Question

The length of time needed to complete a certain test is normally distributed with mean 74 minutes and standard deviation 12 minutes. Find the probability that it will take between 72 and 77 minutes to complete the test.

Answer 1

There is a 16.62% probability that it will take between 72 and 77 minutes to complete the test.

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)`

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.

The length of time needed to complete a certain test is normally distributed with mean 74 minutes and standard deviation 12 minutes, so
`\mu = 74, \sigma = 12`.

Find the probability that it will take between 72 and 77 minutes to complete the test.

We have to subtract the pvalue of Z when X = 77 by the pvalue of Z when X = 72.

So

X = 77

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

`Z = (77 - 74)/(12)`

`Z = 0.25`

`Z = 0.25` has a pvalue of 0.5987.

X = 72

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

`Z = (72 - 74)/(12)`

`Z = -0.17`

`Z = -0.17` has a pvalue of 0.4325.

So, there is a 0.5987 - 0.4325 = 0.1662 = 16.62% probability that it will take between 72 and 77 minutes to complete the test.