A tire manufacturer wishes to set a minimum mileage guarantee on a new set of tires. The mean is 67,900 with a standard deviation of 2,050. The manufacturer wants to set a minimum guaranteed mileage so that no more than 4% of the tires will have to be replaced. What should the minimum mileage be?
71490 miles nearly.
http://davidmlane.com/hyperstat/z_table.html
Use the second applet, put in mean, std deviation, set above to .04
To find the minimum mileage that the tire manufacturer should set as a guarantee, we need to determine the value of the standard score, also known as the z-score, that corresponds to the desired percentage.
1. Start by finding the z-score corresponding to a cumulative percentage of 96% (100% - 4%) using a standard normal distribution table or a calculator. This z-score will tell us the number of standard deviations above or below the mean.
In this case, we want to find the z-score corresponding to a cumulative percentage of 96%. From the standard normal distribution table, we find that the z-score is approximately 1.75.
2. Once we have the z-score, we can use the formula for z-score to find the corresponding value in the original data set.
z = (X - μ) / σ
Here, X is the desired minimum mileage guarantee, μ is the mean mileage, and σ is the standard deviation of the mileage.
Rearranging the formula, we have:
X = (z * σ) + μ
Substituting the values we have, we get:
X = (1.75 * 2,050) + 67,900
Calculating this, we find:
X ≈ 71,662.5
Therefore, the tire manufacturer should set a minimum mileage guarantee of approximately 71,662.5 miles to ensure that no more than 4% of the tires will have to be replaced.