Question

2. In a survey of a sample of 200 guests (100 male and 100 female) on a major cruise line A, it

   was found that 45 male guests and 55 female quests were satisfied. However, the cruise line advertised that 65% of its guests (overall) were satisfied. Undertake a hypothesis test at a 5% significance level to examine if the claim made by the cruise line can be supported with the data.

3. From the results of the survey in part b, do you think female guests are more satisfied than

   male guests on the cruise line? (Note: Undertake the test at a 5% significance level.)

Answer 1

1st Answer

The following information is provided: The sample size is N=200, the number of favorable cases is X= 45 + 55 = 100, 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:

Ha:

This corresponds to a two-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 two-tailed test is zc​=1.96.

The rejection region for this two-tailed test is R={z:∣z∣>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=4.447>zc​=1.96, it is then concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p is different than p0​, at the α=0.05 significance level.

2nd Answer

Let Sample 1 be females while Sample 2 is males

For sample 1, we have that the N1​=100, the number of favorable cases is X1​=55, so then the sample proportion is

For sample 2, we have that the sample size is N2​=100, the number of favorable cases is X2​=45, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha: ​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

(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 zc​=1.64.

The rejection region for this right-tailed test is R={z:z>1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=1.414≤zc​=1.64, it is then 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 p1​ is greater than p2​, at the 0.05 significance level.

Graphically

Let me know in comments if anything is not clear. Will reply ASAP. Please do upvote if satisfied.