Question

A psychology professor assigns letter grades on a test according to the following scheme. A: Top 14% of scores B: Scores below the top 14% and above the bottom 65% C: Scores below the top 35% and above the bottom 16% D: Scores below the top 84% and above the bottom 9% F: Bottom 9% of scores Scores on the test are normally distributed with a mean of 68.4 and a standard deviation of 9.7. Find the numerical limits for a B grade. Round your answers to the nearest whole number, if necessary.

Answer 1

The numerical limits for a B grade are 72 and 79, that is, a score between 72 and 79 results in a B grade.

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 68.4 and a standard deviation of 9.7.

This means that
`\mu = 68.4, \sigma = 9.7`

Find the numerical limits for a B grade.

Below the 100 - 14 = 86th percentile and above the 65th percentile.

65th percentile:

X when Z has a p-value of 0.65, so X when Z = 0.385.

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

`0.385 = (X - 68.4)/(9.7)`

`X - 68.4 = 0.385*9.7`

`X = 72`

86th percentile:

X when Z has a p-value of 0.86, so X when Z = 1.08.

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

`1.08 = (X - 68.4)/(9.7)`

`X - 68.4 = 1.08*9.7`

`X = 79`

The numerical limits for a B grade are 72 and 79, that is, a score between 72 and 79 results in a B grade.