To find the guaranteed mileage such that no more than 2% of the tires will fail to meet it, we need to find the value $x$ such that 2% (or 0.02) of the tires have mileages below $x$. This involves determining a particular percentile of the normal distribution.

Given:

• Mean ($\mu$) = 37,402 miles
• Standard deviation ($\sigma$) = 4,135 miles
• We want the mileage $x$ such that 2% of the tires are below this value. This corresponds to the 2nd percentile of the normal distribution.

Step 1: Find the Z-score corresponding to 2%

Using the standard normal distribution, the Z-score corresponding to the 2nd percentile is approximately $-2.054$. This value can be found using a Z-table or a statistical calculator.

Step 2: Use the Z-score formula

The Z-score formula is: $Z = \frac{x - \mu}{\sigma}$ Where:

• $Z$ is the Z-score corresponding to the percentile (in this case, $-2.054$),
• $x$ is the unknown guaranteed mileage,
• $\mu = 37,402$ is the mean mileage,
• $\sigma = 4,135$ is the standard deviation.

Now we solve for $x$: $-2.054 = \frac{x - 37,402}{4,135}$

Step 3: Solve for $x$

$x = -2.054 \times 4,135 + 37,402$ $x \approx -8,494.49 + 37,402$ $x \approx 28,907.51$

Conclusion:

The company should set the guaranteed mileage to approximately 28,908 miles to ensure that no more than 2% of the tires fail to meet the guaranteed mileage.

Would you like further clarification on any of the steps or additional details?

Here are some related questions for further practice:

1. What would the guaranteed mileage be if only 5% of the tires were allowed to fail?
2. How would the guarantee change if the standard deviation of the tire's mileage increased to 5,000 miles?
3. What is the probability that a randomly selected tire lasts between 30,000 and 40,000 miles?
4. What would the company guarantee if they wanted to increase the failure threshold to 10%?
5. How does changing the mean mileage affect the guaranteed mileage for the same failure percentage?

Tip: When working with normal distributions, the Z-score helps to standardize data, making it easier to find probabilities and percentiles.