Question

Determne the area under the standard normal arve that lies between (a) Ζ:-165 and Z-165 (b) Ζ--051 and Ζ:0 and (c)2 (a) The area that lies between Z-165 and Z . 165i (Round to four decimal places as needed) -171 and Z:-001

Answer 1

As a general rule, we want to express all the negative Z values as positive and right tail probabilities (that is Z>)  as left tail probabilities (that is Z< ). We will use the standard normal tables.

a) The area that lies between Z=-1.65 and Z=1.65 is calculated as

$\begin{align*} P(-1.65<Z<1.65)&=P(Z<1.65)-P(Z<-1.65)\\ &=P(Z<1.65)-P(Z>1.65)\quad \text{due to the symmetry of normal distribution, }P(Z<-1.65)=P(Z>1.65)\\ &=P(Z<1.65)-(1-P(Z<1.65))\\ &=2\times P(Z<1.65) -1\\ &=2\times (0.5&plus;0.4505)-1\quad\text{using standard normal tables, get the area for z=1.65}\\ &=0.9010 \end{align*}$

ans: The area that lies between Z=-1.65 and Z=1.65 is 0.9010

b) The area that lies between Z=-0.51 and Z=0 is

$\begin{align*} P(-0.51<Z<0)&=P(Z<0)-P(Z<-0.51)\\ &=P(Z<0)-P(Z>0.51)\quad \text{due to the symmetry of normal distribution, }P(Z<-0.51)=P(Z>0.51)\\ &=P(Z<0)-(1-P(Z<0.51))\\ &=0.5-(1-(0.5&plus;0.1950))\quad\text{using standard normal tables, get the area for z=0.51}\\ &=0.1950 \end{align*}$

ans: The area that lies between Z=-0.51 and Z=0 is 0.1950

c) The area that lies between Z=-1.71 and Z=-0.01 is

$\begin{align*} P(-1.71<Z<-0.01)&=P(Z<-0.01)-P(Z<-1.71)\\ &=P(Z>0.01)-P(Z>1.71)\quad \text{due to the symmetry of normal distribution}\\ &=(1-P(Z<0.01))-(1-P(Z<1.71))\\ &=P(Z<1.71)-P(Z<0.01)\\ &=(0.5&plus;0.4564)-(0.5&plus;0.0040)\quad\text{using standard normal tables, get the area for z=1.71 and z=0.01}\\ &=0.4524 \end{align*}$

ans: The area that lies between Z=-1.71 and Z=-0.01 is 0.4524