A certain drug is used to treat asthma. In a clinical trial of the drug, 15 of 295 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.
1-PropZTest
prop $< 0.09$
$z=-2.349789860$
$p=0.0093920062$
$\hat{p}=0.0508474576$
$n=295$
Decide whether to reject the null hypothesis. Choose the correct answer below.
A. Fail to 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. Reject the null hypothesis because the P-value is less than or equal to the significance level, $\alpha$.
D. Fail to reject the null hypothesis because the P-value is greater than the significance level, $\alpha$.
Answer
Final Answer: \(\boxed{\text{C. Reject the null hypothesis because the P-value is less than or equal to the significance level, } \alpha}\)
Step 1 ：The null hypothesis in this case is that the proportion of treated subjects who experienced headaches is less than 9%. The alternative hypothesis is that the proportion is greater than or equal to 9%.
Step 2 ：The P-value is the probability of observing a result as extreme as the one we got (or more extreme) if the null hypothesis is true.
Step 3 ：In this case, the P-value is 0.0093920062, which is less than the significance level of 0.05.
Step 4 ：This means that if the null hypothesis is true, there is a very low probability of observing a result as extreme as the one we got.
Step 5 ：Therefore, we reject the null hypothesis.
Step 6 ：Final Answer: \(\boxed{\text{C. Reject the null hypothesis because the P-value is less than or equal to the significance level, } \alpha}\)
