The cost of unleaded gasoline in the Bay Area once followed a normal distribution with a mean of $4.74 and a standard deviation of $0.16. Twenty-eight gas stations from the Bay area are randomly chosen. We are interested in the average cost of gasoline for the 28 gas stations. Find the exact probability that the average price for 28 gas stations is less than $4.69.

To find the exact probability that the average price for 28 gas stations is less than $4.69, given that the cost of unleaded gasoline follows a normal distribution with a mean of $4.74 and a standard deviation of $0.16, we can use the **Central Limit Theorem (CLT)**.

### Steps to Solve the Problem:

1. **Identify the parameters:**
   - Population mean ($\mu$) = $4.74
   - Population standard deviation ($\sigma$) = $0.16
   - Sample size ($n$) = 28
   - Sample mean we are interested in ($\bar{x}$) = $4.69

2. **Calculate the standard error of the mean (SEM):**
   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{0.16}{\sqrt{28}}
   \]

3. **Calculate the Z-score:**
   The Z-score for the sample mean is given by:
   \[
   Z = \frac{\bar{x} - \mu}{\text{SEM}} = \frac{4.69 - 4.74}{\text{SEM}}
   \]

4. **Find the probability corresponding to the Z-score:**
   Use the Z-score to find the probability from the standard normal distribution table.

Let's compute these values.The Z-score for the sample mean of $4.69 is approximately $-1.65$. The probability that the average price for 28 gas stations is less than $4.69 is approximately **0.0491**. 

This means there's about a 4.91% chance that the average price of gasoline for the 28 gas stations is less than $4.69.

Would you like more details on the calculations, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. What is the probability that the average price is more than $4.80?
2. How would the probability change if the sample size were 50 gas stations instead of 28?
3. What would be the probability if the standard deviation was $0.20 instead of $0.16?
4. How does the Central Limit Theorem apply when dealing with small sample sizes?
5. Can we approximate the probability using a normal distribution if the sample size was very small, say 5?

**Tip:** When working with normal distributions, always check if the sample size is large enough to apply the Central Limit Theorem for averages.

Calculate the exact probability that the average price of gasoline for 28 gas stations in the Bay Area is less than $4.69 using normal distribution principles. Understand the use of the Central Limit Theorem and Z-score in statistical calculations.

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