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 between 14 and 19 minutes to​ grade?

A.0.4477

B.0.3175

C.0.3804

D.0.5837

Answer 1

A.0.4477

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 between 14 and 19 minutes to​ grade?

This probability is the pvalue of Z when X = 19 subtracted by the pvalue of Z when X = 14. So

X = 19

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

`Z = (19 - 16.3)/(4.2)`

`Z = 0.64`

`Z = 0.64` has a pvalue of 0.7389.

X = 14

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

`Z = (14 - 16.3)/(4.2)`

`Z = -0.55`

`Z = -0.55` has a pvalue of 0.2912

0.7389 - 0.2912 = 0.4477

So the correct answer is:

A.0.4477