Question

Last years freshman class at big state university totaled $5330 students. Of those, $1258 received a merit scholarship to help offset tuition cost their freshman year. The amount a student received was N($3464, 489). If the cost of full tuition was $4450 last year, what percentage of students who received a merit scholarship did not receive enough to cover full tuition?

Answer 1

2.17% of students who received a merit scholarship did not receive enough to cover full tuition

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 question, we have that:

`\mu = 3464, \sigma = 489`

If the cost of full tuition was $4450 last year, what percentage of students who received a merit scholarship did not receive enough to cover full tuition?

This is 1 subtracted by the pvalue of Z when X = 4450. So

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

`Z = (4450 - 3464)/(489)`

`Z = 2.02`

`Z = 2.02` has a pvalue of 0.9783.

1 - 0.9783 = 0.0217

2.17% of students who received a merit scholarship did not receive enough to cover full tuition