Question

at one university, the mean distance commuted to campus by students is 16.0 miles, with a standard deviation of 5.1 miles. suppose that the commute distances are normally distributed. complete the following statements.
(a) approximately of the students have commute distances between 5.8 miles and 26.2 miles.
(b) approximately 99.7% of the students have commute distances between miles and miles.

Explanation:

Step1: Calculate the number of standard deviations for part (a)

For \(x_1 = 5.8\), \(z_1=\frac{5.8 - 16.0}{5.1}=\frac{- 10.2}{5.1}=- 2\)
For \(x_2 = 26.2\), \(z_2=\frac{26.2 - 16.0}{5.1}=\frac{10.2}{5.1}=2\)
By the empirical rule (68 - 95 - 99.7 rule), approximately 95% of the data lies within \(z=-2\) and \(z = 2\) in a normal distribution.

Step2: Calculate the values for part (b)

By the empirical rule, approximately 99.7% of the data lies within \(z=-3\) and \(z = 3\)
For \(z=-3\), \(x_1=\mu+z\sigma=16.0+(-3)\times5.1=16.0 - 15.3 = 0.7\)
For \(z = 3\), \(x_2=\mu+z\sigma=16.0+3\times5.1=16.0 + 15.3=31.3\)

Answer:

(a) Approximately 95% of the students have commute distances between 5.8 miles and 26.2 miles.
(b) Approximately 99.7% of the students have commute distances between 0.7 miles and 31.3 miles.