A manufacturer knows that their items have a normally distributed lifespan, with a mean of 12.4 years, and standard deviation of 2.1 years.

If 25 items are picked at random, 10% of the time their mean life will be less than how many years?

Round to one decimal place.

To determine the mean lifespan of 25 items such that it is less than a certain number of years 10% of the time, we follow these steps:

1. **Identify the given information:**
   - Population mean (\( \mu \)) = 12.4 years
   - Population standard deviation (\( \sigma \)) = 2.1 years
   - Sample size (\( n \)) = 25

2. **Determine the standard error of the mean (SEM):**
   The standard error of the mean is calculated using the formula:
   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}}
   \]
   Substituting the given values:
   \[
   \text{SEM} = \frac{2.1}{\sqrt{25}} = \frac{2.1}{5} = 0.42
   \]

3. **Find the z-score corresponding to the 10th percentile:**
   For a normal distribution, the z-score corresponding to the 10th percentile (P10) can be found using a z-table or standard normal distribution table. The z-score for the 10th percentile is approximately -1.28.

4. **Calculate the mean life corresponding to the 10th percentile:**
   The formula to find this mean is:
   \[
   \text{mean} = \mu + (z \times \text{SEM})
   \]
   Substituting the values:
   \[
   \text{mean} = 12.4 + (-1.28 \times 0.42) = 12.4 - 0.5376 \approx 11.9
   \]

So, 10% of the time, the mean lifespan of 25 items will be less than approximately 11.9 years.

Would you like more details or have any questions?

Here are 5 questions you can ask next:
1. How do you find the z-score for different percentiles?
2. What is the significance of the standard error in sampling distributions?
3. How would you calculate a confidence interval for the sample mean?
4. What happens to the standard error if the sample size is increased?
5. Can you explain how to use a z-table to find percentiles?

**Tip:** When working with sampling distributions, the mean of the sample means (x-bar) will be the same as the population mean, but the standard deviation (standard error) will be reduced by a factor of the square root of the sample size.

Understand how to calculate the mean lifespan of 25 items with a normal distribution, given a mean of 12.4 years and standard deviation of 2.1 years. Discover how to determine the mean lifespan that is less than a certain number of years 10% of the time. Includes step-by-step solution and key statistical formulas.

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