To solve this, let’s go step-by-step.

Step 1: Point Estimate for the Population Proportion

The point estimate for the population proportion of successes ($\hat{p}$) is calculated as: $\hat{p} = \frac{\text{number of successes}}{\text{total sample size}} = \frac{27}{100} = 0.27$ So, the point estimate for the population proportion is: $\hat{p} = 0.27$

Step 2: Confidence Intervals for the Population Proportion

Formula for Confidence Interval

The confidence interval for the population proportion is given by: $\hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ where:

• $\hat{p}$ is the sample proportion (0.27),
• $Z$ is the Z-score corresponding to the desired confidence level,
• $n$ is the sample size (100).

95% Confidence Interval

For a 95% confidence level, the Z-score ($Z$) is approximately 1.96.

Plugging in the values: $0.27 \pm 1.96 \times \sqrt{\frac{0.27 \times (1 - 0.27)}{100}}$ Calculating inside the square root first: $\sqrt{\frac{0.27 \times 0.73}{100}} = \sqrt{0.001971} \approx 0.0444$ Now, calculate the margin of error: $1.96 \times 0.0444 \approx 0.087$ So, the 95% confidence interval is: $0.27 \pm 0.087 \Rightarrow (0.183, 0.357)$

90% Confidence Interval

For a 90% confidence level, the Z-score ($Z$) is approximately 1.645.

Using the same steps: $0.27 \pm 1.645 \times 0.0444$ Calculating the margin of error: $1.645 \times 0.0444 \approx 0.073$ Thus, the 90% confidence interval is: $0.27 \pm 0.073 \Rightarrow (0.197, 0.343)$

Summary of Results

• Point Estimate for Population Proportion: $\hat{p} = 0.27$
• 95% Confidence Interval: (0.183, 0.357)
• 90% Confidence Interval: (0.197, 0.343)

Would you like additional details or have any questions?

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Here are some follow-up questions you might consider:

1. How would the confidence interval change with a larger sample size?
2. Why does the confidence interval get narrower as the confidence level decreases?
3. How is the point estimate affected by the sample size?
4. What assumptions are made when constructing confidence intervals for proportions?
5. How do we interpret confidence intervals in the context of statistical inference?

Tip: Remember, a higher confidence level widens the interval, giving more certainty but less precision.