chance. This is crucial for validating genetic models and understanding inheritance patterns.
Using a chi-square test on the progeny color data from a summer squash cross, the calculated chi-square statistic (0.69046) is less than the critical value (5.991). Therefore, we do not reject the null hypothesis, suggesting that there is not enough evidence to claim a deviation from the expected 12:3:1 ratio. The conclusion is option C: We do NOT reject H 0 .
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