Question

a genetic experiment involving peas yielded one sample of offspring consisting of 420 green peas and 156 yellow peas. use a 0.05 significance level to test the claim that under the same circumstances, 25% 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. h₁: p ≠ 0.25 h₁: p = 0.25 c. h₀: p ≠ 0.25 d. h₀: p = 0.25 h₁: p < 0.25 h₁: p < 0.25 e. h₀: p ≠ 0.25 f. h₀: p = 0.25 h₁: p > 0.25 h₁: p > 0.25 what is the test statistic?

Explanation:

Step1: Calculate sample proportion

The sample size \(n = 420 + 156=576\). The number of yellow peas \(x = 156\). The sample proportion \(\hat{p}=\frac{x}{n}=\frac{156}{576}\approx0.271\).

Step2: Calculate test - statistic

The formula for the test - statistic \(z\) in a hypothesis test for a proportion is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), where \(p = 0.25\) (the proportion in the null hypothesis), \(\hat{p}\approx0.271\), and \(n = 576\).

Substitute the values:

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Step3: Calculate P - value

Since this is a two - tailed test (the claim is about a proportion not equal to \(0.25\)), the P - value is \(2\times(1 - P(Z<|z|))\).

Using a standard normal table or calculator, \(P(Z < 1.16)=0.8770\).

So the P - value \(=2\times(1 - 0.8770)=2\times0.1230 = 0.246\)

Step4: Conclusion about the null hypothesis

Since the P - value \(=0.246>0.05\) (the significance level \(\alpha = 0.05\)), we fail to reject the null hypothesis \(H_0:p = 0.25\)

Step5: Final conclusion

There is not sufficient evidence at the \(0.05\) significance level to warrant rejection of the claim that \(25\%\) of offspring peas will be yellow.

Answer:

• Null hypothesis \(H_0:p = 0.25\)
• Alternative hypothesis \(H_1:p

eq0.25\)

• Test statistic \(z\approx1.16\)
• P - value \(= 0.246\)
• Conclusion about the null hypothesis: Fail to reject \(H_0\)
• Final conclusion: There is not sufficient evidence to reject the claim that \(25\%\) of offspring peas will be yellow.