Final answer:
Approximately 95% of students commute between 7.4 and 28.6 miles, as this range is within two standard deviations from the mean, and the distribution is normal.
Step-by-step explanation:
Approximately 95% of the students have commute distances between 7.4 miles and 28.6 miles. This is because these distances are within two standard deviations of the mean, and according to the empirical rule, about 95% of data in a normal distribution falls within two standard deviations of the mean.
The mean commute distance is 18 miles, with a standard deviation of 5.3 miles. The low end of the range (7.4 miles) is approximately two standard deviations below the mean, while the high end of the range (28.6 miles) is approximately two standard deviations above the mean.
To determine the approximate percentage of students with commute distances between 7.4 miles and 28.6 miles, we need to calculate the z-scores for these distances using the formula: (value - mean) / standard deviation.
For 7.4 miles, the z-score is (7.4 - 18) / 5.3 = -2.02. For 28.6 miles, the z-score is (28.6 - 18) / 5.3 = 1.98. We can then use a standard normal distribution table to find the corresponding probabilities.
The area between -2.02 and 1.98 is approximately 0.9564 or 95.64%. Therefore, approximately 95.64% of the students have commute distances between 7.4 miles and 28.6 miles.