Question

Grades in large classes are sometimes approximately Normally distributed—a fact that serves as the justification for "grading on a bell curve." A common practice for very large classes is to give 16% of students A grades, 34% B grades, 34% C grades, and 16% D and F grades. Assuming a Normal distribution of grades, what are the z-scores for these letter grades?

Answer 1

Grade A:
`Z \geq 1`

Grade B:
`0 \leq Z < 1`

Grade C:
`-1 \leq Z < 0`

Grade D:
`Z < -1`

Explanation:

Problems of normally distributed samples can be solved using the Z score table.

The Z score of a measure represents how many standard deviations it is above or below the mean of all the measures.

Each Z score has a pvalue. This represents the percentile of the measure.

In this problem, we have that:

The upper 16% of the class get A grades. The upper 16% has a pvalue of at least 100% = 16% = 84% = 0.84. This is
`Z \geq 1`.

The middle 34% of the class get B grades. The middle 34% has a pvalue of at least 84%-35% = 50% = 0.5 and at most 0.84. This is
`0\leqZ < 1`.

Those between a pvalue of 0.5-0.34 = 0.16 and 0.5 get get grade C.
`Z = -1` has a pvalue of 0.16. So a grade C is in the interval
`-1 \leq Z < 0`.

Those with Z lesser than -1 get grades D and F