Rhino viruses typically cause common colds. In a test of the effectiveness of echinacea, 44 of the 49 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 81 of the 98 subjects developed rhinovirus infections. Use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections. Complete parts (a) through (c) below.
D.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
E. $H_{0}: p_{1} \neq p_{2}$
$H_{1}: p_{1}=p_{2}$
\[
\text { F. } \begin{array}{r}
H_{0}: p_{1} \leq p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
Identify the test statistic.
$z=1.14$
(Round to two decimal places as needed.)
Identify the P-value.
P-value $=0.252$
(Round to three decimal places as needed.)
What is the conclusion based on the hypothesis test?
The P-value is greater than the significance level of $\alpha=0.05$, so fail to reject the null hypothesis. There is not sufficient evidence to support the claim that echinacea treatment has an effect.
b. Test the claim by constructing an appropriate confidence interval.
The $95 \%$ confidence interval is $\square< \left(p_{1}-p_{2}\right)< \square$.
(Round to three decimal places as needed.)
Answer
State the final answer. The $95\%$ confidence interval for the difference in proportions is \(-0.042<p_{1}-p_{2}<0.185\). This means that we are $95\%$ confident that the true difference in proportions of rhinovirus infections between the echinacea group and the placebo group is between \(-0.042\) and \(0.185\). Since this interval includes $0$, we cannot conclude that there is a significant difference between the two proportions. This is consistent with the result of the hypothesis test, which also failed to reject the null hypothesis that the proportions are equal. Therefore, there is not sufficient evidence to support the claim that echinacea treatment has an effect on rhinovirus infections. \(\boxed{-0.042<p_{1}-p_{2}<0.185}\)
Step 1 ：State the hypotheses. The null hypothesis is that the proportions are equal, and the alternative hypothesis is that they are not equal. In mathematical terms, this is: \(H_{0}: p_{1}=p_{2}\) and \(H_{1}: p_{1} \neq p_{2}\)
Step 2 ：Calculate the test statistic. The test statistic is given as \(z=1.14\)
Step 3 ：Calculate the P-value. The P-value is given as \(P-value =0.252\)
Step 4 ：Compare the P-value to the significance level. Since the P-value is greater than the significance level of \(\alpha=0.05\), we fail to reject the null hypothesis. This means that there is not sufficient evidence to support the claim that echinacea treatment has an effect.
Step 5 ：Calculate the confidence interval for the difference in proportions. The formula for the confidence interval for the difference in proportions is \((p1 - p2) \pm z*\sqrt{(p1(1 - p1)/n1) + (p2(1 - p2)/n2)}\), where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and z is the z-score for the desired confidence level (1.96 for a 95% confidence level).
Step 6 ：Substitute the given values into the formula. The sample proportions are \(p1 = 0.898\) and \(p2 = 0.827\), the sample sizes are \(n1 = 49\) and \(n2 = 98\), and the z-score for a 95% confidence level is \(z = 1.96\).
Step 7 ：Calculate the standard error (se). The standard error is \(se = 0.058\)
Step 8 ：Calculate the lower and upper bounds of the confidence interval. The lower bound is \(ci_{lower} = -0.042\) and the upper bound is \(ci_{upper} = 0.185\)
Step 9 ：State the final answer. The $95\%$ confidence interval for the difference in proportions is \(-0.042
