Question

For each question please state your null and alternative hypothesis, your p-value and conclusion of your hypothesis test. You may use your calculator to perform the hypothesis.

1. Research conducted a few years ago showed that 35% of UCLA students had travelled outside the US. UCLA has recently implemented a new study abroad program and results of a new survey show that out of the 100 randomly sampled students 40 have travelled abroad. Is there significant evidence to suggest that the proportion of students at UCLA who have travelled abroad has increased after the implementation of the study abroad program? Use a .01 significance level.

   2. Sleep experts believe that sleep apnea is more likely to occur in men than in the general population. In other words, they claim the percentage of men who suffer from sleep apnea is greater than 5.8%. To test this claim, one sleep expert examines a simple random sample of 90 men and determines 9 of these men suffer from sleep apnea. Does this evidence support the claim that the percentage of men who suffer from sleep apnea not longer equals 5.8%? Use a 0.05 level of significance.

Answer 1

Answering the first question

The following information is provided: The sample size is N=100, the number of favorable cases is X=40, and the sample proportion is

and the significance level is α=0.01

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p=0.35

Ha: p>0.35

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=0.01, and the critical value for a right-tailed test is zc​=2.33.

The rejection region for this right-tailed test is R={z:z>2.33}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=1.048≤zc​=2.33, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.1473, and since p=0.1473≥0.01, 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=0.01 significance level.

Confidence Interval

The 99% confidence interval for p is: 0.274<p<0.526.

Graphically

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