A certain drug is used to treat asthma. In a clinical trial of the drug, 29 of 261 treated subjects experienced headaches (based on data from the manufacturer). The accompanying calculator display shows results from a test of the claim that less than $9 \%$ of treated subjects experienced headaches. Use the normal distribution as an approximation to the binomial distribution and assume a 0.05 significance level to complete parts (a) through (e) below.
\[
\begin{array}{l}
\text { 1-PropzTest } \\
\text { prop }< 0.09 \\
z=1.191761618 \\
p=0.8833226352 \\
\hat{p}=0.1111111111 \\
n=261
\end{array}
\]
C. $H_{0}: p< 0.09$
D. $H_{0}: p \neq 0.09$
Decide whether to reject the null hypothesis. Choose the correct answer below.
A. Fail to reject the null hypothesis because the P-value is greater than the significance level, $\alpha$.
B. Reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$.
c. Fail to reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$.
D. Reject the null hypothesis because the P-value is greater than the significance level, $\alpha$.
Answer
\(\boxed{\text{Final Answer: Fail to reject the null hypothesis because the P-value is greater than the significance level, } \alpha.}\)
Step 1 ：Given that the P-value is 0.8833226352 and the significance level is 0.05.
Step 2 ：The P-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. A smaller P-value means that there is stronger evidence in favor of the alternative hypothesis.
Step 3 ：The significance level, denoted by \(\alpha\), is a threshold below which the null hypothesis is rejected. Commonly used values are 0.05 and 0.01.
Step 4 ：In this case, if the P-value is less than or equal to the significance level, we reject the null hypothesis. If the P-value is greater than the significance level, we fail to reject the null hypothesis.
Step 5 ：Since the P-value is greater than the significance level, we fail to reject the null hypothesis.
Step 6 ：\(\boxed{\text{Final Answer: Fail to reject the null hypothesis because the P-value is greater than the significance level, } \alpha.}\)
