Question

A professor gives her 100 students an exam; scores are normally distributed. The section has an average exam score of 80 with a standard deviation of 6.5. What percentage of the class has an exam score of A- or higher (defined as at least 90)? Type your calculations along with your answer for full credit; round your final percentage to two decimal places.

Answer 1

6.18% of the class has an exam score of A- or higher.

Explanation:

When the distribution is normal, we use 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 question, we have that:

`\mu = 80, \sigma = 6.5`

What percentage of the class has an exam score of A- or higher (defined as at least 90)?

This is 1 subtracted by the pvalue of Z when X = 90. So

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

`Z = (90 - 80)/(6.5)`

`Z = 1.54`

`Z = 1.54` has a pvalue of 0.9382

1 - 0.9382 = 0.0618

6.18% of the class has an exam score of A- or higher.