Exercise 8- 5. Perception and Reality In a presidential election, 308 out of 611 voters surveyed said that they voted for the candidate who won (based on data from ICR Survey Research Group). Use a 0.05 significance level to test the claim that among all voters, the percentage who believe that they voted for the winning candidate is equal to 43%, which is the actual percentage of votes for the winning candidate. What does the result suggest about voter perceptions?

The summary of the results obtained from the survey is as follows.

Sample size $n=611$.

Votes in favor of winning candidate $x=308$.

Level of significance $\alpha =0.05$.

It is claimed that the proportion of voters in favor of the winning candidate is 43% or 0.43.

Null hypothesis: The votes for the winning candidate are equal to 43%.

Alternative hypothesis: The votes for the winning candidate are not equal to 43%

Let p be the true proportion of voters who vote for the candidate who won.

Mathematically,

$$\begin{gathered} H_0:p=0.43 \\ {H}_1:p\ne 0.43 \end{gathered}$$

In testing claims for the population proportion, the z-test is used.

The sample proportion is given as follows.

$$\begin{gathered} \hat{p}=\frac{308}{611} \\ =0.5041 \end{gathered}$$

From the given information,

$$\begin{gathered} p=0.43 \\ q=1-p \\ =1-0.43 \\ =0.57 \end{gathered}$$

The test statistic is computed as follows.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.5041-0.43}{\sqrt{\frac{0.43\times 0.57}{611}}} \\ =3.70 \end{gathered}$$

From the standard normal table, the critical values for a two-tailed test with a 0.05 significance level are obtained as follows.

The left-tailed critical value will have an area of 0.025, corresponding to row -1.9 and column 0.06.

The right-tailed critical value will have an area of 0.975, corresponding to row 1.9 and column 0.06.

Thus, the two critical values are -1.96 and 1.96.

If the test statistic lies between the critical values, the null hypothesis is failed to be rejected. Otherwise, the null hypothesis is rejected.

Here, 3.70 does not fall between the critical values -1.96 and 1.96.

Thus, the null hypothesis is rejected.

The result suggests that there is not sufficient evidence to support the claim that the actual percentage of voters who favored the winning candidate is 43%.

Asthe proportion of voters in the survey is quite high, it suggests that the voters did not reveal the true votes in the survey.