Question

Optimal-Eats blender has a mean time before failure of 40 months with a standard deviation of 6 months, and the failure times are normally distributed. What should be the warranty period, in months, so that the manufacturer will not have more than 10% of the blenders returned

Answer 1

The warranty period should be of 32 months.

Explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean time before failure of 40 months with a standard deviation of 6 months.

This means that
`\mu = 40, \sigma = 6`

What should be the warranty period, in months, so that the manufacturer will not have more than 10% of the blenders returned?

The warranty period should be the 10th percentile, which is X when Z has a p-value of 0.1, so X when Z = -1.28.

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

`-1.28 = (X - 40)/(6)`

`X - 40 = -1.28*6`

`X = 32`

The warranty period should be of 32 months.