Question

a certain drug is used to treat asthma. in a clinical trial of the drug, 17 of 282 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 12% 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 (d) below.
1 - propztest
prop < 0.12
z = - 3.08596472
p = 0.001015950
\\( \hat { p } = 0.0602836873 \\)
n = 282
d. what is the null hypothesis, and what do you conclude about it?
identify the null hypothesis.
o a. \\( h _ { 0 } : p < 0.12 \\)
o b. \\( h _ { 0 } : p > 0.12 \\)
o c. \\( h _ { 0 } : p \
eq 0.12 \\)
o d. \\( h _ { 0 } : p = 0.12 \\)
decide whether to reject the null hypothesis. choose the correct answer below

Explanation:

Step1: Identify the null hypothesis

The null hypothesis \(H_0\) is a statement of no difference or no effect. In a hypothesis test for a proportion, the null hypothesis is of the form \(H_0:p = p_0\), where \(p_0\) is the hypothesized proportion. Here, the claim is about the proportion of subjects experiencing headaches. The hypothesized proportion \(p_0 = 0.12\). So, \(H_0:p=0.12\) (Option D).

Step2: Decide whether to reject the null hypothesis

We are given a significance level \(\alpha = 0.05\). The \(p - value\) (from the calculator result \(prop < 0.12\) with \(z=-3.08596472\), \(p = 0.001045950\), \(\hat{p}=0.0602836879\), \(n = 282\)) is \(p=0.001045950\).
Since the \(p - value(0.001045950)<\alpha(0.05)\), we reject the null hypothesis.

Answer:

• For identifying the null hypothesis: D. \(H_0:p = 0.12\)
• For the decision: Reject the null hypothesis \(H_0:p = 0.12\) because the \(p - value(0.001045950)\) is less than the significance level \(\alpha(0.05)\)