Question

Question 2.6 [3, 3, 3]

Suppose that the height of seven-year old children is a normal random variable with mean 1.1m and

standard deviation 0.1m.

a) What proportion of seven-year old children are shorter than 0.95m?

b) What is the probability that a seven-year old child is taller than 1m but shorter than 1.175m?

c) Suppose that a family restaurant has a children’s play-place. The facilities in the play-place

can only be used by children below a certain height. What should the cut-off height be to

ensure that 99% of seven-year olds have access to the play-place?

Answer 1

a)

$$P(X<0.95)=P(Z<\dfrac{0.95-1.1}{0.1})$$

$$=P(Z<-1.5)=0.0668$$

b)

$$P(1<X<1.175)=P(Z<\dfrac{1.175-1.1}{0.1})$$

$$-P(Z\le\dfrac{1-1.1}{0.1})=P(Z<0.75)-P(Z\le-1)$$

$$=0.6147$$

c)

$$P(X<x)=P(Z<\dfrac{x-1.1}{0.1})=0.99$$

$$\dfrac{x-1.1}{0.1}=2.3263$$

$$x=0.23263+1.1$$

$$x=1.333\ m$$