Question

On a 100 point scale, suppose the grades are approximately normally distributed with a mean of 60 and a

standard deviation of 12. The professor plans to give a letter grade of a "B" to some of her students,
beginning with a lower cutoff score of a 52. What should the upper score cutoff be set at if she wishes for
exactly 40% of her students to receive a "B"? Show your work.

Answer 1

64.7

Explanation:

You want the upper cutoff in a normal distribution with a mean of 60 and a standard deviation of 12 such that the fraction of the distribution between 52 and that upper cutoff is 40%.

Lower cutoff

The cutoff of 52 leaves about 25.25% of the distribution below the cutoff. This means the upper cutoff must be chosen so that 25.25% +40% is below the upper cutoff.

The cutoff that has 65.25% of the distribution below it is 64.7.

The professor must set the upper cutoff at 64.7 for 40% of students to get a grade of "B."

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Additional comment

The attached calculator display shows the use of the normalCDF and invNorm functions to find the fraction below 52 and the cutoff that has the fraction 0.6525 below it. We don't need to find or use Z-scores, because these functions can work with the given mean and standard deviation.

The last two lines of the display show a check of the result.

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