Question

Suppose ACT Reading scores are normally distributed with a mean of 21.421.4 and a standard deviation of 5.95.9. A university plans to award scholarships to students whose scores are in the top 9%9%. What is the minimum score required for the scholarship

Answer 1

The minimum score required for the scholarship is 29.306.

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 = 21.4, \sigma = 5.9`

What is the minimum score required for the scholarship

This is X when Z has a pvalue of 1-0.09 = 0.91. So it is X when Z = 1.34.

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

`1.34 = (X - 21.4)/(5.9)`

`X - 21.4 = 1.34*5.9`

`X = 29.306`

The minimum score required for the scholarship is 29.306.