Question

Suppose that replacement times for washing machines are normally distributed with a mean of 8.6 years and a standard deviation of 1.6 years. Find the replacement time that separates the top 18% from the bottom 82%.

Answer 1

Replacement time of 10.064 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 problem, we have that:

`\mu = 8.6, \sigma = 1.6`

Find the replacement time that separates the top 18% from the bottom 82%.

This is the value of X when Z has a pvalue of 0.82. So it is X when Z = 0.915.

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

`0.915 = (X - 8.6)/(1.6)`

`X - 8.6 = 0.915*1.6`

`X = 10.064`

Replacement time of 10.064 years.