Question

You score a 65% on your AP Statistics Exam and a 78% on your Economics Exam. The AP Statistics scores were normally distributed with an average of 60% and a standard deviation of 6%. The Economics scores were also normally distributed with an average of 75% and standard deviation of 2.5%. In which class did you perform better relative to the class average? Be sure to justify your answer.

Answer 1

His economics grade had a higher z-score, so you performed better relative to the class average in economics.

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 which class did you perform better relative to the class average?

In whichever class you had the higher z-score.

Statistics:

Scored 65, mean 60, standard deviation 6. So
`X = 65, \mu = 60, \sigma = 6`

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

`Z = (65 - 60)/(6)`

`Z = 0.83`

Economics:

Scored 78, Mean 75, Standard deviation 2.5. So
`X = 78, \mu = 75, \sigma = 2.5`

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

`Z = (78 - 75)/(2.5)`

`Z = 1.2`

His economics grade had a higher z-score, so you performed better relative to the class average in economics.