The average age of a vehicle registered in Canada is about 96 months and the standard deviation for the population is 15..

If a random sample of 43 vehicles is selected, find the probability (as percent) that the mean of their age is between 99 and 102 months.

The probability is:
Number
%

(Round your answer to the nearest whole percent)

(Omit the percent sign)

To find the probability that the mean age of a sample of 43 vehicles is between 99 and 102 months, we can use the **Central Limit Theorem**. Here’s the step-by-step process:

1. **Given Data:**
   - Population mean (\(\mu\)) = 96 months
   - Population standard deviation (\(\sigma\)) = 15 months
   - Sample size (\(n\)) = 43
   - Sample mean range: 99 to 102 months

2. **Calculate the standard error of the mean (SEM):**
   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{43}}
   \]

3. **Calculate the z-scores for 99 and 102 months:**
   - For 99 months:
     \[
     z_{99} = \frac{99 - 96}{\text{SEM}}
     \]
   - For 102 months:
     \[
     z_{102} = \frac{102 - 96}{\text{SEM}}
     \]

4. **Find the probability corresponding to these z-scores using the standard normal distribution.**

5. **Subtract the lower probability from the higher probability to get the final answer.**

Let's compute this.The probability that the mean age of the selected sample of 43 vehicles is between 99 and 102 months is approximately **9%**.

Would you like any further details or have any questions?

Here are some related questions you might explore:
1. How would the probability change if the sample size were larger?
2. What is the significance of the Central Limit Theorem in this context?
3. How does the standard deviation of the population affect the probability?
4. What is the z-score and how is it interpreted?
5. How would the calculation change if the population distribution were not normal?
6. What if the sample mean range was different, say between 100 and 105 months?
7. How do confidence intervals relate to this probability?
8. What would happen if the sample size were very small, like 5 or 10?

**Tip:** Understanding the concept of the standard error is crucial when dealing with sample means. It provides insight into how much variability to expect in the sample mean relative to the population mean.

Calculate the probability that the mean age of a sample of 43 vehicles in Canada falls between 99 and 102 months. Understand the application of the Central Limit Theorem and normal distribution in statistical analysis. Population mean: 96 months, standard deviation: 15 months.

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