Chapter 15.3, Problem 3WE

To determine

To calculate: The percent of data that is between 3 and 4 standard deviations below the mean in a standard normal distribution.

Answer to Problem 3WE

The percent of data that is between 3 and 4 standard deviations below the mean is $0.13\%$ .

Explanation of Solution

Given information:

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

In a standard normal curve the area under it and between $x=0\text{ and }x=4$ is $0.5000$ .

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 $0.4987$ .In a standard normal curve the area under it and between $x=0\text{ and }x=4$ is $0.5000$ .

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.

Therefore, percent of data that is between 3 and 4 standard deviations below the mean

  $\begin{array}{c}A\left(3\le x\le 4\right)=A\left(4\right)-A\left(3\right)\\ =0.5000-0.4987\\ =0.0013\\ =0.13\%\end{array}$

Thus, the percent of data that is between 3 and 4 standard deviations below the mean is $0.13\%$ .