Question

A philosophy professor assigns letter grades on a test according to the following scheme. A: Top 13% of scores B: Scores below the top 13% and above the bottom 55% C: Scores below the top 45% and above the bottom 23% D: Scores below the top 77% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 76 and a standard deviation of 7.9. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

Answer 1

The minimum score required for an A grade is 85.

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 = 76, \sigma = 7.9`

Find the minimum score required for an A grade.

The top 13% of the scores are A, so the minimum is the 100-13 = 87th percentile, which is X when Z has a pvalue of 0.87. So X when Z = 1.127.

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

`1.127 = (X - 76)/(7.9)`

`X - 76 = 7.9*1.127`

`X = 84.9`

Rounding to the nearest whole number:

The minimum score required for an A grade is 85.