Question

Three students were applying to the same graduate school. They came from schools with different grading systems. Student GPA School Average GPA School Standard Deviation Thuy 2.5 3.2 0.8 Vichet 88 75 20 Kamala 8.9 8 0.4 Which student had the best GPA when compared to other students at his school

Answer 1

Due to the higher Z-score, Kamala had the best GPA when compared to other students at her school.

Explanation:

Z-score:

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

In this question:

Whichever student's GPA had the higher z-score had the best GPA compared to other students.

Thuy 2.5 3.2 0.8

This means that
`X = 2.5, \mu = 3.2, \sigma = 0.8`

The z-score is:

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

`Z = (2.5 - 3.2)/(0.8)`

`Z = -0.875`

Vichet 88 75 20

This means that
`X = 88, \mu = 75, \sigma = 20`

The z-score is:

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

`Z = (88 - 75)/(20)`

`Z = 0.65`

Kamala 8.9 8 0.4

This means that
`X = 8.9, \mu = 8, \sigma = 0.4`

The z-score is:

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

`Z = (8.9 - 8)/(0.4)`

`Z = 2.25`

Due to the higher Z-score, Kamala had the best GPA when compared to other students at her school.