Answer :
To answer this questions we need to remember the standard score formula given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]where x is the value we are looking for, mu si the mean and sigma is the standard deviation.
a.
We need the probability:
[tex]P(IQ>95)[/tex]using the standard score this is equivalent to:
[tex]\begin{gathered} P(IQ>95)=P(z>\frac{95-100}{15}) \\ =P(z>-0.3333) \end{gathered}[/tex]Using a normal distribution table we have:
[tex]P(z>-0.3333)=0.6306[/tex]Therefore the probability to select a person with more than 95 IQ points is 63.1%.
b.
Following the same reasoning as before we have:
[tex]\begin{gathered} P(IQ<125)=P(z<\frac{125-100}{15}) \\ =P(z<1.6667) \\ =0.9522 \end{gathered}[/tex]Therefore the probability to select a person with less than 125 IQ points is 95.2%
c.
To find how many people of this sample have more less than 110 points we need to find that probability:
[tex]\begin{gathered} P(IQ<110)=P(z<\frac{100-110}{15}) \\ =P(z<-0.666) \\ =0.7475 \end{gathered}[/tex]Multiplying this value with the sample size we have
[tex](800)(0.7475)=598[/tex]Therefore 598 people will have an IQ less than 110.
d.
By the same reasoning as before we have:
[tex]\begin{gathered} P(IQ>140)=P(z>\frac{140-100}{15}) \\ =P(z>2.6667) \\ =0.0038 \end{gathered}[/tex]Multiplying this value with the sample size we have
[tex](800)(0.0038)=3[/tex]Therefore 3 people will have an IQ greater than 140.
