Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 2.1 years, and standard deviation of 0.6 years.

The 8% of items with the shortest lifespan will last less than how many years?

Answer 1

They will last less than 1.257 years.

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 = 2.1, \sigma = 0.6`

The 8% of items with the shortest lifespan will last less than how many years?

Less than the 8th percentile, which is X when Z has a pvalue of 0.08. So X when Z = -1.405.

So

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

`-1.405 = (X - 2.1)/(0.6)`

`X - 2.1 = -1.405*0.6`

`X = 1.257`

They will last less than 1.257 years.