Acrotime is a manufacture of quartz crystal watches. Acrotime Acromine researchers have shown that the watches average life of 27 months before certain electronic companies deteriorate causing the watch to become unreliable. standard deviation of watch life time is 5 months and the disturbution of lifetime is normal. If acrotime guarantees a full refund or any defective watch for two years after purchase what percentage of total production will the company expect to replace? 

To solve this problem, we need to calculate the probability that a watch fails within 24 months (2 years). This involves finding the **cumulative probability** for a normally distributed random variable with the given mean and standard deviation.

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### Step 1: Define the known parameters
- Mean lifetime (\(\mu\)) = 27 months  
- Standard deviation (\(\sigma\)) = 5 months  
- The threshold for failure (refund period) = 24 months  

We are asked to find the **percentage of watches that the company expects to replace**. This corresponds to the **probability** that a watch fails within 24 months.

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### Step 2: Standardize the 24-month threshold using a Z-score
The Z-score is used to standardize the normal distribution to a standard normal distribution (mean = 0, standard deviation = 1). The formula for the Z-score is:

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

Where:  
- \(X = 24\) (the threshold for failure)  
- \(\mu = 27\)  
- \(\sigma = 5\)  

\[
Z = \frac{24 - 27}{5} = \frac{-3}{5} = -0.6
\]

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### Step 3: Find the cumulative probability for \(Z = -0.6\)
Using a **standard normal distribution table**, the cumulative probability corresponding to a Z-score of \(-0.6\) is:

\[
P(Z \leq -0.6) \approx 0.2743
\]

This means that approximately 27.43% of the watches are expected to fail within the first 24 months.

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### Step 4: Final Answer
The company can expect to replace **27.43%** of its total production due to the two-year warranty guarantee.

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Would you like more details on how to use Z-tables, or do you have any further questions?

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#### 5 Related Questions:
1. What is the probability that a watch lasts more than 30 months?
2. How many watches (in a batch of 1,000) are expected to fail within 18 months?
3. What is the lifetime percentile of a watch that lasts exactly 27 months?
4. How would the percentage of refunds change if the standard deviation was 3 months?
5. What refund percentage would result if the warranty period was extended to 3 years?

#### Tip:
If you need cumulative probabilities often, try using statistical software or calculators to speed up the lookup process for Z-scores.

Acrotime is a manufacturer of quartz crystal watches. Acrotime Acromine researchers have shown that the watches average life of 27 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetime is 5 months, and the distribution of lifetime is normal. If Acrotime guarantees a full refund or any defective watch for two years after purchase, what percentage of total production will the company expect to replace?

Learn how to calculate the expected replacement rate for watches under a two-year warranty. This problem covers statistical concepts, including normal distribution, Z-score calculations, and cumulative probability, using a mean lifetime of 27 months and a standard deviation of 5 months. Suitable for students in Grades 10-12.

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