Question: Numeric 4 points The average one-way commuting time in Richmond, Virginia is 21.9 minutes with a…

Numeric 4 points

The average one-way commuting time in Richmond, Virginia is 21.9 minutes with a standard deviation of 6.4 minutes.

If commuting times are normally distributed, what is the probability that a commuter in Richmond will have a commuting time of less than 15 minutes?

0.1401

Solution

To solve this problem, we need to find the probability that a commuter in Richmond will have a commuting time of less than 15 minutes, given that the times are normally distributed with a mean of 21.9 minutes and a standard deviation of 6.4 minutes. Step 1: Calculate the z-score for 15 minutes using the formula: \[ z = \frac{x - \mu}{\sigma} \] where \( x = 15 \), \( \mu = 21.9 \), \( \sigma = 6.4 \). \[ z = \frac{15 - 21.9}{6.4} \] Step 2: Compute the z-score: \[ z = \frac{-6.9}{6.4} \approx -1.0781 \] Step 3: Look up the z-score in a standard normal distribution table, or use a calculator to find the probability. The z-score of \(-1.0781\) corresponds to a probability of approximately \(0.1401\). Thus, the probability that a commuter will have a commuting time of less than 15 minutes is \(0.1401\).