Question

Find the indicated IQ score. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

Answer 1

Concepts and reason

A Z-score indicates the number of standard deviations an element is deviated from the mean. Z-scores may be positive or negative. The positive value indicates that the score is above the mean value. The negative value indicates that the score is below the mean value. The normal probability values can be determined with the help of Z-score.

Fundamentals

A Z-score can be calculated by using the following formula:

$Z = \frac{{X - \mu }}{\sigma }$

Here,

The value of the element is denoted by X.

The mean of the population is denoted by$\mu$.

The standard deviation of the population is denoted by$\sigma$.

The z scores for the standard normal distribution are given in table of areas under the normal distribution or using the Excel function ${\rm{ = NORM}}{\rm{.S}}{\rm{.INV(}}p{\rm{)}}$

The IQ scores are normally distributed with mean $\mu = 100$ and standard deviation$\sigma = 15$.

The x value in the graph covers the area of 0.9 to the right of the curve.

The z score obtained at the point of x is,

$\begin{array}{l}\\P\left( {Z > z} \right) = 0.9\\\\ \Rightarrow P\left( {Z \le z} \right) = 1 - 0.9\\\\ \Rightarrow P\left( {Z \le z} \right) = 0.1\\\\ \Rightarrow \Phi \left( z \right) = 0.1\\\\ \Rightarrow z = {z_{0.1}}\\\\ \Rightarrow z = - 1.28\\\end{array}$

The z score that covers 90% of the area to the right is obtained as –1.28.

The IQ score that corresponds to the z score obtained is,

Ans:

The IQ score that covers 90% of the data to its right is 80.8.