Question

Your company manufactures hot water heaters. The life spans of your product are known to be normally distributed with a mean of 13 years and a standard deviation of 1.5 years. You want to set the warranty on your product so that you do not have to replace more than 5% of the hot water heaters that you sell. How many years should you claim on your warranty

Answer 1

You should claim 10.5325 years on your warranty.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 13, \sigma = 1.5`

Want's to replace no more than 5% of the products.

This means that the warranty should be 5th percentile, that is, the value of X when Z has a pvalue of 0.05. So X when Z = -1.645.

`Z = (X - \mu)/(\sigma)`

`-1.645 = (X - 13)/(1.5)`

`X - 13 = -1.645*1.5`

`X = 10.5325`

You should claim 10.5325 years on your warranty.