Question

The mayor of a town has proposed a plan for the annexation of an adjoining community. A political study took a sample of 1300 voters in the town and found that 50% of the residents favored annexation. Using the data, a political strategist wants to test the claim that the percentage of residents who favor annexation is over 47%. Testing at the 0.02 level, is there enough evidence to support the strategist's claim?

Step 1 of 6: State the null and alternative hypotheses.

Step 2 of 6:

Find the value of the test statistic. Round your answer to two decimal places.

Step 3 of 6:

Specify if the test is one-tailed or two-tailed.

Step 4 of 6:

Determine the decision rule for rejecting the null hypothesis, Ho​​​​​​Ho.

Step 5 of 6:

Make the decision to reject or fail to reject the null hypothesis.

Step 6 of 6:

State the conclusion of the hypothesis test.

Answer 1

Solution:

Given: A political study took a sample of 1300 voters in the town and found that 50% of the residents favored annexation.

That is: n = sample size = 1300 and Sample proportion = $\hat{p}=0.50$

Claim: the percentage of residents who favor annexation is over 47%.

Level of Significance = $\alpha = 0.02$

Step 1) State the null and alternative hypotheses

$H_{0}: p = 0.47\: \: \: \: Vs\: \: \: \: H_{1}: p >0.47$

Step 2) Find the value of the test statistic.

$z= \frac{\hat{p}-p}{\sqrt{\frac{p\times (1-p)}{n}}}$

$z= \frac{0.50-0.47}{\sqrt{\frac{0.47\times (1-0.47)}{1300}}}$

$z= \frac{0.03}{\sqrt{\frac{0.47\times 0.53}{1300}}}$

$z= \frac{0.03}{\sqrt{0.0001916 }}$

$z= \frac{0.03}{0.0138425 }$

$z=2.17$

Step 3) Specify if the test is one-tailed or two-tailed.

Since claim is: the percentage of residents who favor annexation is over 47%. which means we have to test proportion p is more than 47%. Thus this is one tailed test.

Step 4) Determine the decision rule for rejecting the null hypothesis, Ho​

Find z critical value for given level of Significance = $\alpha = 0.02$

Since this is right tailed test, find Area = 1 - 0.02 = 0.98 and look in z table for Area = 0.9800 or its closest area and find z value.

Area 0.9798 is closest to 0.9800, which corresponds to 2.0 and 0.05

thus z critical value = 2.05

Thus decision rule is:

" Reject null hypothesis H0, if z test statistic value > z critical value = 2.05 , otherwise we fail to reject H0."

Step 5) Make the decision to reject or fail to reject the null hypothesis.

Since z test statistic value = 2.17 > z critical value= 2.05 , we reject null hypothesis H0.

Step 6) State the conclusion of the hypothesis test.

Since we have rejected null hypothesis H0, we conclude that: There is sufficient evidence to support the claim that: the percentage of residents who favor annexation is over 47%.