Chapter A.9, Problem 118AYU

To determine

To find: A standard intelligence test has an average score of 100. According to statistical theory, of the people who take the test, the $2.5\%$ with the highest scores will have scores of more than $1.96\sigma$ above the average, where $\sigma$ (sigma, a number called the standard deviation) depends on the nature of the test. If $\sigma = 12$ for this test and there is (in principle) no upper limit to the score possible on the test. Write the interval of possible test scores of the people in the top $2.5\%$.

Answer to Problem 118AYU

$x > 123.52$

Explanation of Solution

Given:

A standard intelligence test has an average score of 100. According to statistical theory, of the people who take the test, the $2.5\%$ with the highest scores will have scores of more than $1.96\sigma$ above the average, where $\sigma$ (sigma, a number called the standard deviation) depends on the nature of the test. If $\sigma = 12$ for this test and there is (in principle) no upper limit to the score possible on the test.

Calculation:

Given that $2.5\%$ with the highest scores will have scores of more than $1.96\sigma$ above the average.

This means that $x$ has to be greater than $x > 100 + 1.96\sigma$

$\begin{array}{l}x > 100 + 1.96\left(12\right)\\ x > 100 + 23.52\\ x > 123.52\end{array}$

The interval of possible test scores of the people in the top $2.5\%$ is $x > 123.52$