Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level.95% confidence; n = 2888, x = 1872

To find the margin of error \( E \) for the given statistics and confidence level, we can use the formula for the margin of error for a proportion:

\[
E = Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \( Z_{\alpha/2} \) is the z-score corresponding to the confidence level (for 95% confidence, \( Z_{\alpha/2} = 1.96 \)),
- \( \hat{p} \) is the sample proportion,
- \( n \) is the sample size.

### Step 1: Calculate the sample proportion
The sample proportion \( \hat{p} \) is calculated as:

\[
\hat{p} = \frac{x}{n} = \frac{1872}{2888} \approx 0.648
\]

### Step 2: Apply the formula for the margin of error
Now, plug the values into the formula for the margin of error:

\[
E = 1.96 \cdot \sqrt{\frac{0.648 \cdot (1 - 0.648)}{2888}}
\]

Let me calculate that.The margin of error \( E \) is approximately \( 0.0174 \).

### Summary:
For a 95% confidence level, the margin of error is about 0.0174 (or 1.74%).

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How does the margin of error change with different confidence levels?
2. What is the relationship between sample size and the margin of error?
3. How do we interpret the confidence interval for the population proportion?
4. What happens to the margin of error if the sample proportion is closer to 0.5?
5. How is the z-score for different confidence levels determined?

**Tip**: Increasing the sample size decreases the margin of error, making your estimate more precise.

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. 95% confidence; n = 2888, x = 1872

Learn how to calculate the margin of error for a population proportion with a 95% confidence level, using sample data where n = 2888 and x = 1872. This step-by-step guide covers statistical concepts like proportions, confidence intervals, and the margin of error. Suitable for high school and college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation