https://file.aiquickdraw.com/mathbot/file/1731947296852fg7yyzhx.jpg

The given confidence interval is \( (0.655, 0.683) \). To solve for the **margin of error (E)** and the **sample proportion (\( \hat{p} \))**, follow these steps:

### Step 1: Calculate the Sample Proportion (\( \hat{p} \)):
The sample proportion is the midpoint of the confidence interval. Use the formula:
\[
\hat{p} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}
\]
\[
\hat{p} = \frac{0.655 + 0.683}{2} = 0.669
\]

### Step 2: Calculate the Margin of Error (E):
The margin of error is the distance from the sample proportion to either end of the interval. Use the formula:
\[
E = \text{Upper Limit} - \hat{p}
\]
\[
E = 0.683 - 0.669 = 0.014
\]

### Final Answer:
- **Margin of Error (E):** \( 0.014 \)
- **Sample Proportion (\( \hat{p} \)):** \( 0.669 \)

Would you like me to explain any part of this calculation in more detail or clarify the formulas?

### Related Questions:
1. What is the significance of the margin of error in a confidence interval?
2. How can a confidence interval be used to make predictions about a population proportion?
3. What happens to the margin of error when the confidence level increases?
4. How is sample size related to the width of a confidence interval?
5. What are the assumptions required for constructing a confidence interval for a population proportion?

### Tip:
Always ensure that your confidence interval aligns with the population parameter and meets the conditions for approximation (e.g., \( np \geq 5 \) and \( n(1-p) \geq 5 \)).

Use the given confidence interval (0.655, 0.683) to find the margin of error and the sample proportion.

Learn how to calculate the margin of error and sample proportion from the confidence interval (0.655, 0.683). This detailed explanation includes the necessary formulas and statistical concepts, ideal for students in Grades 10-12 or introductory college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation