he shape of the distribution of the time required to get an oil change at a 20​-minute ​oil-change facility is skewed right.​ However, records indicate that the mean time is 21.3 minutes​, and the standard deviation is 3.8 minutes.What is the probability that a random sample of nequals45 oil changes results in a sample mean time less than 20 ​minutes?
The probability is approximately

enter your response here.
​(Round to four decimal places as​ needed.)

To solve this problem, we need to find the probability that the sample mean time for 45 oil changes is less than 20 minutes, given that the population mean is 21.3 minutes and the standard deviation is 3.8 minutes. We can approach this problem using the Central Limit Theorem (CLT).

### Steps:
1. **Determine the population mean and standard deviation:**
   - Mean (\(\mu\)) = 21.3 minutes
   - Standard deviation (\(\sigma\)) = 3.8 minutes

2. **Determine the sample size:**
   - \(n = 45\)

3. **Calculate the standard error (SE) of the sample mean:**
   \[
   \text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{3.8}{\sqrt{45}} \approx \frac{3.8}{6.708} \approx 0.5662
   \]

4. **Find the z-score for the sample mean of 20 minutes:**
   \[
   z = \frac{\text{Sample Mean} - \mu}{\text{SE}} = \frac{20 - 21.3}{0.5662} \approx \frac{-1.3}{0.5662} \approx -2.296
   \]

5. **Look up the z-score in the standard normal distribution table** to find the probability associated with that z-score.

   - A z-score of -2.296 corresponds to a cumulative probability of approximately 0.0108.

### Conclusion:
The probability that a random sample of 45 oil changes results in a sample mean time less than 20 minutes is approximately **0.0108**.

Let me know if you'd like more details or have any questions! 

Here are 8 related questions:
1. How does the Central Limit Theorem apply to sample means?
2. What happens to the standard error as the sample size increases?
3. What does a right-skewed distribution imply about the median relative to the mean?
4. How would the probability change if the sample size were larger?
5. What is the importance of knowing the shape of the distribution?
6. How would you interpret a z-score of -2.296 in a real-world context?
7. What are the implications of a small probability in this context?
8. How would the result differ if the standard deviation were larger?

**Tip:** The Central Limit Theorem allows us to use the normal distribution to approximate probabilities for the sample mean, even if the underlying population distribution is not normal, as long as the sample size is sufficiently large.

Explore the probability that a sample mean time for 45 oil changes is less than 20 minutes. Given a population mean of 21.3 minutes and a standard deviation of 3.8 minutes, learn how the Central Limit Theorem is applied to calculate this probability. Get step-by-step solutions and understand the statistical concepts involved.

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