Question

Final grades for all of the sections of the data analysis class (WCOB 1033) for the spring semester are normally distributed with a mean (µ) of 75 and a standard deviation (σ) of 13. What is the approximate cutoff value for the top 5% of all the grades?

Answer 1

The cutoff value for the top 5% of all the grades is 96.385.

Explanation:

Problems of normally distributed samples are 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.

In this problem, we have that:

`\mu = 75, \sigma = 13`

What is the approximate cutoff value for the top 5% of all the grades?

This is the value of X when Z has a pvalue of 0.95. So it is X when Z = 1.645.

So

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

`1.645 = (X - 75)/(13)`

`X - 75 = 13*1.645`

`X = 96.385`

The cutoff value for the top 5% of all the grades is 96.385.