Question

For the most recent year available, the mean annual cost to attend a private university in the United States was $20,207. Assume the distribution of annual costs follows the normal probability distribution and the standard deviation is $4,375. Use Appendix B.3. Ninety percent of all students at private universities pay less than what amount

Answer 1

Ninety percent of all students at private universities pay less than $25,807.

Explanation:

When the distribution is normal, we use 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 question, we have that:

`\mu = 20207, \sigma = 4375`

Ninety percent of all students at private universities pay less than what amount

Less than the 90th percentile, which is X when Z has a pvalue of 0.9. So X when Z = 1.28.

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

`1.28 = (X - 20207)/(4375)`

`X - 20207 = 1.28*4375`

`X = 25807`

Ninety percent of all students at private universities pay less than $25,807.