Question

The time for a professor to grade an exam is normally distributed with a mean of 16.3 minutes and a standard deviation of 4.2 minutes. What is the probability that a randomly selected exam will require more than 15 minutes to grade

Answer 1

62.17% probability that a randomly selected exam will require more than 15 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 = 16.3, \sigma = 4.2`

What is the probability that a randomly selected exam will require more than 15 minutes to grade

This is 1 subtracted by the pvalue of Z when X = 15. So

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

`Z = (15 - 16.3)/(4.2)`

`Z = -0.31`

`Z = -0.31` has a pvalue of 0.3783.

1 - 0.3783 = 0.6217

62.17% probability that a randomly selected exam will require more than 15 minutes to grade