Question

a philosophy 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 57 %
c: scores below the top 43 % and above the bottom 17 %
d: scores below the top 83 % and above the bottom 9 %
f: bottom 9 % of scores
scores on the test are normally distributed with a mean of 81.2 and a standard deviation of 8.1. find the minimum score required for an a grade. round your answer to the nearest whole number, if necessary.

Explanation:

Step1: Determine the z-score for the top 12%

The top 12% means we need the z-score corresponding to a cumulative probability of \(1 - 0.12 = 0.88\). Using a standard normal distribution table or calculator, the z-score for a cumulative probability of 0.88 is approximately \(z = 1.175\) (this value can be found from z-tables or using invNorm function in a calculator).

Step2: Use the z-score formula to find the score

The z-score formula is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the score we want to find, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We know \(z = 1.175\), \(\mu = 81.2\), and \(\sigma = 8.1\). Rearranging the formula to solve for \(x\) gives \(x=\mu + z\sigma\).
Substituting the values: \(x = 81.2+1.175\times8.1\)
First, calculate \(1.175\times8.1 = 9.5175\)
Then, \(x = 81.2 + 9.5175 = 90.7175\)

Step3: Round to the nearest whole number

Rounding 90.7175 to the nearest whole number gives 91.

Answer:

91