The confidence interval is given as (1.61, 1.97).

Step 1: Find the Sample Mean ($\bar{x}$)

The sample mean is the midpoint of the confidence interval:

$\bar{x} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

$\bar{x} = \frac{1.61 + 1.97}{2} = \frac{3.58}{2} = 1.79$

Step 2: Find the Margin of Error (E)

The margin of error is the distance from the sample mean to either endpoint of the confidence interval:

$E = \text{Upper Bound} - \bar{x}$

$E = 1.97 - 1.79 = 0.18$

Final Answer:

• Sample Mean: 1.79
• Margin of Error: 0.18

Would you like a detailed explanation on how the confidence interval is related to sampling distributions?

Here are some related questions you might find useful:

1. How do you calculate a confidence interval given a sample mean and standard deviation?
2. What is the significance of the margin of error in statistics?
3. How does the confidence level affect the width of the confidence interval?
4. What is the relationship between the sample size and the margin of error?
5. How can you interpret a confidence interval in real-world scenarios?

Tip: The wider the confidence interval, the more uncertainty there is in the estimate. Reducing the margin of error requires increasing the sample size.