A battery company 12-volt car batteries After many years of product testing the company knows that the average life of its battery is normally disturbuted with a mean of 45 months and a standard deviation of 7 months. If a company guarantees a full refund on any battery that fails within a 36 month period after purchase what percentage of its batteries will the company expect to replace?

To determine the percentage of batteries that will need to be replaced, we calculate the probability that a battery fails within 36 months. Given a normally distributed lifespan, we can use the z-score formula:

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

where:
- \( X = 36 \) months (the failure period),
- \( \mu = 45 \) months (the mean lifespan),
- \( \sigma = 7 \) months (the standard deviation).

1. **Calculate the z-score:**

   \[
   z = \frac{36 - 45}{7} = \frac{-9}{7} \approx -1.29
   \]

2. **Find the probability associated with \( z = -1.29 \):**

   Using standard normal distribution tables, the probability \( P(Z \leq -1.29) \approx 0.0985 \).

This value, 0.0985, indicates that approximately **9.85%** of batteries will fail within 36 months, meaning the company would expect to replace about 9.85% of its batteries under the guarantee policy.

Would you like a more detailed explanation on z-scores and normal distribution?

Here are some related questions:
1. How would the replacement percentage change if the guarantee was extended to 40 months?
2. What would happen to the replacement percentage if the standard deviation increased to 9 months?
3. How is the replacement percentage impacted if the average lifespan increases?
4. How does altering the guarantee period affect company costs?
5. What is the likelihood of a battery lasting more than 50 months?

**Tip:** Familiarizing yourself with z-score tables can simplify calculating probabilities for normally distributed data.

A battery company 12-volt car batteries. After many years of product testing, the company knows that the average life of its battery is normally distributed with a mean of 45 months and a standard deviation of 7 months. If a company guarantees a full refund on any battery that fails within a 36-month period after purchase, what percentage of its batteries will the company expect to replace?

Learn how to calculate the percentage of 12-volt car batteries that a company expects to replace under a 36-month guarantee. Given a normal distribution with a mean lifespan of 45 months and a standard deviation of 7 months, this solution explores the z-score calculation and probability associated with battery lifespan.

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