Question

he length of time to perform an oil change at a certain automobile repair shop is ap-proximately normally distributed with mean 29.5 minutes and standard deviation 3 minutes.What is the probability that a mechanic can complete 16 oil changes in an eight-hour day

Answer 1

56.75% probability that a mechanic can complete 16 oil changes in an eight-hour day

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 = 29.5, \sigma = 3`

What is the probability that a mechanic can complete 16 oil changes in an eight-hour day

16 oil changes in 8 hours is 2 changes per hour, that is, one each 30 minutes.

This probability is the pvalue of Z when X = 30. So

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

`Z = (30 - 29.5)/(3)`

`Z = 0.17`

`Z = 0.17` has a pvalue of 0.5675

56.75% probability that a mechanic can complete 16 oil changes in an eight-hour day