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.
what are the null and alternative hypotheses?
a. $h_0:p = 0.26$
$h_1:p\
eq0.26$
b. $h_0:p\
eq0.26$
$h_1:p = 0.26$
c. $h_0:p\
eq0.26$
$h_1:p>0.26$
d. $h_0:p\
eq0.26$
$h_1:p<0.26$
e. $h_0:p = 0.26$
$h_1:p<0.26$
f. $h_0:p = 0.26$
$h_1:p>0.26$
what is the test statistic?

Explanation:

Step1: Calculate sample proportion

The sample size \(n = 439+171=610\). The number of yellow peas \(x = 171\). The sample proportion \(\hat{p}=\frac{x}{n}=\frac{171}{610}\approx0.2803\).

Step2: Calculate test - statistic

The formula for the test - statistic \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), where \(p = 0.26\), \(n = 610\), \(\hat{p}=0.2803\).

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Step3: Calculate P - value

Since \(H_1:p
eq0.26\), this is a two - tailed test. The P - value is \(2\times(1 - P(Z<|z|))\). From the standard normal table, \(P(Z < 1.14)=0.8729\). So the P - value \(=2\times(1 - 0.8729)=2\times0.1271 = 0.2542\).

Step4: Conclusion about the null hypothesis

Since the P - value \(=0.2542>0.01\) (significance level \(\alpha = 0.01\)), we fail to reject the null hypothesis \(H_0:p = 0.26\).

Step5: Final conclusion

There is not sufficient evidence at the \(0.01\) significance level to reject the claim that \(26\%\) of offspring peas will be yellow.

Answer:

• Null hypothesis \(H_0:p = 0.26\), alternative hypothesis \(H_1:p

eq0.26\) (Option A).

• Test statistic \(z\approx1.14\).
• P - value \(=0.2542\).
• Conclusion about the null hypothesis: Fail to reject \(H_0\).
• Final conclusion: There is not sufficient evidence to reject the claim that \(26\%\) of offspring peas will be yellow.