The tread life a certain brand of tires is known to be normally distributed with a mean of 60,000 miles and a standard deviation of 5,000 miles. The company wishes to set the warranty so that only 4% of the tires will need to be replaced. What should the warranty mileage be?

To find the warranty mileage, we need to find the point on the distribution that corresponds to the 4th percentile. Since the distribution is normal, we can use z-scores to find this point.

First, we need to find the z-score corresponding to the 4th percentile. This can be found using a standard normal distribution table or calculator. The z-score corresponding to the 4th percentile is approximately -1.75.

The formula for calculating the z-score is given by:

z = (x - μ) / σ

Where:
- z is the z-score
- x is the value we want to find (warranty mileage)
- μ is the mean of the distribution (60,000 miles)
- σ is the standard deviation of the distribution (5,000 miles)

Rearranging the formula to solve for x, we have:

x = z * σ + μ

Substituting the known values, we have:

x = -1.75 * 5,000 + 60,000

x = -8,750 + 60,000

x = 51,250

Therefore, the warranty mileage should be set at 51,250 miles so that only 4% of the tires will need to be replaced.

To determine the warranty mileage, we need to find the value at which only 4% of the tires will need to be replaced. This value is known as the "warranty threshold" or "cutoff point."

Here's how you can find the warranty mileage:

Step 1: Convert the given percentile to a standard score (Z-score).

The given percentile is 4%, which means we are looking for the value that falls below 4% of the tires. To convert this percentile to a Z-score, we use the standard normal distribution table or a statistical calculator.

Using a standard normal distribution table, we find the Z-score corresponding to a cumulative probability of 0.04. The Z-score comes out to be approximately -1.75.

Step 2: Calculate the warranty mileage using the Z-score.

The Z-score formula is: Z = (X - μ) / σ

Given:
Mean (μ) = 60,000 miles
Standard deviation (σ) = 5,000 miles
Z-score (Z) = -1.75

Solving the formula for X (warranty mileage):
X = Z * σ + μ

Substituting the values:
X = -1.75 * 5,000 + 60,000
X ≈ 52,500 miles

Therefore, the recommended warranty mileage should be approximately 52,500 miles. This means that the tire manufacturer should warranty their tires up to 52,500 miles to cover only 4% of the tires that may need replacement.