Question

The toasters produced by a company have a normally distributed life span with a mean of 5.8 years and a standard deviation of 0.9 years, what warranty should be provided so that the company is replacing at most 5% of their toasters sold?

Answer 1

A warranty of 4.32 years should be provided.

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 question, we have that:

`\mu = 5.8, \sigma = 0.9`

What warranty should be provided so that the company is replacing at most 5% of their toasters sold?

The warranty should be the 5th percentile, which is X when Z has a pvalue of 0.05. So it is X when Z = -1.645.

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

`-1.645 = (X - 5.8)/(0.9)`

`X - 5.8 = -1.645*0.9`

`X = 4.32`

A warranty of 4.32 years should be provided.