Question

a genetic experiment involving peas yielded one sample of offspring consisting of 438 green peas and 120 yellow peas. use a 0.01 significance level to test the claim that under the same circumstances, 24% 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.

what is the conclusion about the null hypothesis?
○ a. reject the null hypothesis because the p - value is less than or equal to the significance level, α.
○ b. fail to reject the null hypothesis because the p - value is greater than the significance level, α.
○ c. fail to reject the null hypothesis because the p - value is less than or equal to the significance level, α.
○ d. reject the null hypothesis because the p - value is greater than the significance level, α.

what is the final conclusion?
○ a. there is not sufficient evidence to warrant rejection of the claim that 24% of offspring peas will be yellow.
○ b. there is not sufficient evidence to support the claim that less than 24% of offspring peas will be yellow.
○ c. there is sufficient evidence to support the claim that less than 24% of offspring peas will be yellow.
○ d. there is sufficient evidence to warrant rejection of the claim that 24% of offspring peas will be yellow.

Explanation:

Step1: Calculate the sample proportion

The total number of peas \(n = 438+120=558\). The sample proportion of yellow peas \(\hat{p}=\frac{120}{558}\approx0.215\)

Step2: State the hypotheses

The null hypothesis \(H_0:p = 0.24\) (claim). The alternative hypothesis \(H_1:p
eq0.24\)

Step3: Calculate the test - statistic

The formula for the test - statistic \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.24\), \(\hat{p}=0.215\), \(n = 558\)

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Step4: Calculate the P - value

Since \(H_1:p
eq0.24\), it is a two - tailed test.
\(P - value=2\times P(Z\lt - 1.38)\)
From the standard normal table, \(P(Z\lt - 1.38)=0.0838\)
\(P - value=2\times0.0838 = 0.1676\)

Step5: Conclusion about the null hypothesis

The significance level \(\alpha = 0.01\)
Since \(P - value=0.1676\gt\alpha = 0.01\), we fail to reject the null hypothesis.

Step6: Final conclusion

Since we fail to reject \(H_0:p = 0.24\), there is not sufficient evidence to warrant rejection of the claim that \(24\%\) of offspring peas will be yellow.

Answer:

For the conclusion about the null hypothesis: B. Fail to reject the null hypothesis because the P - value is greater than the significance level, \(\alpha\).
For the final conclusion: A. There is not sufficient evidence to warrant rejection of the claim that \(24\%\) of offspring peas will be yellow.