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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!

Los Angeles workers have an average commute of 29 minutes. Suppose the LA commute time is normally distributed with a standard deviation of 12 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? b. Find the probability that a randomly selected LA worker has a commute that is longer than 30 minutes. c. Find the 90th percentile for the commute time of LA workers.

This problem covers normal distribution concepts, including Z-scores and percentiles, in the context of LA workers' commute time. Learn how to calculate probabilities and percentiles for normally distributed data with a mean of 29 minutes and a standard deviation of 12 minutes.

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