the combined city/highway fuel economy of a 2023 lexus UX 250 Hybrid is a normally distributed random variables with a mean of u=19.75 mpg and a standard deviation of o=1.75 mpg. within what intervals would you expect the sample mean to fall, with 98%  probability?

To find the interval within which we expect the sample mean of the fuel economy to fall with 98% probability, we use the concept of confidence intervals.

Given:
- Mean (\(\mu\)) = 19.75 mpg
- Standard deviation (\(\sigma\)) = 1.75 mpg
- Confidence level = 98%

The formula for the confidence interval for the sample mean is:

\[
\text{Confidence Interval} = \mu \pm Z_{\alpha/2} \times \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution for a 98% confidence level.
- \( \sigma \) is the standard deviation.
- \( n \) is the sample size.

Since the sample size \( n \) is not provided, we will assume \( n \) is sufficiently large, which allows us to focus on the margin of error without scaling for \( n \) (since the question is about the interval of the sample mean itself, not a specific sample size).

For a 98% confidence level, the critical value \( Z_{\alpha/2} \) corresponds to the z-score for 1% in each tail of the normal distribution. From the z-tables:

\[
Z_{\alpha/2} \approx 2.33
\]

The margin of error (E) is calculated as:

\[
E = Z_{\alpha/2} \times \sigma = 2.33 \times 1.75 = 4.0775
\]

Thus, the 98% confidence interval for the sample mean fuel economy is:

\[
\text{Confidence Interval} = 19.75 \pm 4.0775 = (15.6725, 23.8275) \text{ mpg}
\]

This interval means that with 98% probability, the sample mean of the fuel economy is expected to fall within the range of **15.67 mpg** to **23.83 mpg**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the confidence interval change if the confidence level were 95%?
2. What happens to the confidence interval if the sample size increases?
3. How is the width of the confidence interval affected by the standard deviation?
4. How would you calculate the confidence interval if the sample size was given?
5. What is the interpretation of a confidence interval in the context of this problem?

**Tip:** When calculating confidence intervals, always ensure you have the correct critical value (\( Z_{\alpha/2} \)) based on the confidence level.

Learn how to calculate the 98% confidence interval for the combined city/highway fuel economy of a 2023 Lexus UX 250 Hybrid. Understand the statistical concepts involved, including confidence intervals and the normal distribution. Suitable for advanced high school students and beyond.

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