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 $< 0.09$
$z=-2.349789860$
$p=0.0093920062$
$\hat{p}=0.0508474576$
$n=295$
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, $\alpha$.
e. What is the final conclusion?
A. There is not sufficient evidence to support the claim that less than $9 \%$ of treated subjects experienced headaches.
B. There is sufficient evidence to warrant rejection of the claim that less than $9 \%$ of treated subjects experienced headaches.
C. There is not sufficient evidence to warrant rejection of the claim that less than $9 \%$ of treated subjects experienced headaches.
D. There is sufficient evidence to support the claim that less than $9 \%$ of treated subjects experienced headaches.
Answer
Final Answer: \(\boxed{\text{B. There is sufficient evidence to warrant rejection of the claim that less than 9 \% of treated subjects experienced headaches.}}\)
Step 1 ：The problem provides us with the following information: the proportion of treated subjects who experienced headaches is less than 0.09 (or 9%), the sample size is 295, the observed proportion of subjects who experienced headaches is 0.0508474576, and the P-value is 0.0093920062. The significance level is 0.05.
Step 2 ：We are asked to test the null hypothesis that less than 9% of treated subjects experienced headaches. The alternative hypothesis is that 9% or more of treated subjects experienced headaches.
Step 3 ：We compare the P-value to the significance level. 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 4 ：In this case, the P-value is 0.0093920062, which is less than the significance level of 0.05. Therefore, we reject the null hypothesis.
Step 5 ：By rejecting the null hypothesis, we conclude that there is sufficient evidence to warrant rejection of the claim that less than 9% of treated subjects experienced headaches.
Step 6 ：Final Answer: \(\boxed{\text{B. There is sufficient evidence to warrant rejection of the claim that less than 9 \% of treated subjects experienced headaches.}}\)
