Question

Using the z-table, find the corresponding area between 𝑧 = 0

and each of the following:

1. z = 0.92

2. z = 1.29

3. z = 2.73

4. z = −0.50

5. z = −2.98

Answer 1

As we know from calculus, the area under the graph of a positive function $f(x)$ between two values $x_1 <x_2$ is equal to the integral $\int\limits_{x_1}^{x_2}f(x)dx =F(x_2)-F(x_1)$ , where F(x) is any antidevative of the function $f(x)$. If we take $f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ then its antiderivative is $\Phi(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{x}e^{-t^2/2}dt$, which is a cumulative function of distribution for standard normal distribution.

Using the table of CFD for standard normal distribution, we have

$\begin{matrix} z& -0.50 & 0.00 & 0.92& 1.29 & 2.73 & 2.98 \\ \Phi(z)& 0.3085 & 0.5000 & 0.8212 & 0.9015 & 0.9968 & 0.9986 \end{matrix}$

The area under the graph of density of standard normal distribution between 𝑧 = 0 and $z=a$ is:

if $a=-0.50$ then $S=\Phi(0.00)-\Phi(-0.50)=0.5000-0.3085=0.1915$

if $a=0.92$ then $S=\Phi(0.92)-\Phi(0.00)=0.8212-0.5000=0.3212$

if $a=1.29$ then $S=\Phi(1.29)-\Phi(0.00)=0.9015-0.5000=0.4015$

if $a=2.73$ then $S=\Phi(2.73)-\Phi(0.00)=0.9968-0.5000=0.4968$

if $a=2.98$ then $S=\Phi(2.98)-\Phi(0.00)=0.9986-0.5000=0.4986$