Suppose you are told that a 95% confidence interval for the average price of a gallon of regular gasoline in your state is from $4.06 to $4.56. Use the fact that the confidence interval for the mean is in the form *x* − *E* to *x* + *E* to compute the sample mean and the maximal margin of error *E*.

What is the sample mean (in dollars)?

$ 

What is the maximal margin of error (in dollars)?

$ 

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?

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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.

Learn how to calculate the sample mean and maximal margin of error in a confidence interval for the average price of gasoline. Explore step-by-step calculations and understand the significance of confidence intervals in statistical analysis.

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