12.	A factory produces washing machines, and the lifetime of a washing machine follows a normal distribution with a mean of 5 years and a standard deviation of 1 year. The factory now wants to set the warranty period of the washing machine such that the probability that a washing machine is still in function after the warranty period is 0.96. Find the warranty period to be set.

To find the warranty period for the washing machines, we need to determine the time period such that the probability a washing machine is still functioning after that time is 0.96. This means that we need to find the time \( T \) such that the probability of a washing machine failing before time \( T \) is 0.04 (since \( 1 - 0.96 = 0.04 \)).

The lifetime of the washing machines follows a normal distribution with:

- Mean \( \mu = 5 \) years
- Standard deviation \( \sigma = 1 \) year

We are looking for the time \( T \) such that:
\[
P(X \leq T) = 0.04
\]
Where \( X \) is the lifetime of the washing machine.

This is a problem of finding the z-score corresponding to a cumulative probability of 0.04 under the normal distribution. Once we find the z-score, we can convert it back to the actual time \( T \) using the formula:
\[
T = \mu + z \cdot \sigma
\]

### Step-by-step solution:

1. **Find the z-score for the cumulative probability of 0.04.**
   Using a standard normal distribution table or calculator, the z-score corresponding to a cumulative probability of 0.04 is approximately \( z = -1.75 \).

2. **Convert the z-score to the actual time \( T \):**
   \[
   T = \mu + z \cdot \sigma = 5 + (-1.75) \cdot 1 = 5 - 1.75 = 3.25 \text{ years}
   \]

Thus, the factory should set the warranty period to approximately **3.25 years** in order to ensure that the probability of a washing machine still functioning after this period is 0.96.

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Do you need any further details or clarifications?

Here are 5 related questions to expand your understanding:
1. How would the warranty period change if the standard deviation was 2 years instead of 1 year?
2. What would the warranty period be if the factory wanted the probability of a machine still functioning after the warranty period to be 0.90 instead of 0.96?
3. Can we use a similar method to calculate the warranty period for a different distribution, like an exponential distribution?
4. How does the z-score change if we want a warranty period that covers a higher percentage of machines, such as 99%?
5. What is the probability that a washing machine will last exactly 5 years?

**Tip:** In problems involving normally distributed data, using z-scores to find probabilities and critical values is key when shifting between cumulative probabilities and actual values.

A factory produces washing machines, and the lifetime of a washing machine follows a normal distribution with a mean of 5 years and a standard deviation of 1 year. The factory now wants to set the warranty period of the washing machine such that the probability that a washing machine is still in function after the warranty period is 0.96. Find the warranty period to be set.

Discover how to calculate the warranty period for washing machines when the lifetime follows a normal distribution with a mean of 5 years and a standard deviation of 1 year. The factory seeks a 0.96 probability that machines will last beyond the warranty period. Learn how to use z-scores and the normal distribution formula to solve this problem, ideal for Grades 11-12.

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