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 [tex]$12: 3: 1$[/tex] ratio predicted by a certain genetic model? Use the chi-square test at alpha [tex]$=0.05$[/tex].
(a) Write [tex]$H _0$[/tex] and [tex]$H _1$[/tex].
(b) Calculate the "expected" frequency for each color.
A. [tex]$H _0$[/tex]: Ratio among three colors [tex]$=12: 3: 1$[/tex] vs. [tex]$H _1$[/tex]: Ratio [tex]$\neq 12: 3: 1$[/tex] expected values [tex]$=146.75,31.4375,15.8125$[/tex]
B. [tex]$H _0$[/tex]: Ratio among three colors [tex]$=12: 3: 1$[/tex] vs. [tex]$H _1$[/tex]: Ratio [tex]$\neq 12: 3: 1$[/tex] expected values [tex]$=155,40,10$[/tex]
C. [tex]$H _0$[/tex]: Ratio among three colors [tex]$=12: 3: 1$[/tex] vs. [tex]$H _1$[/tex]: Ratio [tex]$\neq 12: 3: 1$[/tex] expected values [tex]$=0.75,0.1875,0.0625$[/tex]
D. [tex]$H _0$[/tex]: Ratio among three colors [tex]$=12: 3: 1$[/tex] vs. [tex]$H _1$[/tex]: Ratio [tex]$\neq 12: 3: 1$[/tex] expected values [tex]$=153.75,38.4375,12.8125$[/tex]
Answer (2)
Calculate the expected frequencies for each color based on the 12:3:1 ratio: White = 153.75, Yellow = 38.4375, Green = 12.8125.
Calculate the chi-square test statistic: χ 2 ≈ 0.6903428 .
Determine the degrees of freedom: df = 2.
Compare the calculated chi-square statistic with the critical value (5.991). Since 0.6903428 < 5.991, fail to reject the null hypothesis. The data are consistent with the 12:3:1 ratio.
Explanation
Analyze the problem and data We are given data from a cross between white and yellow summer squash, with progeny of white, yellow, and green colors. The observed numbers are 155 white, 40 yellow, and 10 green. We want to test if these data are consistent with a 12:3:1 ratio predicted by a genetic model, using a chi-square test at a significance level of alpha = 0.05.
State the hypotheses First, we state the null and alternative hypotheses:
H 0 : The ratio among the three colors is 12:3:1. H 1 : The ratio among the three colors is not 12:3:1.
Calculate the total number of progeny Next, we calculate the expected frequencies for each color based on the 12:3:1 ratio. The total number of progeny is:
155 + 40 + 10 = 205
Calculate the total ratio parts The total ratio parts are:
12 + 3 + 1 = 16
Calculate the expected frequencies Now, we calculate the expected frequencies for each color:
Expected White = 16 12 × 205 = 153.75 Expected Yellow = 16 3 × 205 = 38.4375 Expected Green = 16 1 × 205 = 12.8125
Calculate the chi-square statistic We calculate the chi-square test statistic using the formula:
χ 2 = ∑ E i ( O i − E i ) 2
where O i is the observed frequency and E i is the expected frequency 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 7.900390625
χ 2 ≈ 0.0101626 + 0.0635135 + 0.6166667 ≈ 0.6903428
Determine the degrees of freedom The degrees of freedom (df) are calculated as:
df = number of categories - 1 = 3 - 1 = 2
Find the critical value The critical value from the chi-square distribution table for alpha = 0.05 and df = 2 is 5.991.
Compare the chi-square statistic with the critical value We compare the calculated chi-square statistic (0.6903428) with the critical value (5.991). Since 0.6903428 < 5.991, we fail to reject the null hypothesis.
State the conclusion Therefore, the data are consistent with the 12:3:1 ratio predicted by the genetic model.
Examples
In genetics, the chi-square test is often used to determine if observed data from experiments involving genetic crosses are consistent with theoretical expectations. For example, if we cross two pea plants and observe the traits of the offspring, we can use a chi-square test to see if the observed ratios of traits match the ratios predicted by Mendel's laws. This helps us validate or reject genetic models and understand inheritance patterns.
We performed a chi-square test to determine if the progeny color ratios fit a 12:3:1 ratio. We calculated the expected frequencies and compared the calculated chi-square statistic with the critical value. Since the statistic was less than the critical value, we fail to reject the null hypothesis, indicating the data are consistent with the predicted ratio.
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