Question

In a survey of 480 childless married couples who were asked if they plan to have children in the next 55 years, 22% of the men and 26% of the women responded "Yes". Based on this survey, can it be concluded that there is a difference in the proportion of men ( p1 ) and women ( p2 ) responding "Yes"? Use a significance level of α=0.01 for the test.

Step 1 of 5: State the null and alternative hypotheses for the test.

Step 2 of 5: Compute the weighted estimate of p, p‾p‾. Round your answer to three decimal places

Step 3 of 5: Compute the value of the test statistic. Round your answer to two decimal places.

Step 4 of 5: Find the P-value for the hypothesis test. Round your answer to four decimal places.

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

Answer 1

Two-Proportion Z test

The following information is provided: (a) Sample 1 - The sample size is N1 = 480, and the sample proportion is p^1​​=0.22 (b) Sample 2 - The sample size is N2 = 480, and the sample proportion is p^2​​=0.26 and the significance level is α=0.01 Pooled Proportion The value of the pooled proportion is computed as (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = p2 Ha: p1 ≠ p2 This corresponds to a Two-tailed test, for which a z-test for two population proportions needs to be conducted. (2a) Critical Value Based on the information provided, the significance level is α=0.01, therefore the critical value for this Two-tailed test is Zc​=2.5758. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Two-tailed test is |Z|>2.5758 i.e. Z>2.5758 or Z<-2.5758 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(|Z|>1.451)=0.1468 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=1.451 < Zc​=2.5758, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.1468, and since p=0.1468>0.01, it is concluded that the null hypothesis is Not rejected. (6) Conclusion It is concluded that the null hypothesis Ho is Not rejected. Therefore, there is Not enough evidence to claim that the population proportion p1 is different than p2, at the 0.01 significance level.

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