Question

The time it takes a customer service complaint to be settled at a small department store is normally distributed with a mean of 10 minutes and a standard deviation of 3 minutes. Find the probability that a randomly selected complaint takes more than 15 minutes to be settled.

Answer 1

0.0475 = 4.75% probability that a randomly selected complaint takes more than 15 minutes to be settled.

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 10 minutes and a standard deviation of 3 minutes

This means that
`\mu = 10, \sigma = 3`

Find the probability that a randomly selected complaint takes more than 15 minutes to be settled.

This is 1 subtracted by the p-value of Z when X = 15, so:

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

`Z = (15 - 10)/(3)`

`Z = 1.67`

`Z = 1.67` has a p-value of 0.9525.

1 - 0.9525 = 0.0475.

0.0475 = 4.75% probability that a randomly selected complaint takes more than 15 minutes to be settled.