At one university, the mean distance commuted to campus by students is 16.0 miles, with a standard deviation of 4.2 miles. Suppose that the commute distances are normally distributed. Approximately 95% of the students have commute distances between ___ miles and ___ miles

To find the range that covers approximately 95% of the data in a normally distributed dataset, we use the empirical rule, also known as the 68-95-99.7 rule. This rule states that:

- About 68% of the data falls within one standard deviation of the mean.
- About 95% of the data falls within two standard deviations of the mean.
- About 99.7% of the data falls within three standard deviations of the mean.

Given:
- Mean (\(\mu\)) = 16.0 miles
- Standard deviation (\(\sigma\)) = 4.2 miles

To find the range for approximately 95% of the students, we calculate two standard deviations from the mean:

\[
\text{Lower limit} = \mu - 2\sigma = 16.0 - 2(4.2) = 16.0 - 8.4 = 7.6 \text{ miles}
\]

\[
\text{Upper limit} = \mu + 2\sigma = 16.0 + 2(4.2) = 16.0 + 8.4 = 24.4 \text{ miles}
\]

Thus, approximately 95% of the students have commute distances between **7.6 miles and 24.4 miles**.

Would you like a more detailed explanation, or do you have any questions?

### Related Questions:
1. What is the probability that a student commutes more than 24.4 miles?
2. How would the range change for 68% of the students?
3. What percentage of students commute less than 7.6 miles?
4. How does the standard deviation affect the range of the normal distribution?
5. What if the mean commute distance changed to 18 miles? How would that affect the range?

### Tip:
In a normal distribution, the area under the curve represents the probability of occurrences. Use the empirical rule as a quick approximation for ranges.

At one university, the mean distance commuted to campus by students is 16.0 miles, with a standard deviation of 4.2 miles. Suppose that the commute distances are normally distributed. Approximately 95% of the students have commute distances between ___ miles and ___ miles.

Discover how to apply the empirical rule to a normal distribution problem. In this case, we find that approximately 95% of students commute between 7.6 miles and 24.4 miles, given a mean of 16.0 miles and a standard deviation of 4.2 miles. Perfect for high school students studying statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation