Let's break down the questions based on the image you provided:

a. What is the distribution of $X$?

This part asks for the distribution of $X$, where $X$ represents the commute time for LA workers. The normal distribution is typically written as:

$X \sim N(\mu, \sigma^2)$

where:

• $\mu$ is the mean commute time (29 minutes),
• $\sigma$ is the standard deviation (12 minutes).

Thus, the distribution is:

$X \sim N(29, 12^2)$

b. Find the probability that a randomly selected LA worker has a commute that is longer than 30 minutes.

This is a probability question asking for $P(X > 30)$, where $X \sim N(29, 12^2)$.

1. First, we calculate the Z-score: $Z = \frac{X - \mu}{\sigma} = \frac{30 - 29}{12} = \frac{1}{12} \approx 0.0833$
2. Now, we use the Z-score to find the probability using standard normal distribution tables or a calculator.

$P(Z > 0.0833)$

Using a Z-table or a calculator, we find that: $P(Z > 0.0833) \approx 0.4660$

So, the probability that a worker has a commute longer than 30 minutes is approximately $0.4660$ or 46.60%.

c. Find the 90th percentile for the commute time of LA workers.

The 90th percentile corresponds to a Z-score of approximately 1.2816 (from standard normal tables).

We use the formula for the Z-score: $X = \mu + Z \cdot \sigma$ Substituting the known values: $X = 29 + 1.2816 \cdot 12 \approx 29 + 15.3792 \approx 44.3792$

So, the 90th percentile commute time is approximately 44.3792 minutes.

Would you like more details on any of these steps?

Five related questions:

1. How do we interpret the result of a normal distribution in real-life scenarios?
2. What is the significance of the Z-score in calculating probabilities?
3. How can we use the 68-95-99.7 rule in understanding normal distributions?
4. How do we find the 50th percentile of a normal distribution?
5. What is the difference between a probability and a percentile in statistics?

Tip:

Always remember to check if your distribution is normal before applying Z-scores. Normality is essential for the correct use of Z-tables!