Question

A psychology professor assigns letter grades on a test according to the following scheme. A: Top 9% of scores B: Scores below the top 9% and above the bottom 56% C: Scores below the top 44% and above the bottom 20% D: Scores below the top 80% and above the bottom 6% F: Bottom 6% of scores Scores on the test are normally distributed with a mean of 79.3 and a standard deviation of 8.4. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Answer 1

You get a D if you have a grade between 66 and 72.

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 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.

Scores on the test are normally distributed with a mean of 79.3 and a standard deviation of 8.4.

This means that
`\mu = 79.3, \sigma = 8.4`

D: Scores below the top 80% and above the bottom 6%

So between the 6th and the 20th percentile.

6th percentile:

X when Z has a pvalue of 0.06. So X when Z = -1.555.

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

`-1.555 = (X - 79.3)/(8.4)`

`X - 79.3 = -1.555*8.4`

`X = 66.2`

Rounds to 66

20th percentile:

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

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

`-0.84 = (X - 79.3)/(8.4)`

`X - 79.3 = -0.84*8.4`

`X = 72.2`

Rounds to 72

You get a D if you have a grade between 66 and 72.