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We are given the following information in this problem:

- The lifespan of items is normally distributed with a **mean** (\(\mu\)) of 10.7 years and a **standard deviation** (\(\sigma\)) of 1.3 years.
- You purchase 6 items, and the goal is to find the probability that their **mean life** will be longer than 11 years.

### Step-by-Step Approach:

1. **Find the standard error of the mean (SEM)** for the sample:
   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{1.3}{\sqrt{6}}
   \]
   where \( n = 6 \) is the number of items purchased.

2. **Find the z-score** for the sample mean of 11 years:
   \[
   z = \frac{\text{sample mean} - \mu}{\text{SEM}} = \frac{11 - 10.7}{\text{SEM}}
   \]
   
3. **Use the z-score to find the probability** from the standard normal distribution. This will give the probability that the sample mean is less than 11 years, and subtracting that value from 1 will give the probability that the sample mean is more than 11 years.

Let me calculate the exact probability for you.The z-score for a sample mean of 11 years is approximately **0.57**. Using this z-score, the probability that the sample mean lifespan will be **longer than 11 years** is approximately **0.286** (or 28.6%).

Would you like any further details or have questions?

Here are 5 related questions for further exploration:
1. How would the probability change if the sample size increased to 10?
2. What if the standard deviation were smaller, say 1.0 years?
3. How can we interpret the z-score in this context?
4. What is the probability that the mean lifespan is less than 10 years?
5. How does the Central Limit Theorem apply to this problem?

**Tip:** When working with the normal distribution, increasing the sample size reduces the standard error, making estimates more precise.

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 10.7 years, and standard deviation of 1.3 years. If you randomly purchase 6 items, what is the probability that their mean life will be longer than 11 years?

Calculate the probability that the mean lifespan of 6 items exceeds 11 years, given a normal distribution with mean 10.7 years and standard deviation 1.3 years. This problem applies statistical concepts such as the Z-score and standard error of the mean.

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