A manufacturer knows that their items have a normally distributed lifespan, with a mean of 6.2 years, and standard deviation of 2 years.

If you randomly purchase one item, what is the probability it will last longer than 8 years? Round answer to three decimal places.

To solve this problem, we can use the properties of the normal distribution. The lifespan of the items is normally distributed with a mean (\(\mu\)) of 6.2 years and a standard deviation (\(\sigma\)) of 2 years. We are asked to find the probability that a randomly chosen item will last longer than 8 years.

### Steps:

1. **Standardize the value**: We first convert the raw value (8 years) to a z-score using the formula:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   where:
   - \(x = 8\) (the value we are interested in),
   - \(\mu = 6.2\) (mean lifespan),
   - \(\sigma = 2\) (standard deviation).

   Substituting the values:
   \[
   z = \frac{8 - 6.2}{2} = \frac{1.8}{2} = 0.9
   \]

2. **Find the probability**: We now need to find the probability that the lifespan is greater than 8 years, which corresponds to finding the probability that the z-score is greater than 0.9.

   Using standard normal distribution tables or a calculator:
   - The cumulative probability for \(z = 0.9\) is approximately 0.8159.
   - Since we want the probability of the item lasting *longer* than 8 years, we need to subtract this from 1:
     \[
     P(X > 8) = 1 - P(Z \leq 0.9) = 1 - 0.8159 = 0.1841
     \]

### Final Answer:
The probability that a randomly selected item will last longer than 8 years is approximately **0.184** (rounded to three decimal places).

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Would you like more details on any part of the solution, or do you have any questions?

Here are five related questions to extend your understanding:

1. What is the probability that the item will last less than 5 years?
2. What is the probability that an item will last between 4 and 7 years?
3. How long will an item last if you want the lifespan to be in the top 5%?
4. What is the probability that an item will last between 6 and 9 years?
5. How does the probability change if the standard deviation increases to 3 years?

**Tip**: The z-score allows you to compare data points from different normal distributions or the same distribution.

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 6.2 years, and standard deviation of 2 years. If you randomly purchase one item, what is the probability it will last longer than 8 years? Round answer to three decimal places.

Discover how to calculate the probability of an item lasting more than 8 years, given a normal distribution with a mean lifespan of 6.2 years and a standard deviation of 2 years. This problem involves the use of z-scores and normal distribution properties, suitable for Grades 10-12.

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