Question: Numeric 5 points The average one-way commuting time in Richmond, Virginia is 21.9 minutes with a…

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 15 minutes?

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Solution

The problem asks for the probability that the average commuting time of 8 commuters is less than 15 minutes, given that commuting times are normally distributed. 1. Identify the given parameters: - Mean (\(\mu\)) = 21.9 minutes - Standard deviation (\(\sigma\)) = 6.4 minutes - Sample size (\(n\)) = 8 2. Find the standard error of the mean (\(SE\)): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{6.4}{\sqrt{8}} \] 3. Calculate the standard error: \[ SE = \frac{6.4}{2.828} \approx 2.26 \] 4. Calculate the z-score for the average time of 15 minutes using the formula: \[ z = \frac{\bar{x} - \mu}{SE} = \frac{15 - 21.9}{2.26} \] 5. Calculate the z-score: \[ z = \frac{-6.9}{2.26} \approx -3.05 \] 6. Look up the z-score in a standard normal distribution table or use a calculator to find the probability: - \(P(Z < -3.05)\) 7. The probability for \(z = -3.05\) is approximately 0.0011. Thus, the probability that the average commute time for 8 commuters is less than 15 minutes is approximately 0.0011, or 0.11%.