Question

According to a report 49% of all eligible soldiers use their GI Bill for themselves. A spouse believes it is lower than that. She did a survey of 33 soldiers and of that, 12 have used their benefits already and 5 are currently using their benefits. The others have transferred their education benefits to their spouse or children. Does the evidence support the spouse’s claim that using a 0.05 level of significance, more than 51% of eligible soldiers transfer their benefits?   Draw a conclusion using the p-value compared to the level of significance and Interpret the decision

Answer 1

Number of soldiers  transferred their education benefits to their spouse or children = 33 - 12 - 5 = 16

The following information is provided: The sample size is N = 33 , the number of favorable cases is X = 16 , and the sample proportion is , and the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p ≤ 0.51

Ha: p > 0.51

This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is z_c = 1.64

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = -0.289 ≤ zc​ = 1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.6137 , and since p = 0.6137 ≥ 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p is greater than p0​, at the α = 0.05 significance level.