A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11.6 years, and standard deviation of 1.3 years.

If you randomly purchase one item, what is the probability it will last longer than 14 years?



Round answer to 4 decimal places.


To find the probability that a randomly selected item will last **longer than 14 years**, we will use the properties of the **normal distribution** with the following given parameters:

- Mean (\(\mu\)) = 11.6 years
- Standard Deviation (\(\sigma\)) = 1.3 years

We want to find:

\[
P(X > 14)
\]

### Step 1: Calculate the Z-score
The Z-score formula is:

\[
Z = \frac{X - \mu}{\sigma}
\]

Substitute the values:

\[
Z = \frac{14 - 11.6}{1.3} = \frac{2.4}{1.3} \approx 1.8462
\]

### Step 2: Find the Probability Using the Z-score
Now, we need to find the probability that \( Z \) is greater than 1.8462, which is represented as \( P(Z > 1.8462) \).

Using a **Z-table** or a **calculator**, we find the **cumulative probability** for \( Z \leq 1.8462 \):

\[
P(Z \leq 1.8462) \approx 0.9678
\]

The probability of \( Z > 1.8462 \) is:

\[
P(Z > 1.8462) = 1 - 0.9678 = 0.0322
\]

### Final Answer:
The probability that a randomly purchased item will last longer than **14 years** is approximately:

\[
\boxed{0.0322}
\]

(rounded to 4 decimal places).

---

Would you like more details or have any questions? Here are 5 related questions you might find interesting:

1. What is the probability that the item will last less than 10 years?
2. How long will an item last if it is at the 90th percentile of lifespans?
3. What is the mean lifespan for items if the standard deviation changes to 2 years?
4. How would the probability change if you wanted to find the likelihood of an item lasting between 11.6 and 14 years?
5. What Z-score corresponds to a lifespan of 12 years?

**Tip:** For probabilities involving normally distributed data, finding the Z-score helps transform the problem into one that uses the standard normal table or function.

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 11.6 years, and standard deviation of 1.3 years. If you randomly purchase one item, what is the probability it will last longer than 14 years?

Learn how to calculate the probability that an item will last longer than 14 years using the properties of normal distribution. With a mean lifespan of 11.6 years and a standard deviation of 1.3 years, this step-by-step solution involves finding the Z-score and using the cumulative probability table. Suitable for students in Grades 10-12.

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