To solve this problem, we need to use the information provided about the confidence interval. The confidence interval is given as $4.06 to $4.56, and we know it is in the form $\bar{x} - E$ to $\bar{x} + E$, where:

• $\bar{x}$ is the sample mean.
• $E$ is the maximal margin of error.

Step 1: Calculate the sample mean ($\bar{x}$)

The sample mean $\bar{x}$ is the midpoint of the confidence interval. We calculate it by taking the average of the lower and upper bounds of the interval:

$\bar{x} = \frac{4.06 + 4.56}{2} = \frac{8.62}{2} = 4.31$

So, the sample mean $\bar{x}$ is $4.31.

Step 2: Calculate the maximal margin of error ($E$)

The maximal margin of error $E$ is the difference between the sample mean $\bar{x}$ and either endpoint of the confidence interval. We can calculate it by subtracting the sample mean from the upper bound:

$E = 4.56 - 4.31 = 0.25$

Thus, the maximal margin of error $E$ is $0.25.

Final Answers:

• Sample mean: $4.31
• Maximal margin of error: $0.25

Would you like to explore any details or have any questions about this problem?

---

Here are 8 related questions you might consider:

1. How does the confidence level affect the width of the confidence interval?
2. What happens to the margin of error if the sample size increases?
3. How would a 99% confidence interval for the same data compare in width to the 95% interval?
4. What role does the standard deviation play in determining the margin of error?
5. How can we use this interval to estimate the population mean with a higher or lower confidence level?
6. What is the formula for the margin of error in a confidence interval?
7. How does the confidence interval change if the variability of the data increases?
8. What assumptions are made about the data when calculating a confidence interval?

Tip: The margin of error is directly influenced by the sample size and variability of the data; larger sample sizes typically result in smaller margins of error.