Question

The time for a professor to grade a student's homework in statistics is normally distributed with a mean of 12.6 minutes and a standard deviation of 2.5 minutes. What is the probability that randomly selected homework will require between 8 and 12 minutes to grade?

Answer 1

37.23% probability that randomly selected homework will require between 8 and 12 minutes to grade

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 = 12.6, \sigma = 2.5`

What is the probability that randomly selected homework will require between 8 and 12 minutes to grade?

This is the pvalue of Z when X = 12 subtracted by the pvalue of Z when X = 8. So

X = 12

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

`Z = (12 - 12.6)/(2.5)`

`Z = -0.24`

`Z = -0.24` has a pvalue of 0.4052

X = 8

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

`Z = (8 - 12.6)/(2.5)`

`Z = -1.84`

`Z = -1.84` has a pvalue of 0.0329

0.4052 - 0.0329 = 0.3723

37.23% probability that randomly selected homework will require between 8 and 12 minutes to grade