In a study of treatments for very painful "cluster" headaches, 157 patients were treated with oxygen and 145 other patients were given a placebo consisting of ordinary air. Among the 157 patients in the oxygen treatment group, 108 were free from headaches 15 minutes after treatment. Among the 145 patients given the placebo, 21 were free from headaches 15 minutes after treatment. Use a 0.01 significance level to test the claim that the oxygen treatment is effective. Complete parts (a) through (c) below.
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of patients treated with oxygen and the second sample to be the sample of patients given a placebo. What are the null and alternative hypotheses for the hyothesis test?
A.
\[
\begin{array}{l}
H_{0}: p_{1} \leq p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: p_{1} \neq p_{2} \\
H_{1}: p_{1}=p_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}>p_{2}
\end{array}
\]
E.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
c.
\[
\begin{array}{l}
H_{0}: p_{1} \geq p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
$F$.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}
Solution
Step 1 ：Consider the first sample to be the sample of patients treated with oxygen and the second sample to be the sample of patients given a placebo. What are the null and alternative hypotheses for the hypothesis test?
Step 2 ：The null hypothesis (H0) is usually a statement of no effect or no difference. In this case, it would be that the proportion of patients free from headaches after 15 minutes is the same for both the oxygen treatment group and the placebo group.
Step 3 ：The alternative hypothesis (H1) is what we are testing for, in this case, that the proportion of patients free from headaches after 15 minutes is greater for the oxygen treatment group than the placebo group.
Step 4 ：Therefore, the correct hypotheses would be: \[H_{0}: p_{1}=p_{2}\] \[H_{1}: p_{1}>p_{2}\]
Step 5 ：So, the correct answer is B.
Step 6 ：Final Answer: The correct hypotheses are \[\boxed{H_{0}: p_{1}=p_{2}}\] \[\boxed{H_{1}: p_{1}>p_{2}}\]
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Solution
Step 1 ：Consider the first sample to be the sample of patients treated with oxygen and the second sample to be the sample of patients given a placebo. What are the null and alternative hypotheses for the hypothesis test?
Step 2 ：The null hypothesis (H0) is usually a statement of no effect or no difference. In this case, it would be that the proportion of patients free from headaches after 15 minutes is the same for both the oxygen treatment group and the placebo group.
Step 3 ：The alternative hypothesis (H1) is what we are testing for, in this case, that the proportion of patients free from headaches after 15 minutes is greater for the oxygen treatment group than the placebo group.
Step 4 ：Therefore, the correct hypotheses would be: \[H_{0}: p_{1}=p_{2}\] \[H_{1}: p_{1}>p_{2}\]
Step 5 ：So, the correct answer is B.
Step 6 ：Final Answer: The correct hypotheses are \[\boxed{H_{0}: p_{1}=p_{2}}\] \[\boxed{H_{1}: p_{1}>p_{2}}\]
