Use the given confidence interval to find the margin of error and the sample proportion.

To determine the **margin of error** and the **sample proportion** from a given confidence interval, you can follow these general steps: ### 1. Understanding the Confidence Interval A confidence interval for a population proportion is typically expressed in the form: \[ (\text{Lower Limit}, \text{Upper Limit}) \] For example, a 95% confidence interval might be: \[ (0.40, 0.60) \] ### 2. Calculating the Sample Proportion (\(\hat{p}\)) The **sample proportion** is the point estimate around which the confidence interval is constructed. It is calculated as the **midpoint** of the confidence interval: \[ \hat{p} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \] **Example:** Given the confidence interval (0.40, 0.60): \[ \hat{p} = \frac{0.40 + 0.60}{2} = \frac{1.00}{2} = 0.50 \] So, the sample proportion \(\hat{p}\) is **0.50**. ### 3. Calculating the Margin of Error (ME) The **margin of error** represents the range above and below the sample proportion and is calculated as **half the width** of the confidence interval: \[ \text{ME} = \frac{\text{Upper Limit} - \text{Lower Limit}}{2} \] **Example:** Using the same confidence interval (0.40, 0.60): \[ \text{ME} = \frac{0.60 - 0.40}{2} = \frac{0.20}{2} = 0.10 \] So, the margin of error is **0.10**. ### 4. Summary - **Sample Proportion (\(\hat{p}\))**: The midpoint of the confidence interval. - **Margin of Error (ME)**: Half the width of the confidence interval. ### 5. Applying to Your Confidence Interval If you provide a specific confidence interval, you can apply the above steps to find the exact values for the margin of error and the sample proportion. --- **Quick Reference:** Given a confidence interval \((L, U)\): \[ \hat{p} = \frac{L + U}{2} \] \[ \text{ME} = \frac{U - L}{2} \]