Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 5.8 years, and

standard deviation of 1.7 years.
The 10% of items with the shortest lifespan will last less than how many years?
Round your answer to one decimal place.

Answer 1

Less than 3.6 years.

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 of 5.8 years, and standard deviation of 1.7 years.

This means that
`\mu = 5.8, \sigma = 1.7`

The 10% of items with the shortest lifespan will last less than how many years?

Less than 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 - 5.8)/(1.7)`

`X - 5.8 = -1.28*1.7`

`X = 3.6`

Less than 3.6 years.