Chapter 15.4, Problem 4ST

To determine

To calculate: The percent of data that is between $x=-2.0\text{ and }x=1.0$ in a standard normal distribution.

Answer to Problem 4ST

The percent of data that is between $x=-2.0\text{ and }x=1.0$ in a standard normal distributionis $81.85\%$ .

Explanation of Solution

Given information:

In a standard normal curve the area under it and between $x=0\text{ and }x=1$ is $0.3413$ .

In a standard normal curve the area under it and between $x=0\text{ and }x=2$ is $0.4772$ .

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 $x=0\text{ and }x=1$ is $0.3413$ .

In a standard normal curve the area under it and between $x=0\text{ and }x=2$ is $0.4772$ .

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.

Area between $x=0\text{ and }x=2$ is same as area between $x=0\text{ and }x=-2$ .

Therefore, percent of data that is between $x=-2.0\text{ and }x=1.0$ in a standard normal distribution is,

  $\begin{array}{c}A\left(-2\le x\le 1\right)=A\left(1\right)-A\left(2\right)\\ =0.3414-\left(-0.4772\right)\\ =0.8185\\ =81.85\%\end{array}$

Thus, the percent of data that is between $x=-2.0\text{ and }x=1.0$ in a standard normal distribution is $81.85\%$ .