Hypothesis Tests for a Single Population Proportion

Question

Test the claim that the proportion of people who own cats is significantly different than 90% at the 0.1 significance level.

The null and alternative hypothesis would be:H0:p≥0.9H0:p≥0.9

H1:p<0.9H1:p<0.9

H0:μ=0.9H0:μ=0.9

H1:μ≠0.9H1:μ≠0.9

H0:μ≤0.9H0:μ≤0.9

H1:μ>0.9H1:μ>0.9

H0:p≤0.9H0:p≤0.9

H1:p>0.9H1:p>0.9

H0:μ≥0.9H0:μ≥0.9

H1:μ<0.9H1:μ<0.9

H0:p=0.9H0:p=0.9

H1:p≠0.9H1:p≠0.9

The test is:right-tailed

two-tailed

left-tailed

Based on a sample of 400 people, 97% owned cats

The test statistic is:  (to 2 decimals)

The p-value is:  (to 2 decimals)

Based on this we:

1. Fail to reject the null hypothesis
2. Reject the null hypothesis

Answer 1

The problem statement tells you to test for what proportion of the population who owns cats. That should indicate whether you use mu or p. In addition, the claim is interested in whether or not this proportion is different from 0.9 (i.e. it does not care about whether or not the proportion is greater than or less than 0.9). That should help you determine the null and alternative hypotheses along with determining if the test is right/left/two-tailed.

As for the test statistic, you will want to use the central limit theorem. From your sample data of 0.97, you have an estimate for either mu or p already. Furthermore, the null has something to do with 0.9. So, for the numerator of your test statistic, you can do 0.97 - 0.9 = 0.07.

The next part is finding the standard deviation. How you find the standard deviation will depend on whether or not you use mu or p. As a hint, if you use mu, you probably will need a sample standard deviation somewhere. If you use p, the variance is p(1-p)/n, where n is the number of samples. You may plug in your sample data of 0.97 for p.

The test statistic will take the form of 0.07/(standard deviation). From there, you can use a z-table (or some statistical software) to determine a p-value and come to a conclusion.