Question

Last year’s freshman class at Big State University totaled 5,307 students. Of those, 1,264 received a merit scholarship to help offset tuition costs their freshman year (although the amount varied per student). The amount a student received was N($3,454, $490). If the cost of full tuition was $4,300 last year, what percentage of students who received a merit scholarship did notreceive enough to cover full tuition?

Answer 1

Approximately 95.73% of students who received a merit scholarship at Big State University did not receive enough to cover the cost of full tuition, which was $4,300.

Step-by-step explanation:

To determine the percentage of students who received a merit scholarship but did not receive enough to cover full tuition, we need to analyze the normal distribution N($3,454, $490).

The full tuition cost is $4,300. Students who received less than this amount in merit scholarships will be considered as not having enough to cover full tuition.

First, we calculate the z-score for $4,300 using the formula: z = (X - μ) / σ, where X is the full tuition amount ($4,300), μ is the mean scholarship amount ($3,454), and σ is the standard deviation ($490). Substituting the values, z = ($4,300 - $3,454) / $490 ≈ 1.7245.

Next, we look up the z-score in the standard normal distribution table to find the proportion of students who received less than $4,300.

A z-score of 1.7245 corresponds to a cumulative probability of approximately 0.9573, meaning that about 95.73% of students received scholarship amounts less than or equal to $4,300.

Since we want to find the percentage who did not receive enough to cover full tuition, we subtract the cumulative probability from 1 (1 - 0.9573 = 0.0427).

Therefore, about 4.27% of the students who received a merit scholarship received enough to cover full tuition. Conversely, approximately 95.73% did not receive enough to cover full tuition.