Question

Politics A county clerk in Essex County, New Jersey, selected candidates for positions on election ballots. Democrats were selected first in 40 of 41 ballots. Because he was supposed to use a method of random selection, Republicans claimed that instead of using randomness, he used a method that favored Democrats. Use a 0.01 significance level to test the claim that the ballot selection method favors Democrats.

Short answer

The decision is to reject the hypotheses at a 0.01 level of significance, and hence, there is sufficient evidence to say that the ballot selection method favors the Democrats.

Step-by-step solution

Given information

The number of times Democrats were selected first out of 41 ballots is 40.

The significance level is 0.01.

The claim requires testing if the ballot selection favors Democrats.

State the hypotheses

Let p be the proportion of selections that favors the Democrats.

Null hypothesis: The ballot selection method does not support the Democrats.

Alternative hypothesis: The ballot selection method favors the Democrats.

Mathematically,

$$\begin{gathered} H_0:p=0.5 \\ {H}_1:p>0.5 \end{gathered}$$

Compute the test statistic

In testing the population proportion based on the sample proportion, the z-test is used.

The sample proportion is given as follows.

$$\begin{gathered} \hat{p}=\frac{40}{41} \\ =0.9756 \end{gathered}$$

From the given information,

$$\begin{gathered} p=0.5 \\ q=1-p \\ =1-0.5 \\ =0.5 \end{gathered}$$

The test statistic is as follows.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.9756-0.5}{\sqrt{\frac{0.5\times 0.5}{41}}} \\ =6.0908 \end{gathered}$$

Thus, the test statistic is 6.09.

Obtain the critical value

The critical value for the right-tailed test is expressed as follows.

$$\begin{gathered} P\left(Z>z_{\alpha }\right)=\alpha \\ P\left(Z>z_{0.01}\right)=0.01 \end{gathered}$$

From the z-table, the critical value for a one-tailed test with a 0.01 significance level is obtained as 2.33 (corresponding to rows 2.3 and 0.03).

State the decision rule

If the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

The test statistic value of 6.09 is greater than 2.33. Therefore, reject .

Thus, there is significant evidence to support the claim that the ballot selection method favors the Democrats.