Question

Suppose in a certain primary school, the weight of boys are normally distributed with mean 28kg and standard
deviation 2.25kg. A boy is picked out at random and his weight is measured.
a) Find the probability that his weight lies between 25kg and 30kg. Show all your work!
b) Find the probability that his weight is greaterthan 32.5kg. Show all your work!
c) Find the probability that his weight is lessthan 25kg or greaterthan 32.5kg. Additionally draw a
standard normal distribution curve and shade the area under the curve to represent these probabilities.
Show all your work!
Hint: You might have to convert the weight values to the Z values to obtain appropriate probabilities.

Answer 1

We are given:

$\mu =28,\:\sigma =2.25$

(a) We are required to find:(a) We are required to find:

$P(25 < x< 30 )$

Using the z-score formula, we have:

$P\left(25<x<30\right)=P\left(\frac{25-28}{2.25}<x<\frac{30-20}{2.25}\right)$

$=P\left(-1.33<x<0.89\right)$

$=P\left(z<0.89\right)-P\left(z<-1.33\right)$

Now using the standard normal table, we have:

$P\left(25<x<30\right)=P\left(z<0.89\right)-P\left(z<-1.33\right)$

$=0.8133-0.0918=0.7215$

(b) We are required to find:

$P\left(x>32.5\right)$

Now using the z-score formula, we have:

$P\left(x>32.5\right)=P\left(z>\frac{32.5-28}{2.25}\right)$

$=P\left(z>2\right)$

Now using the standard normal table, we have:

$P(x>32.5)=P(z>2)=0.0228$

(c) We are required to find:

$P\left(x<25\cup x>32.5\right)=P\left(x<25\right)+P\left(x>32.5\right)$

Using the z-score formula, we have:

$P\left(x<25\right)+P\left(x>32.5\right)=P\left(z<\frac{25-28}{2.25}\right)+P\left(z>\frac{32.5-28}{2.25}\right)\:$

$=P\left(z<-1.33\right)+P\left(z>2\right)$

Now using the standard normal table, we have:

$P\left(x<25\right)+P\left(x>32.5\right)=P\left(z<-1.33\right)+P\left(z>2\right)\:\:$

$=0.0918+0.0228=0.1146$