A certain drug is used to treat asthma. In a clinical trial of the drug, 29 of 280 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 $10 \%$ of treated subjects experienced headaches. Use the normal distribution as an approximation to the binomial distribution and assume a
- 0.01 significance level to complete parts (a) through (e) below.
\begin{tabular}{|l|}
\hline 1-PropZTest \\
prop $< 0.1$ \\
$z=0.199204768$ \\
$p=0.5789487151$ \\
$\hat{p}=0.1035714286$ \\
$n=280$
\end{tabular}
Decide whether to reject the null hypothesis. Choose the correct answer below.
A. Reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$.
B. Reject the null hypothesis because the P-value is greater than the significance level, $\alpha$.
C. Fail to reject the null hypothesis because the P-value is greater than the significance level, $\alpha$.
D. Fail to reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$.
Answer
Final Answer: \(\boxed{\text{C. Fail to reject the null hypothesis because the P-value is greater than the significance level, } \alpha}\).
Step 1 ：The null hypothesis in this case is that the proportion of treated subjects who experienced headaches is less than 10%.
Step 2 ：The P-value is the probability that we would observe a result as extreme as, or more extreme than, the result we actually observed, given that the null hypothesis is true.
Step 3 ：In this case, the P-value is 0.5789487151, which is greater than the significance level of 0.01.
Step 4 ：This means that we would expect to see a result as extreme as, or more extreme than, the one we observed more than 5.78% of the time if the null hypothesis were true.
Step 5 ：Because the P-value is greater than the significance level, we do not have sufficient evidence to reject the null hypothesis.
Step 6 ：Therefore, the correct answer is C. Fail to reject the null hypothesis because the P-value is greater than the significance level, α.
Step 7 ：Final Answer: \(\boxed{\text{C. Fail to reject the null hypothesis because the P-value is greater than the significance level, } \alpha}\).
