https://file.aiquickdraw.com/mathbot/file/1730334436327pdiunpjs.jpg

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?

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### 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.

According to a recent census, 14.4% of all housing units in a certain country are vacant. A county supervisor wonders if her county is different from this. She randomly selects 797 housing units in her county and finds that 144 of the housing units are vacant. Complete parts a through d below.

Learn how to test if a county's housing vacancy rate is different from the national rate of 14.4% using a one-proportion z-test. The problem involves formulating null and alternative hypotheses, selecting the correct test, and applying statistical methods. Suitable for college or AP Statistics students.

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