To compute the confidence intervals for the population proportion based on Sample 1, we'll use the following formula for the confidence interval of a population proportion:

$\text{CI} = \hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $\hat{p}$ is the sample proportion (0.37 in this case),
• $n$ is the sample size (85 in this case),
• $z$ is the critical value (provided for 75% and 90% confidence levels),
• $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ is the standard error of the sample proportion.

Step 1: Calculate the Standard Error (SE)

The standard error (SE) is calculated as: $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.37(1 - 0.37)}{85}} = \sqrt{\frac{0.37 \times 0.63}{85}} \approx \sqrt{\frac{0.2331}{85}} \approx \sqrt{0.002742} \approx 0.05237$

Step 2: Calculate the 75% Confidence Interval

For the 75% confidence interval, the critical value $z = 1.150$.

The confidence interval formula is: $CI = 0.37 \pm 1.150 \times 0.05237$ Calculate the margin of error: $1.150 \times 0.05237 \approx 0.06023$

Now calculate the lower and upper bounds:

• Lower bound: $0.37 - 0.06023 = 0.30977 \approx 0.31$
• Upper bound: $0.37 + 0.06023 = 0.43023 \approx 0.43$

Thus, the 75% confidence interval is $(0.31, 0.43)$.

Step 3: Calculate the 90% Confidence Interval

For the 90% confidence interval, the critical value $z = 1.645$.

The confidence interval formula is: $CI = 0.37 \pm 1.645 \times 0.05237$ Calculate the margin of error: $1.645 \times 0.05237 \approx 0.08614$

Now calculate the lower and upper bounds:

• Lower bound: $0.37 - 0.08614 = 0.28386 \approx 0.28$
• Upper bound: $0.37 + 0.08614 = 0.45614 \approx 0.46$

Thus, the 90% confidence interval is $(0.28, 0.46)$.

Summary of Results:

• 75% Confidence Interval: (0.31, 0.43)
• 90% Confidence Interval: (0.28, 0.46)

If plotted on a graph, you'd mark the population proportion $p = 0.45$ alongside these intervals.

Would you like more details on any of these calculations, or have any questions?

Here are five questions to further explore this topic:

1. How does increasing the sample size affect the width of the confidence intervals?
2. Why is the 90% confidence interval wider than the 75% confidence interval?
3. What would happen to the intervals if the sample proportion was closer to 0.45?
4. How is the choice of confidence level related to the margin of error?
5. What are the implications if the population proportion falls outside of the calculated confidence intervals?

Tip: The width of a confidence interval decreases as the sample size increases, which means more precise estimates for larger samples.