Question

You wish to test the following claim ( H a ) at a significance level of α = 0.001 . H o : p 1 = p 2 H a : p 1 < p 2 You obtain 22.7 % successes in a sample of size n 1 = 775 from the first population. You obtain 26.7 % successes in a sample of size n 2 = 795 from the second population.

Calculator tip test statistic = What is the p-value for this sample? p-value =

The p-value is... less than (or equal to) α greater than α This test statistic leads to a decision to...

reject the null accept the null fail to reject the null As such, the final conclusion is that...

There is sufficient evidence to warrant rejection of the claim that the first population proportion is less than the second population proportion.

There is not sufficient evidence to warrant rejection of the claim that the first population proportion is less than the second population proportion.

The sample data support the claim that the first population proportion is less than the second population proportion. There is not sufficient sample evidence to support the claim that the first population proportion is less than the second population proportion.

Answer 1

Solution:

Given:

We have to test the hypothesis :

H_o : p₁ = p₂ Vs H_a : p₁ < p₂

Alternative hypothesis H_a is the claim.

first population

Sample Size = n₁ = 775

Sample proportion =

Second population

Sample Size = n₂ = 795

Sample proportion =

Find test statistic value:

where

Thus we get:

Thus test statistic value = z = -1.84

What is the p-value for this sample?

p-value = P( Z < z test statistic value)

p-value = P( Z < -1.84)

Look in z table for z = -1.8 and 0.04 and find corresponding area.

P( Z < -1.84 ) = 0.0329

Thus

p-value = P( Z < -1.84)

p-value = 0.0329

Given significance level =

Since p-value = 0.0329 > significance level = , we fail to reject null hypothesis.

Thus

The p-value is greater than α. This test statistic leads to a decision to fail to reject the null.

As such, the final conclusion is that:

There is not sufficient sample evidence to support the claim that the first population proportion is less than the second population proportion.