A simple random sample of size n equals 250 individuals who are currently employed is asked if they work at home at least once per week. Of the 250 employed individuals​ surveyed, 33 responded that they did work at home at least once per week. Construct a​ 99% confidence interval for the population proportion of employed individuals who work at home at least once per week.
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Part 1
The lower bound is

. 077. ​(Round to three decimal places as​ needed.)
Part 2
The upper bound is

enter your response here. ​(Round to three decimal places as​ needed.)

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

\[
\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

where:
- \( \hat{p} \) is the sample proportion.
- \( Z_{\alpha/2} \) is the critical value for the 99% confidence level (for 99%, \( Z_{\alpha/2} \approx 2.576 \)).
- \( n \) is the sample size.

### Step 1: Calculate the sample proportion (\( \hat{p} \))
\[
\hat{p} = \frac{33}{250} = 0.132
\]

### Step 2: Find the critical value \( Z_{\alpha/2} \)
For a 99% confidence level, the critical value is:
\[
Z_{\alpha/2} = 2.576
\]

### Step 3: Compute the standard error
\[
\text{Standard Error} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.132(1 - 0.132)}{250}}
\]
\[
\text{Standard Error} = \sqrt{\frac{0.132 \times 0.868}{250}} = \sqrt{\frac{0.114576}{250}} = \sqrt{0.0004583} = 0.02142
\]

### Step 4: Calculate the margin of error
\[
\text{Margin of Error} = Z_{\alpha/2} \times \text{Standard Error} = 2.576 \times 0.02142 = 0.0552
\]

### Step 5: Find the confidence interval
\[
\hat{p} \pm \text{Margin of Error} = 0.132 \pm 0.0552
\]

- **Lower bound**: \( 0.132 - 0.0552 = 0.0768 \)
- **Upper bound**: \( 0.132 + 0.0552 = 0.1872 \)

### Final Answer
- **Lower bound**: \( 0.077 \) (rounded to three decimal places)
- **Upper bound**: \( 0.187 \) (rounded to three decimal places)

Let me know if you have further questions or need more details!

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Here are 5 follow-up questions to deepen understanding:
1. How does changing the confidence level affect the width of the interval?
2. What is the interpretation of this 99% confidence interval?
3. How would a larger sample size affect the confidence interval?
4. What assumptions are made when constructing this confidence interval?
5. How can we determine the required sample size for a desired margin of error?

**Tip:** The critical value \(Z_{\alpha/2}\) depends on the desired confidence level. Make sure you use the correct value for your calculations!

A simple random sample of size n equals 250 individuals who are currently employed is asked if they work at home at least once per week. Of the 250 employed individuals surveyed, 33 responded that they did work at home at least once per week. Construct a 99% confidence interval for the population proportion of employed individuals who work at home at least once per week.

Learn how to construct a 99% confidence interval for the population proportion of employed individuals who work at home at least once per week. This statistical problem covers key concepts such as sample proportion, standard error, and confidence intervals, providing a step-by-step solution using a sample size of 250 and a response count of 33.

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