The scoring of modern IQ tests is such that Intelligence Quotients (IQs) have No

Step 1
Given:
${\displaystyle \mu =Mean=100}$
${\displaystyle \sigma =\text{Standard deviation}=15}$
a) The z-score is the value decreased by the mean, divivded by the standard deviation.
${\displaystyle z=\frac{x-\mu }{\sigma }=\frac{80-100}{15}\approx -1.33}$
Determine the corresponding probability using the normal probability table in the appendix. ${\displaystyle P(Z<-1.33)}$ is given in the row starting with -1.3 and in the cilumn starting with 0.03 of the standard normal probability table in the appendix.
${\displaystyle P\left(X<80\right)=P(Z<-1,33)}$
${\displaystyle =0.0918}$
${\displaystyle =9.18\mathrm{\%}}$
Thus 9.18% of the people are expected to have an IQ less than 80.
Step 2
b) The z-score is the value decreased by the mean, divivded by the standard deviation.
${\displaystyle z=\frac{x-\mu }{\sigma }=\frac{120-100}{15}\approx 1.33}$
Determine the corresponding probability using the normal probability table in the appendix. ${\displaystyle P\left(Z<1.33\right)}$ is given in the row starting with 1.3 and in the cilumn starting with 0.03 of the standard normal probability table in the appendix.
${\displaystyle P\left(X>120\right)=P(Z>1,33)}$
${\displaystyle =1-P\left(Z<1.33\right)}$
${\displaystyle =1-0.9082}$
${\displaystyle =9.18\mathrm{\%}}$ Thus 9.18% of the people are expected to have an IQ less than 120.
c) The boundary of the top 2% has the property that ${\displaystyle 100\mathrm{\%}-2\mathrm{\%}=98\mathrm{\%}}$ of the people below this boundary.
Determine the critical value corresponding with a probability of 98% or 0.98 using the normal probability table in the appendix. We note that the closest probability 0.9798 is given in the row starting with "2.0" and in the column starting with "0.05:
${\displaystyle z=2.05}$
The corresponding value is then obtained by adding the product of the z-score and the standard deviation to the mean:
${\displaystyle x=\mu +z\sigma =100+2.05\left(15\right)=130.75}$
Thus the top 2% have an IQ of at least 130.75.