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

Answer:

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

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{12 - 12.6}{2.5}$

$Z = -0.24$

$Z = -0.24$ has a pvalue of 0.4052

X = 8

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{8 - 12.6}{2.5}$

$Z = -1.84$

$Z = -1.84$ 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