A tire manufacturer warranties its tires to last at least 20 comma 000 miles or​ "you get a new set of​ tires." In its​ experience, a set of these tires last on average 28 comma 000 miles with SD 5 comma 000 miles. Assume that the wear is normally distributed. What is the probability that a set of tires wears out before 20 comma 000 ​miles?

We are given that the mileage of the tires is normally distributed with a mean (\(\mu\)) of 28,000 miles and a standard deviation (\(\sigma\)) of 5,000 miles. We are asked to find the probability that a set of tires wears out before 20,000 miles, which can be written as \(P(X < 20,000)\), where \(X\) is the mileage of the tires.

### Steps:

1. **Standardize the variable**:
   We need to convert the value of 20,000 miles to a standard normal variable \(Z\) using the following formula for the z-score:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   Here, \(X = 20,000\), \(\mu = 28,000\), and \(\sigma = 5,000\).

   Substituting the values:
   \[
   Z = \frac{20,000 - 28,000}{5,000} = \frac{-8,000}{5,000} = -1.6
   \]

2. **Find the probability**:
   Now, we need to find the probability corresponding to \(Z = -1.6\) in the standard normal distribution table. This gives us the cumulative probability \(P(Z < -1.6)\).

   From the standard normal distribution table, \(P(Z < -1.6) \approx 0.0548\).

Thus, the probability that a set of tires wears out before 20,000 miles is approximately \(0.0548\), or about **5.48%**.

### Summary:
- The probability that a set of tires will wear out before 20,000 miles is approximately 5.48%.

Would you like more details or have any questions on this?

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Here are 5 related questions to expand the topic:
1. What is the probability that the tires last more than 30,000 miles?
2. How would the probability change if the standard deviation were reduced to 3,000 miles?
3. What is the probability that the tires last between 25,000 and 35,000 miles?
4. What is the z-score for a tire mileage of 25,000 miles?
5. How does the shape of the normal distribution curve affect warranty predictions?

**Tip**: Z-scores help compare different values from distributions with varying means and standard deviations by standardizing them to a common scale.

A tire manufacturer warranties its tires to last at least 20,000 miles or 'you get a new set of tires.' In its experience, a set of these tires lasts on average 28,000 miles with a standard deviation of 5,000 miles. Assume that the wear is normally distributed. What is the probability that a set of tires wears out before 20,000 miles?

Learn how to calculate the probability that a tire wears out before 20,000 miles using normal distribution concepts. This example explores the Z-score formula and standard normal distribution table with a detailed step-by-step solution, ideal for high school students in Grades 10-12.

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