Question

a genetic experiment involving peas yielded one sample of offspring consisting of 439 green peas and 171 yellow peas. use a 0.01 significance level to test the claim that under the same circumstances, 26% of offspring peas will be yellow. identify the null hypothesis, alternative hypothesis, test statistic, p - value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. use the p - value method and the normal distribution as an approximation to the binomial distribution.
significance level, α.
d. fail to reject the null hypothesis because the p - value is less than or equal to the significance level, α.
what is the final conclusion?
a. there is not sufficient evidence to support the claim that less than 26% of offspring peas will be yellow.
b. there is sufficient evidence to support the claim that less than 26% of offspring peas will be yellow.
c. there is not sufficient evidence to warrant rejection of the claim that 26% of offspring peas will be yellow.
d. there is sufficient evidence to warrant rejection of the claim that 26% of offspring peas will be yellow.

Explanation:

Step1: Calculate sample proportion

The total number of offspring peas \(n = 439+171=610\). The number of yellow peas \(x = 171\). The sample proportion \(\hat{p}=\frac{x}{n}=\frac{171}{610}\approx0.28\)

Step2: State hypotheses

The null hypothesis \(H_0:p = 0.26\) (claim). The alternative hypothesis \(H_1:p
eq0.26\)

Step3: Calculate test - statistic

The formula for the test - statistic \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.26\), \(\hat{p}=0.28\), \(n = 610\)

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Step4: Calculate P - value

Since it is a two - tailed test, \(P - value=2\times(1 - P(Z<|z|))\)
From the standard normal table, \(P(Z < 1.13)=0.8708\)
\(P - value=2\times(1 - 0.8708)=2\times0.1292 = 0.2584\)

Step5: Conclusion about null hypothesis

Since \(P - value=0.2584>0.01=\alpha\), we fail to reject the null hypothesis

Step6: Final conclusion

Since we fail to reject \(H_0:p = 0.26\), there is not sufficient evidence to warrant rejection of the claim that \(26\%\) of offspring peas will be yellow

Answer:

C. There is not sufficient evidence to warrant rejection of the claim that \(26\%\) of offspring peas will be yellow.