Question: The average one-way commuting time in Richmond, Virginia is 21.9 minutes with a standard deviation…

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?

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Solution

The problem involves finding the probability of a certain commuting time using the normal distribution. Given: - Mean \((\mu) = 21.9\) minutes - Standard deviation \((\sigma) = 6.4\) minutes - We need to find the probability of commuting time less than 15 minutes. Step 1: Calculate the \(z\)-score for 15 minutes using the formula: \[ z = \frac{X - \mu}{\sigma} \] where \(X = 15\). Step 2: Substitute the given values: \[ z = \frac{15 - 21.9}{6.4} \] \[ z = \frac{-6.9}{6.4} \] \[ z \approx -1.078125 \] Step 3: Use the standard normal distribution table to find the probability corresponding to \(z = -1.078125\). Looking up the \(z\)-score of \(-1.078125\) in the standard normal distribution table gives a probability of approximately \(0.1401\). So, the probability that a commuter in Richmond will have a commuting time of less than 15 minutes is approximately \(0.1401\).