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.