A marketing company selected a sample of 25 lawyers and 25 doctors to determine if they had voted in the last election. At $= .05$, test the claim that the proportions of lawyers and doctors who voted are equal . $\begin{array}{lcc} & \text { Lawyers } & \text { Doctors } \ \text { Voted } & 17 & 8 \ \text { Did not vote } & 8 & 17 \ \text { Total } & 25 & 25 \end{array}$ A) There is not evidence to reject the claim that the proportions are equal because the test value $3.841 3.841$ C) There is evidence to reject the claim that the proportions are equal because the test value $6.480 9.488$ D) There is not evidence to reject the claim that the proportions are equal because the test value $9.488

Question

A marketing company selected a sample of 25 lawyers and 25 doctors to determine if they had voted in the last election. At $= .05$, test the claim that the proportions of lawyers and doctors who voted are equal . $\begin{array}{lcc} & \text { Lawyers } & \text { Doctors } \ \text { Voted } & 17 & 8 \ \text { Did not vote } & 8 & 17 \ \text { Total } & 25 & 25 \end{array}$ A) There is not evidence to reject the claim that the proportions are equal because the test value $3.841 <$ $1.440$ B) There is evidence to reject the claim that the proportions are equal because the test value $6.480 > 3.841$ C) There is evidence to reject the claim that the proportions are equal because the test value $6.480 > 9.488$ D) There is not evidence to reject the claim that the proportions are equal because the test value $9.488 <$ $1.440$

Quizplus · Accepted Answer

We need to perform a two-proportion z-test to compare the proportions of lawyers and doctors who voted. The null hypothesis is that the proportions are equal, and the alternative hypothesis is that they are not equal.

The test statistic is calculated as:
$$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\frac{\hat{p}(1 - \hat{p})}{n_1} + \frac{\hat{p}(1 - \hat{p})}{n_2}}}$$
where $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions of lawyers and doctors who voted, $\hat{p}$ is the pooled sample proportion, and $n_1$ and $n_2$ are the sample sizes.

Plugging in the numbers, we get:
$$\hat{p}_1 = 17/25 = 0.68, \hat{p}_2 = 8/25 = 0.32$$
$$\hat{p} = (17 + 8)/(25 + 25) = 0.50$$
$$z = \frac{(0.68 - 0.32) - 0}{\sqrt{\frac{0.50(1 - 0.50)}{25} + \frac{0.50(1 - 0.50)}{25}}} = 2.848$$

Using a standard normal distribution table, we find that the critical value for a two-tailed test with $\alpha = 0.05$ is 1.96. Since the calculated test statistic of 2.848 is greater than 1.96, we reject the null hypothesis and conclude that there is evidence to support the claim that the proportions of lawyers and doctors who voted are not equal.

Therefore, the answer is B.

A Marketing Company Selected a Sample of 25 Lawyers and 25