Question

Researchers are studying rates of homeownership in a certain town. They believe that the proportion of people ages 36−50 who own homes is significantly greater than the proportion of people ages 21−35 who own homes and want to test this claim. The results of their survey are shown in the table.

Homeowners
- Ages 21-35: 18
- Ages 36-50: 40
- Total: 58

Renters
- Ages 21-35: 38
- Ages 36-50: 22
- Total: 60

Total
- Ages 21-35: 56
- Ages 36-50: 62
- Total: 118

What are the null hypothesis and alternative hypothesis for this situation?

A) null: p_1 - p_2 = 0; alternative: p_1 - p_2 > 0
B) null: p_1 - p_2 > 0; alternative: p_1 - p_2 = 0
C) null: p_2 - p_1 = 0; alternative: p_2 - p_1 ≠ 0
D) null: p_1 - p_2 > 0; alternative: p_1 - p_2 < 0

Answer 1

In this scenario, the researchers are interested in comparing the proportions of homeownership between two different age groups:

- \( p_1 \): Proportion of homeowners in the age group 36-50
- \( p_2 \): Proportion of homeowners in the age group 21-35

The researchers believe that the proportion of homeowners in the 36-50 age group is significantly greater than in the 21-35 age group. Hence, the hypotheses can be formulated as follows:

- **Null Hypothesis (H0):** There is no difference in the proportions of homeownership between the two age groups (\( p_1 - p_2 = 0 \)).
- **Alternative Hypothesis (H1):** The proportion of homeowners in the 36-50 age group is greater than that in the 21-35 age group (\( p_1 - p_2 > 0 \)).

Therefore, the correct choice is:

**A) null: \( p_1 - p_2 = 0 \); alternative: \( p_1 - p_2 > 0 \)**.