Exercise 20 Page 744

The grades of a test in a class of 25 students are normally distributed with a mean of 75 and a standard deviation of 5. We are told that one of the students scored 100 points. To determine if that score is an outlier, let's first make a sketch of the probability distribution.

In a normally distributed data set, 99.7 % of data is within 3 standard deviations of the mean.

From the graph we can see that a score of 100 points is more than 3 standard deviations away form the mean. Let's calculate the probability that the score of a student is more than 3 standard deviations away from the mean.

Probability=100 %-99.7 %

SubTerm

Subtract term

Probability=0.3 %

The probability that a student would score more than 3 standard deviations away from the mean (for example 100 points) is very low — 0.3 %. Therefore, a score of 100 points is an outlier. To be even more certain about this, we can calculate the expected number of students from the class of 25 students that would score more than 3 standard deviations away from the mean.

Number of students=0.3 %* 25

WriteDec

Write as a decimal

Number of students=0.003* 25

Multiply

Multiply

Number of students=0.075

RoundInt

Round to nearest integer

Number of students≈ 0

We found that on average no student would score more than 3 standard deviations away from the mean. This means that a score of 100 points — which is more than 3 standard deviations away from the mean — is an outlier.