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 $n(1-p) \geq 5$).