Question

The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under guarantee. If she is willing to replace only 3% of the motors that fail, how long a guarantee should be offered? Assume that the lifetime of a motor follows a normal distribution.

Answer 1

A guarantee of 6.24 years should be offered.

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 = 10, \sigma = 2`

If she is willing to replace only 3% of the motors that fail, how long a guarantee should be offered?

The 3rd percentile, that is, the value of X when Z has a pvalue of 0.03. So X when Z = -1.88.

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

`-1.88 = (X - 10)/(2)`

`X - 10 = -1.88*2`

`X = 6.24`

A guarantee of 6.24 years should be offered.