Question

Suppose a preliminary screening is given to prospective student athletes at a university to determine whether they would qualify for a scholarship. The scores are approximately normal with a mean of 85 and a standard deviation of 20. If the range of possible scores is 0 to 100, what percentage of students has a score less than 85?

Answer 1

80

Explanation:

Answer 2

50% of students has a score less than 85

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 = 85, \sigma = 20`

What percentage of students has a score less than 85?

This is the pvalue of Z when X = 85. So

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

`Z = (85 - 85)/(20)`

`Z = 0`

`Z = 0` has a pvalue of 0.5

50% of students has a score less than 85