To find a 99% confidence interval for the true population proportion of employees who prefer to provide their own retirement benefits, we can use the formula for the confidence interval of a proportion:

$\text{Confidence Interval} = \hat{p} \pm Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Where:

• $\hat{p}$ is the sample proportion (0.4 in this case).
• $Z_{\alpha/2}$ is the critical value for the desired confidence level (for 99% confidence, $Z_{\alpha/2} = 2.576$).
• $n$ is the sample size (75 in this case).

Step 1: Calculate the standard error (SE)

$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.4(1-0.4)}{75}} = \sqrt{\frac{0.4 \times 0.6}{75}} = \sqrt{\frac{0.24}{75}} \approx \sqrt{0.0032} \approx 0.0566$

Step 2: Calculate the margin of error (ME)

$ME = Z_{\alpha/2} \cdot SE = 2.576 \cdot 0.0566 \approx 0.1458$

Step 3: Determine the confidence interval

$\text{Confidence Interval} = \hat{p} \pm ME = 0.4 \pm 0.1458$ $\text{Confidence Interval} = (0.4 - 0.1458, 0.4 + 0.1458) = (0.2542, 0.5458)$

Conclusion

The 99% confidence interval for the true population proportion of employees who prefer to provide their own retirement benefits is approximately (0.2542, 0.5458).

This means that management can be 99% confident that the true proportion of all employees in the organization who prefer to provide their own retirement benefits lies between 25.42% and 54.58%.

Would you like any more details or have any further questions?

Here are 5 related questions for further exploration:

1. How would the confidence interval change if we wanted a 95% confidence level instead of 99%?
2. What would happen to the confidence interval if the sample size increased to 150?
3. How does the sample proportion ($\hat{p}$) affect the width of the confidence interval?
4. Can you explain why a larger sample size results in a narrower confidence interval?
5. How would the confidence interval change if the sample proportion were 0.5 instead of 0.4?

Tip: Remember that the width of a confidence interval depends on the sample size, the level of confidence, and the sample proportion. Increasing the sample size decreases the margin of error, leading to a narrower confidence interval.