Part 3 of 6
Points: 0 of 1
In randomized, double-blind clinical trials of a new vaccine, rats were randomly divided into two groups. Subjects in group 1 received the new vaccine while subjects in group 2 received a control vaccine. After the second dose, 108 of 713 subjects in the experimental group (group 1) experienced drowsiness as a side effect. After the second dose, 65 of 594 of the subjects in the control group (group 2) experienced drowsiness as a side effect. Does the evidence suggest that a higher proportion of subjects in group 1 experienced drowsiness as a side effect than subjects in group 2 at the $\alpha=0.10$ level of significance?
A. The sample size is less than $5 \%$ of the population size for each sample.
B. The sample size is more than $5 \%$ of the population size for each sample.
C. The data come from a population that is normally distributed.
D. The samples are independent.
E. The samples are dependent.
F. $n_{1} \hat{p}_{1}\left(1-\hat{p}_{1}\right) \geq 10$ and $n_{2} \hat{p}_{2}\left(1-\hat{p}_{2}\right) \geq 10$
Determine the null and alternative hypotheses.
\[
\begin{array}{l}
H_{0}: P_{1}=P_{2} \\
H_{1}: P_{1}> P_{2}
\end{array}
\]
Find the test statistictior this hypothesis test.
(Round to two decimal places as needed.)
lew an example
Get more help .
Clear all
Check answer
Answer
Final Answer: The test statistic for this hypothesis test is \(\boxed{2.23}\).
Step 1 ：Given data: \(x_1 = 108\), \(n_1 = 713\), \(x_2 = 65\), \(n_2 = 594\). These represent the number of successes and the sample size for groups 1 and 2 respectively.
Step 2 ：Calculate the sample proportions for each group: \(p_1 = \frac{x_1}{n_1} = 0.1514726507713885\), \(p_2 = \frac{x_2}{n_2} = 0.10942760942760943\).
Step 3 ：Calculate the pooled proportion: \(p = \frac{x_1 + x_2}{n_1 + n_2} = 0.13236419280795717\).
Step 4 ：Calculate the test statistic: \(z = \frac{p_1 - p_2}{\sqrt{p(1 - p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} = 2.2333715844151727\).
Step 5 ：Round the test statistic to two decimal places: \(z = 2.23\).
Step 6 ：The test statistic for this hypothesis test is approximately 2.23. This value represents how many standard deviations the sample proportion is away from the null hypothesis proportion. A positive z-score indicates that the sample proportion is greater than the null hypothesis proportion, which supports the alternative hypothesis. However, we would need to compare this test statistic to the critical value at the given level of significance (\(\alpha=0.10\)) to make a final decision about the null hypothesis.
Step 7 ：Final Answer: The test statistic for this hypothesis test is \(\boxed{2.23}\).
