The lifetime in miles for a certain brand of tire is normally distributed with a mean of 22,000 miles and a standard deviation of 3,100 miles The tire manufacturer wants to offer a money-back guarantee so that no more than 3% of tires will qualify for a refund. What is the minimum number of miles the manufacturer should guarantee that the tires will last? 27,828 miles 25,007 miles 15,800 miles 16,172 miles
Question
The lifetime in miles for a certain brand of tire is normally distributed with a mean of 22,000 miles and a standard deviation of 3,100 miles The tire manufacturer wants to offer a money-back guarantee so that no more than 3% of tires will qualify for a refund. What is the minimum number of miles the manufacturer should guarantee that the tires will last? 27,828 miles 25,007 miles 15,800 miles 16,172 miles
Solution
To solve this problem, we need to use the concept of z-scores in a normal distribution. The z-score is a measure of how many standard deviations an element is from the mean.
In this case, we want to find the mileage that only the worst 3% of tires will not reach. This corresponds to finding the z-score that has 3% of the distribution to its left.
Looking up this value in a standard normal distribution table, or using a calculator, we find that the z-score corresponding to 3% is approximately -1.88.
We can then use the formula for a z-score, which is (X - μ) / σ, where X is the value we're looking for, μ is the mean, and σ is the standard deviation.
Rearranging the formula to solve for X gives us X = μ + zσ.
Substituting the given values into this formula gives us X = 22,000 + (-1.88 * 3,100) = 16,172 miles.
So, the manufacturer should guarantee that the tires will last at least 16,172 miles.
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