Question

Over the past several years, the proportion of one-person households has been increasing. The Census Bureau would like to test the hypothesis that the proportion of one-person households exceeds 0.27. A random sample of 125 households found that 43 consisted of one person. To conduct the hypothesis test, what distribution would you use to calculate the critical value and the p-value?

Answer 1

To conduct the hypothesis test I would use the standard normal distribution to calculate the critical value and the p-value.

Explanation:

To conduct the hypothesis test I would use the standard normal distribution because there is a large sample size of n = 125 households. This because a point estimator for the true proportion p of one-person households is
`\hat{p} = Y/n` which is normally distributed with mean p and standard error
`√(p(1-p)/n)` when the sample size n is large. Here Y is the random variable that represents the number of one-person households observed. Then the test statistic is
`Z = \frac{\hat{p}-0.27}{√(p(1-p)/n)}` which has a standard normal distribution under the null hypothesis.