Question: If commuting times are normally distributed, what is the probability that a commuter in Richmond…

If commuting times are normally distributed, what is the probability that a commuter in Richmond will have a commuting time of less than 19 minutes?

0.1401

Numeric 5 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 the average of 8 commuters in one Richmond neighborhood would have a commuting time of less than 19 minutes?

0.11

Solution

The problem involves finding the probability that the average commuting time for 8 commuters is less than 20 minutes, given that the average one-way commuting time is 21.9 minutes with a standard deviation of 6.4 minutes. 1. Identify the given values: - Mean (\(\mu\)) = 21.9 minutes - Standard deviation (\(\sigma\)) = 6.4 minutes - Sample size (\(n\)) = 8 - Sample mean (\(\bar{x}\)) = 20 minutes 2. Calculate the standard error of the mean (SEM): \[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{6.4}{\sqrt{8}} \] 3. Calculate the z-score for the sample mean: \[ z = \frac{\bar{x} - \mu}{\text{SEM}} = \frac{20 - 21.9}{\frac{6.4}{\sqrt{8}}} \] 4. Find the probability corresponding to the calculated z-score using the standard normal distribution table. This gives the probability of the average being less than 20 minutes.