A sample of 100 results in 27 successes.

Calculate the point estimate for the population proportion of successes.

Note: Round final answer to 3 decimal places.



Construct the 95% and the 90% confidence intervals for the population proportion.

Note: Round final answers to 3 decimal places.





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?

---

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.

A sample of 100 results in 27 successes. Calculate the point estimate for the population proportion of successes.

Construct the 95% and the 90% confidence intervals for the population proportion.

Learn how to calculate the point estimate and confidence intervals for a population proportion, using a sample of 100 with 27 successes. Detailed solution includes confidence intervals at both 95% and 90% levels, suitable for students in advanced high school or college statistics.

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