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To express the confidence interval \(0.555 < p < 0.777\) in the form \(\hat{p} \pm E\), we need to identify:

1. The point estimate \(\hat{p}\), which is the midpoint of the interval.
2. The margin of error \(E\), which is the distance from the midpoint to either endpoint.

### Step-by-Step Solution:

1. **Calculate \(\hat{p}\) (the midpoint)**:
   \[
   \hat{p} = \frac{0.555 + 0.777}{2} = \frac{1.332}{2} = 0.666
   \]

2. **Calculate \(E\) (the margin of error)**:
   \[
   E = 0.777 - 0.666 = 0.111
   \]

### Answer in the form \(\hat{p} \pm E\):
\[
\hat{p} \pm E = 0.666 \pm 0.111
\]

Would you like more details or have any questions?

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1. How do you calculate the midpoint of any interval?
2. What does the margin of error represent in a confidence interval?
3. How can we interpret a confidence interval in practical terms?
4. What is the formula for calculating a confidence interval for a proportion?
5. How does sample size affect the margin of error in a confidence interval?

**Tip:** The midpoint of an interval provides the best estimate of the parameter based on the given range.

Express the confidence interval 0.555 < p < 0.777 in the form p̂ ± E.

Learn how to express the confidence interval 0.555 < p < 0.777 in the form p̂ ± E. This solution involves finding the point estimate (p̂) as the midpoint and calculating the margin of error (E). Suitable for students in Grades 10-12.

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