Question

A person must score in the upper 7% of the population on an admissions test to qualify for membership in society catering to highly intelligent individuals. If test scores are normally distributed with a mean of 140 and a standard deviation of 15, what is the minimum score a person must have to qualify for the society? (Round your answer to the nearest integer.)

Answer 1

The minimum score a person must have to qualify for the society is 162.05

Explanation:

Problems of normally distributed samples can be 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:

Test scores are normally distributed with a mean of 140 and a standard deviation of 15. This means that
`\mu = 140, \sigma = 15`.

What is the minimum score a person must have to qualify for the society?

Since the person must score in the upper 7% of the population, this is the X when Z has a pvalue of 0.93.

This is
`Z = 1.47`.

So

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

`1.47 = (X - 140)/(15)`

`X - 140 = 15*1.47`

`X = 162.05`

The minimum score a person must have to qualify for the society is 162.05