Human heights are one of the many biological random variable

Human heights are one of the many biological random variables that can be modelled by the normal distribution. The average height of Canadian women aged 18 and older is 163cm, while the average height for men is 177 cm. Assume the standard deviation for said Canadian men is 8cm.

a. What proportion of all men will be taller than 185cm?

b. What is the probability that randomly selected man will be between 170 and 185cm tall?

c. The height of the current PM is 188 cm tall. is this unusual height?

Solution

Mean ( u ) =177
Standard Deviation ( sd )=8
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 185) = (185-177)/8
= 8/8 = 1
= P ( Z >1) From Standard Normal Table
= 0.1587                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 170) = (170-177)/8
= -7/8 = -0.875
= P ( Z <-0.875) From Standard Normal Table
= 0.19079
P(X < 185) = (185-177)/8
= 8/8 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(170 < X < 185) = 0.84134-0.19079 = 0.6506                  
c)
P(X > 188) = (188-177)/8
= 11/8 = 1.375
= P ( Z >1.375) From Standard Normal Table
= 0.0846                  
No, it\'s not. we have above 5% are more than 188