The combined city/highway fuel economy of a 2023 Lexus UX 250 Hybrid is a normally distributed random variable with a mean of μ = 26.75 mpg and a standard deviation of σ = 3.50 mpg. Within what interval would you expect the sample mean to fall, with 99 percent probability?

Note: Round your answers to 4 decimal places.

To find the interval within which the sample mean is expected to fall with 99% probability, we can use the concept of the confidence interval for the mean of a normally distributed population.

Given dataTo find the interval within which the sample mean is expected to fall with 99% probability, we can use the concept of the confidence interval for the mean of a normally distributed population.

Given 
- Mean (μ) = 26.75 mpg
- Standard deviation (σ) = 3.50 mpg
- Confidence level = 99%

For a 99% confidence level, the corresponding z-score (z) is approximately 2.576. This value is obtained from the standard normal distribution table.

The formula for the confidence interval is:

\[
\text{Confidence Interval} = \mu \pm z \times \frac{\sigma}{\sqrt{n}}
\]

Since the problem doesn't specify the sample size \( n \), we can assume that \( n \) is large enough so that the sample mean distribution follows a normal distribution, but we'll leave the confidence interval in terms of \( n \).

Now, let's calculate the confidence interval:

\[
\text{Lower limit} = \mu - z \times \frac{\sigma}{\sqrt{n}} 
\]
\[
\text{Upper limit} = \mu + z \times \frac{\sigma}{\sqrt{n}} 
\]

Plugging in the values:

\[
\text{Lower limit} = 26.75 - 2.576 \times \frac{3.50}{\sqrt{n}} 
\]
\[
\text{Upper limit} = 26.75 + 2.576 \times \frac{3.50}{\sqrt{n}} 
\]

Now, express the interval as:

\[
(26.75 - 2.576 \times \frac{3.50}{\sqrt{n}}, 26.75 + 2.576 \times \frac{3.50}{\sqrt{n}})
\]

This is the interval within which you would expect the sample mean to fall with 99% probability, given a sample size \( n \).

If you have the sample size, you can plug it into the formula to get specific values.

Would you like to explore further details or ask additional questions? Here are some suggestions:

1. How does the sample size \( n \) affect the width of the confidence interval?
2. What if the confidence level was 95% instead of 99%?
3. Can you explain how the z-score is determined for a given confidence level?
4. How would the interval change if the standard deviation was larger?
5. How do we interpret the confidence interval in practical terms?
6. How does the shape of the normal distribution influence the confidence interval?
7. What assumptions are made in constructing this confidence interval?
8. How would the interval change if we considered a different car model with a different mean and standard deviation?

**Tip:** Always check if the sample size is given, as it directly impacts the calculation of the confidence interval.

Learn how to calculate the confidence interval for the mean combined city/highway fuel economy of a 2023 Lexus UX 250 Hybrid, based on a normally distributed random variable. Given a mean of 26.75 mpg and a standard deviation of 3.50 mpg, find the interval within which the sample mean is expected to fall with 99% probability.

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