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Let's solve each part of the question step-by-step.

### Given Information
- Confidence interval for the probability of an event: \( 0.591 \) to \( 0.749 \).
- Number of trials (\( n \)) = 25.

### Part (a): Margin of Error
The margin of error (ME) is half the width of the confidence interval.

1. Find the width of the interval:
   \[
   0.749 - 0.591 = 0.158
   \]

2. Divide by 2 to find the margin of error:
   \[
   \text{Margin of Error} = \frac{0.158}{2} = 0.079
   \]

So, the margin of error is **0.079**.

### Part (b): Success Rate within the Sample
The success rate (sample proportion, \(\hat{p}\)) is the midpoint of the confidence interval.

1. Find the midpoint:
   \[
   \hat{p} = \frac{0.591 + 0.749}{2} = \frac{1.340}{2} = 0.67
   \]

Therefore, the success rate is **0.67**.

### Part (c): Confidence Level Used
To find the confidence level, we can use the margin of error formula for a binomial proportion:

\[
\text{Margin of Error} = z \cdot \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}
\]

where:
- \( z \) is the z-score corresponding to the confidence level,
- \(\hat{p} = 0.67\),
- \( n = 25 \),
- Margin of Error = 0.079 (from part a).

Rearrange to solve for \( z \):

\[
z = \frac{\text{Margin of Error}}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}}
\]

Substitute the values:

\[
z = \frac{0.079}{\sqrt{\frac{0.67 \times (1 - 0.67)}{25}}}
\]

Calculating the denominator:

\[
0.67 \times (1 - 0.67) = 0.67 \times 0.33 = 0.2211
\]

\[
\frac{0.2211}{25} = 0.008844
\]

\[
\sqrt{0.008844} = 0.09405
\]

Now, calculate \( z \):

\[
z = \frac{0.079}{0.09405} \approx 0.84
\]

A \( z \)-score of approximately \( 0.84 \) corresponds to a confidence level close to **80%**.

---

### Summary of Answers
1. **Margin of Error**: 0.079
2. **Success Rate**: 0.67
3. **Confidence Level**: Approximately 80%

Would you like further details on any of these parts or have additional questions?

---

### Related Questions for Deeper Understanding
1. How do we calculate confidence intervals for different sample sizes?
2. What would happen to the confidence interval if the sample size increased?
3. How does changing the confidence level affect the margin of error?
4. Why is the midpoint used to calculate the success rate?
5. How do we interpret confidence levels in real-world contexts?

### Tip
In binomial confidence intervals, a larger sample size generally narrows the interval, increasing the precision of the estimate.

Without further clarifying, a news report provided a binomial confidence interval to describe the probability of an event as between 0.591 and 0.749 based on an experiment with 25 trials. a. What was the margin of error in their interval? b. What was the success rate within the sample? c. What confidence level did they use?

Learn how to determine the margin of error, success rate, and confidence level for a binomial confidence interval. This example uses a confidence interval of 0.591 to 0.749 based on 25 trials, providing a detailed solution using statistical concepts.

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