https://file.aiquickdraw.com/mathbot/file/17288606090312u7g70qx.jpg

To construct a 99% confidence interval for the population proportion \(p\), we will use the following formula:

\[
CI = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \( \hat{p} = \frac{x}{n} \) is the sample proportion.
- \(x = 60\) (number of adults living with their parents).
- \(n = 200\) (sample size).
- \(Z\) is the z-score for a 99% confidence level, which is approximately **2.576**.

---

### Step 1: Calculate the sample proportion \( \hat{p} \).

\[
\hat{p} = \frac{60}{200} = 0.3
\]

---

### Step 2: Calculate the standard error (SE).

\[
SE = \sqrt{\frac{0.3 \cdot (1 - 0.3)}{200}} = \sqrt{\frac{0.3 \cdot 0.7}{200}} = \sqrt{0.00105} \approx 0.0324
\]

---

### Step 3: Find the margin of error (ME).

\[
ME = Z \cdot SE = 2.576 \cdot 0.0324 \approx 0.0835
\]

---

### Step 4: Compute the confidence interval.

\[
CI = 0.3 \pm 0.0835
\]

- Lower bound: \(0.3 - 0.0835 = 0.2165\)
- Upper bound: \(0.3 + 0.0835 = 0.3835\)

---

### Final Answer:

\[
0.2165 < p < 0.3835
\]

Thus, the 99% confidence interval for the true population proportion is **(0.2165, 0.3835)**.

---

Do you have any questions or need further clarification?

---

Here are 5 related questions for further exploration:
1. What is the z-score for a 95% confidence interval?
2. How does increasing the sample size affect the width of the confidence interval?
3. What happens to the margin of error if the confidence level is reduced to 90%?
4. How would the confidence interval change if 80 out of 200 adults lived with their parents?
5. How do you interpret a confidence interval in a practical scenario?

**Tip:** Always round your final answers to the required number of decimal places to avoid small rounding errors in statistical problems!

Out of a sample of 200 adults aged 18 to 30, 60 still lived with their parents. Based on this, construct a 99% confidence interval for the true population proportion of adults ages 18 to 30 that still live with their parents. Give your answers rounded to 4 decimal places.

Learn how to construct a 99% confidence interval for the population proportion of adults aged 18-30 who live with their parents. This solution involves key statistical concepts such as the Z-distribution and the formula for confidence intervals for proportions. Based on a sample size of 200, where 60 adults live with their parents, this guide provides a detailed step-by-step solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation