Question

A philosophy professor assigns letter grades on a test according to the following scheme. A: Top 10% of scores B: Scores below the top 10% and above the bottom 58% C: Scores below the top 42% and above the bottom 20% D: Scores below the top 80% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 67.3 and a standard deviation of 7.3. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer 1

The numerical limits for a D grade are 58(lower limit) and 61(upper limit).

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Scores on the test are normally distributed with a mean of 67.3 and a standard deviation of 7.3.

This means that
`\mu = 67.3, \sigma = 7.3`

D: Scores below the top 80% and above the bottom 9%

This means that lower bound is the 9th percentile and the upper bound is the 100 - 80 = 20th percentile.

9th percentile:

This is X when Z has a pvalue of 0.09. So X when Z = -1.34.

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

`-1.34 = (X - 67.3)/(7.3)`

`X - 67.3 = -1.34*7.3`

`X = 57.52`

Rounding to the nearest whole number, 58.

20th percentile:

This is X when Z has a pvalue of 0.2. So X when Z = -0.84

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

`-0.84 = (X - 67.3)/(7.3)`

`X - 67.3 = -0.84*7.3`

`X = 61.17`

Rounding to the nearest whole number, 61.

The numerical limits for a D grade are 58(lower limit) and 61(upper limit).