A manufacturer knows that their items have a normally distributed lifespan, with a mean of 6.8 years, and standard deviation of 1.4 years.

If you randomly purchase one item, what is the probability it will last longer than 5 years?

Round answer to three decimal places

To solve this problem, we need to find the probability that a randomly selected item will last longer than 5 years. This is a normal distribution problem, so we will use the **Z-score formula**:

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

Where:
- \( X = 5 \) (the value we are interested in),
- \( \mu = 6.8 \) (the mean lifespan),
- \( \sigma = 1.4 \) (the standard deviation).

### Step 1: Calculate the Z-score

\[
Z = \frac{5 - 6.8}{1.4} = \frac{-1.8}{1.4} \approx -1.286
\]

### Step 2: Find the probability corresponding to the Z-score

We now need to look up the probability associated with a Z-score of -1.286 in a standard normal distribution table, or use a calculator. The Z-table gives the cumulative probability for values less than a given Z-score.

The cumulative probability for \( Z = -1.286 \) is approximately 0.0996. This means the probability that the item lasts **less than 5 years** is 0.0996.

### Step 3: Find the probability of lasting longer than 5 years

To find the probability that the item lasts **longer than 5 years**, we subtract the cumulative probability from 1:

\[
P(X > 5) = 1 - 0.0996 = 0.9004
\]

### Final Answer:

The probability that a randomly purchased item will last longer than 5 years is approximately \( \mathbf{0.900} \).

---

Would you like more details or have any questions?

Here are 5 related questions for deeper exploration:
1. What is the probability that an item will last between 5 and 7 years?
2. If the manufacturer guarantees items for 4 years, what percentage of items will last beyond the warranty?
3. What lifespan corresponds to the 95th percentile?
4. How does the probability change if the standard deviation increases to 2 years?
5. What is the Z-score for an item lasting exactly 8 years?

**Tip**: For normally distributed data, the empirical rule states that approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 6.8 years, and standard deviation of 1.4 years. If you randomly purchase one item, what is the probability it will last longer than 5 years? Round answer to three decimal places.

Find the probability that an item lasts longer than 5 years when its lifespan is normally distributed with a mean of 6.8 years and standard deviation of 1.4 years. Learn how to use Z-scores and the normal distribution curve to solve real-world probability problems.

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