Question

An appliance dealer's website features the mean life expectancy values for major household appliances. A standard refrigerator's mean life expectancy is 18 years with a standard deviation of 2 years. What is the probability that your standard refrigerator will last between 12 and 20 years? Express the probability as a percentage rounded to the nearest hundredth.

Answer 1

There is an 84% probability that your standard refrigerator will last between 12 and 20 years.

Explanation:

Problems of normally distributed samples can be 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:

A standard refrigerator's mean life expectancy is 18 years with a standard deviation of 2 years. This means that
`\mu = 18, \sigma = 2`.

What is the probability that your standard refrigerator will last between 12 and 20 years?

This probability is the pvalue of Z when
`X = 20` subtracted by the pvalue of Z when
`X = 12`. So

X = 20

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

`Z = (20 - 18)/(2)`

`Z = 1`

`Z = 1` has a pvalue of 0.8413

X = 12

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

`Z = (12- 18)/(2)`

`Z = -3`

`Z = -3` has a pvalue of 0.0014

This means that there is a 0.8413-0.0014 = 0.8399 = 0.84 = 84% probability that your standard refrigerator will last between 12 and 20 years.