Chapter 15.3, Problem 4WE

To determine

To calculate: The percent of data that is within three standard deviation from mean in a standard normal distribution.

Answer to Problem 4WE

The percent of data that is within three standard deviation from meanis $99.74\%$ .

Explanation of Solution

Given information:

In a standard normal curve the area under it and between $x=0\text{ and }x=3$ is $0.4987$ .

Formula used:

In a standard normal curve the area between two points $x=a\text{ and }x=b$ is,

  $A\left(a\le x\le b\right)=A\left(b\right)-A\left(a\right)$

Calculation:

Consider the provided information that in a standard normal curve the area under it and between $x=0\text{ and }x=3$ is $0.4987$ .

Recall that the in a standard normal curve the area between two points $x=a\text{ and }x=b$ is,

  $A\left(a\le x\le b\right)=A\left(b\right)-A\left(a\right)$

In a standard normal curve mean is 0 and standard deviation is 1.

It is known that standard normal curve is symmetric about y -axis. Therefore, the area is twice the area between 0 and $3$ .

In a standard normal curve the area under it and between $x=0\text{ and }x=3$ is $0.4987$ .

  $\begin{array}{c}A\left(-3\le x\le 3\right)=2\cdot A\left(3\right)\\ =2\left(0.4987\right)\\ =0.9974\\ =99.74\%\end{array}$

Thus, the percent of data that is within three standard deviation from meanis $99.74\%$ .