Question

The U.S. Census Bureau conducts annual surveys to obtain information on the percentage of the voting-age population that is registered to vote. Suppose that 380 employed persons and 487 unemployed persons are independently and randomly selected, and that 221 of the employed persons and 211 of the unemployed persons have registered to vote. Can we conclude that the percentage of employed workers ( p1 ), who have registered to vote, exceeds the percentage of unemployed workers ( p2 ), who have registered to vote? Use a significance level of α=0.05 for the test.

Step 2: Compute the value of the test statistic. Round your answer to two decimal places.

Step 3: Determine the decision rule for rejecting the null hypothesis H0. Round the numerical portion of your answer to three decimal places.

Step 4: Reject or fail to reject the hypothesis?

Step 5 of 5: Make the decision for the hypothesis test.

Answer 1

Solution:

Given:

Employed workers:

n₁ = 380

x₁ = 221

thus

Unemployed workers:

n₂ = 487

x₂ = 211

thus

We have to test if  the percentage of employed workers ( p1 ), who have registered to vote, exceeds  the percentage of unemployed workers ( p2 ), who have registered to vote. That is : p₁ > p₂   

Level of significance = α=0.05

Step 1) State H0 and H1:

H₀: p₁ = p₂ Vs H₁: p₁ > p₂

This is right tailed test.

Step 2: Compute the value of the test statistic.

where

Thus

Step 3: Determine the decision rule for rejecting the null hypothesis H0.

Level of significance = α=0.05

This is right tailed test.

Thus find area = 1 - 0.05 = 0.95

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus Z_critical = 1.645

Thus  Decision Rule:
Reject null hypothesis ,if z  test statistic value > z critical value = 1.645 , otherwise we fail to reject H0.

Or

Reject H0 if  

Step 4: Reject or fail to reject the hypothesis?

Since  z  test statistic value = > z critical value = 1.645 , we reject null hypothesis H0.

Step 5: Make the decision for the hypothesis test.

At 0.05 level of significance, we have sufficient evidence to conclude that the percentage of employed workers ( p₁ ), who have registered to vote, exceeds the percentage of unemployed workers ( p₂ ), who have registered to vote