A humanities professor assigns letter grades on a test according to the following scheme. A: Top 12% of scores B: Scores below the top 12% and above the bottom 62% C: Scores below the top 38% and above the bottom 19% D: Scores below the top 81% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 77.5 and a standard deviation of 8.6. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

The minimum score required for an A grade is 88.

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

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:

Find the minimum score required for an A grade.

This score is the 100 - 12 = 88th percentile, which is X when Z has a pvalue of 0.88. So X when Z = 1.175.

Rounding to the nearest whole number:

The minimum score required for an A grade is 88.