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 387 employed persons and 359 unemployed persons are independently and randomly selected, and that 243 of the employed persons and 202 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=0.01 for the test.

State the null and alternative hypotheses for the test

Find the values of the two sample proportions, pˆ1 and pˆ2. Round to 3 decimal places

Compute the weighted estimate of p, p‾. Round to 3 decimal places

Compute the value of the test statistic. Round to 2 decimal places

Determine the decision rule for rejecting the null hypothesis H_0. Round to 3 decimal places [ (Reject H₀ if (t or absolute value of t) is (< or >)   (value) ]

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

Answer 1

Solution:

Given: 387 employed persons and 359 unemployed persons are independently and randomly selected, and that 243 of the employed persons and 202 of the unemployed persons have registered to vote.

Employed group:

n₁ = 387

x₁ = 243

Unemployed group:

n₂ = 359

x₂ = 202

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 .

Part a) State the null and alternative hypotheses for the test

H0: p₁ = p₂

Vs

H1: p₁ > p₂

Part b) Find the values of the two sample proportions, pˆ1 and pˆ2. Round to 3 decimal places

Part c) Compute the weighted estimate of p, p‾. Round to 3 decimal places

Part d) Compute the value of the test statistic. Round to 2 decimal places

Part e) Determine the decision rule for rejecting the null hypothesis H_0. Round to 3 decimal places

significance level =

Since H1: p₁ > p_{2 ,} this is right tailed test.

Thus find An area = 1 -

Thus look in z table for Area = 0.9900 or its closest area and find corresponding z critical value.

Area 0.9901 is closest to 0.9900 , thus corresponding z value is 2.3 and 0.03

Thus Z_critical = 2.33

Thus decision rule:
Reject H0, if z test statistic value > Z_critical = 2.33

Part f) Make a decision to reject or fail to reject the null hypothesis.

Since z test statistic value = 1.81 < Z_critical = 2.33, we fail to reject null hypothesis H0.

Thus there is not 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