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?

4% lie 1.75 std below the mean.

So, 60000-5000*1.75 = 51250

If the warranty is for 51250, 96% of the tires will exceed that mileage.

You can enter your values and play around with Z table stuff at

http://davidmlane.com/hyperstat/z_table.html

Well, if the mean tread life is 60,000 miles and only 4% of the tires need to be replaced, I'd say the company should set the warranty mileage at... 59,999 miles. That way, if your tire wears out at exactly 60,000 miles, you can still claim the warranty and get a replacement! Just remember to bring your clown nose and oversized shoes when you visit the tire shop!

To find the warranty mileage, we need to determine the cutoff point that includes only 4% of the tires.

First, we need to find the z-score associated with the 4% probability. The z-score represents the number of standard deviations away from the mean.

Using a standard normal distribution table or a calculator, we can find the z-score associated with a cumulative probability of 0.04. The z-score is approximately -1.75.

Next, we can use the formula for standardizing a normally distributed variable:

z = (x - μ) / σ

Where:
z = z-score
x = warranty mileage
μ = mean mileage
σ = standard deviation

Plugging in the values, we can solve for x:

-1.75 = (x - 60,000) / 5,000

Solving for x, we get:

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

-8,750 = x - 60,000

x = 60,000 - 8,750

x ≈ 51,250

Therefore, the warranty mileage should be set at approximately 51,250 miles.

To determine the warranty mileage for the tires, we need to find the value at which only 4% of the tires will need to be replaced. This can be done by finding the cutoff point on the normal distribution curve.

1. Start by converting the 4% probability into a z-score. The z-score represents the number of standard deviations a value is from the mean. We can find the corresponding z-score using a standard normal distribution table or a statistical calculator.

In this case, since we want to find the cutoff point on the left tail of the distribution (where only 4% of the tires will need to be replaced), we need to find the z-score that corresponds to a cumulative probability of 4%.

2. Once you have the z-score, you can use the formula to convert it back into an actual value on the distribution using the formula:

X = μ + (z * σ)

where X is the value we are looking for (warranty mileage), μ is the mean, z is the z-score, and σ is the standard deviation.

Plug in the given values:

μ = 60,000 miles
σ = 5,000 miles

Now, find the z-score that corresponds to a cumulative probability of 4%:

Using a standard normal distribution table or calculator, the z-score is approximately -1.75.

3. Now, substitute the values into the formula:

X = 60,000 + (-1.75 * 5,000)

Calculating this gives us:

X = 60,000 - 8,750

So the warranty mileage should be set at approximately 51,250 miles.

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