Question

A campaign strategist wants to determine whether demographic shifts have caused a drop in allegiance to the Uniformian Party in Bowie County. Historically, around 62% of the county's registered voters have supported the Uniformians. In a survey of 196 registered voters, 57% indicated that they would vote for the Uniformians in the next election. Assuming a confidence level of 95% and conducting a one-sided hypothesis test, which of the following should the strategist do?

a. Accept the hypothesis that the proportion of Uniformian voters has not changed.
b. Accept the hypothesis that the proportion of Uniformian voters has decreased.
c. Conclude that the proportion of Uniformian voters is now between 56% and 62%.
d. There is not enough evidence to support the hypothesis that the proportion of Uniformian voters has decreased.

Answer 1

d. There is not enough evidence to support the hypothesis that the proportion of Uniformian voters has decreased.

Explanation:

This is a hypothesis test for a proportion.

The claim is that there is a significant drop in allegiance to the Uniformian Party in Bowie County.

Then, the null and alternative hypothesis are:

`H_0: \pi=0.62\\\\H_a:\pi<0.62`

The significance level is 0.05.

The sample has a size n=196.

The sample proportion is p=0.57.

The standard error of the proportion is:

`\sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt{(0.62*0.38)/(196)}\\\\\\ \sigma_p=√(0.001202)=0.035`

Then, we can calculate the z-statistic as:

`z=(p-\pi+0.5/n)/(\sigma_p)=(0.57-0.62+0.5/196)/(0.035)=(-0.047)/(0.035)=-1.369`

This test is a left-tailed test, so the P-value for this test is calculated as:

`\text{P-value}=P(z<-1.369)=0.0855`

As the P-value (0.0855) is greater than the significance level (0.05), the effect is not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that there is a significant drop in allegiance to the Uniformian Party in Bowie County.