Let's go through each part of this hypothesis test question.

Part (a): Hypotheses Selection

The problem states:

• The null hypothesis $H_0$ should represent the assumption that the county’s vacancy rate is the same as the national rate, which is 14.4%.
• The alternative hypothesis $H_A$ should represent that the county's vacancy rate is different from this rate.

So:

• Null Hypothesis ($H_0$): $p = 0.144$
• Alternative Hypothesis ($H_A$): $p \neq 0.144$

This matches option (C) from the choices listed.

Part (b): Test Type Selection

To decide on the test type, we need to consider:

1. Model Type: We’re working with a proportion test since the data represents a percentage of vacant housing.
2. Conditions: This is a single sample compared to a known population rate, so it fits a one-proportion z-test for proportions.

Thus, the correct answer for part (b) is:

• One-proportion z-test

Would you like a deeper explanation of these steps or have any questions?

Additional Questions to Consider

1. How do you determine if the sample size is large enough for a one-proportion z-test?
2. What are the assumptions or conditions for using a z-test on proportions?
3. What would the test statistic and p-value look like in this scenario?
4. How would you interpret the results if you conducted this test?
5. What would the conclusions be if you found a p-value below your significance level?

Tip

For hypothesis testing, always clearly define your null and alternative hypotheses and ensure the correct test type aligns with your data and objectives.