Question

A sample of 111 mortgages approved during the current year showed that 32 were issued to a single-earner family or individual. The historical percentage is 24 percent.

At the .05 level of significance in a right-tailed test, has the percentage of single-earner or individual mortgages risen?

Answer 1

To determine if the percentage of single-earner mortgages has risen, we use a one-proportion z-test. The calculated z-score, based on the sample proportion and historical proportion, indicates whether the increase is significant at the .05 level.

Step-by-step explanation:

The task is to determine if the percentage of single-earner or individual mortgages has significantly risen from the historical percentage of 24%. We start with the null hypothesis (H0) that the true percentage of single-earner mortgages is still 24%, and the alternative hypothesis (H1) that the true percentage is greater than 24%.

An appropriate statistical test for this problem is the one-proportion z-test. To perform this test, we calculate the z-score to see how many standard deviations the observed proportion is from the historical proportion.

1. Calculate the standard error (SE) using the formula
   SE = \sqrt{(p(1-p)/n)}, where p is the historical proportion and n is the sample size.
2. Calculate the z-score with the formula z = (p_hat - p) / SE, where p_hat is the sample proportion.
3. Check if the z-score falls in the critical region, which, at a significance level (α) of 0.05 for a right-tailed test, corresponds to a z-score greater than 1.645.

If the calculated z-score exceeds the critical value, we reject the null hypothesis, indicating that the percentage of single-earner mortgages has significantly risen above 24%.