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, 31 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.

To construct a 99% confidence interval for the population proportion of employed individuals who work at home at least once per week, we follow these steps:

### Step 1: Determine the sample proportion (\(\hat{p}\))
The sample proportion \(\hat{p}\) is calculated as:
\[
\hat{p} = \frac{x}{n}
\]
where:
- \(x\) is the number of individuals who responded that they work at home at least once per week.
- \(n\) is the sample size.

Given:
- \(x = 31\)
- \(n = 250\)

\[
\hat{p} = \frac{31}{250} = 0.124
\]

### Step 2: Find the critical value \(z\) for the 99% confidence level
For a 99% confidence level, the critical value \(z\) corresponds to the 99% confidence level, which is typically \(z \approx 2.576\).

### Step 3: Calculate the standard error (SE)
The standard error of the proportion is given by:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
\[
SE = \sqrt{\frac{0.124(1 - 0.124)}{250}} = \sqrt{\frac{0.124 \times 0.876}{250}} \approx 0.021
\]

### Step 4: Construct the confidence interval
The 99% confidence interval is given by:
\[
\hat{p} \pm z \times SE
\]
\[
0.124 \pm 2.576 \times 0.021
\]
\[
0.124 \pm 0.0541
\]
This results in the interval:
\[
(0.0699, 0.1781)
\]

### Conclusion
The 99% confidence interval for the population proportion of employed individuals who work at home at least once per week is approximately \((0.070, 0.178)\).

Would you like more details or have any questions?

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Here are 5 related questions you might explore:
1. How would the confidence interval change if the confidence level was reduced to 95%?
2. What would be the confidence interval if the sample size were increased to 500?
3. How does the standard error affect the width of the confidence interval?
4. How is the critical value \(z\) determined for different confidence levels?
5. How does a higher sample proportion affect the confidence interval?

**Tip:** When increasing the confidence level, the width of the confidence interval also increases because a higher confidence level means we are more certain that the interval contains the population proportion.

Learn how to construct a 99% confidence interval for the population proportion of employed individuals who work from home at least once per week. This detailed solution covers sample proportion calculation, standard error determination, and confidence interval construction. Suitable for advanced high school students and beyond.

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