Questions: Los Angeles workers have an average commute of 29 minutes. Suppose the LA commute time is normally distributed with a standard deviation of 15 minutes. Let X represent the commute time for a randomly selected LA worker. Round all answers to 4 decimal places where possible. a. What is the distribution of X ? X-NC (29, 15) b. Find the probability that a randomly selected LA worker has a commute that is longer than 37 minutes. c. Find the 85th percentile for the commute time of LA workers. minutes

Solution Steps

The random variable \( X \), representing the commute time for a randomly selected LA worker, follows a normal distribution. Specifically, we have:

\[ X \sim N(29, 15) \]

To find the probability that a randomly selected LA worker has a commute time longer than 37 minutes, we calculate:

\[ P(X > 37) = 1 - P(X \leq 37) \]

First, we find the Z-score for \( X = 37 \):

\[ Z = \frac{X - \mu}{\sigma} = \frac{37 - 29}{15} = \frac{8}{15} \approx 0.5333 \]

Using the cumulative distribution function \( \Phi \):

\[ P(X \leq 37) = \Phi(0.5333) \approx 0.2969 \]

Thus, the probability is:

\[ P(X > 37) = 1 - 0.2969 = 0.7031 \]

To find the 85th percentile, we need to determine the value \( x \) such that:

\[ P(X \leq x) = 0.85 \]

Using the inverse of the cumulative distribution function, we find:

\[ x = \Phi^{-1}(0.85) \approx 44.5465 \]

Final Answer