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.