CAIE A-Level Mathematics AS 5.5.2 Normal Distribution Probabilities

[Maximum number: 8]

In a certain country, the heights of the adult population are normally distributed with mean 1.64 m and standard deviation 0.25 m .

(a)

Find the probability that an adult chosen at random from this country will have height greater than 1.93 m.

[ 3 ]

Let $H\sim N(1.64,0.25^2)$ . Then
$P(H>1.93)=P\left(Z>\frac{1.93-1.64}{0.25}\right)=P(Z>1.16).$
Therefore,
P(H>1.93)=1-0.8770=0.123.

(b)

In another country, the heights of the adult population are also normally distributed. 33% of the adult population have height less than 1.56 m. 25% of the adult population have height greater than 1.86 m.

Find the mean and the standard deviation of this distribution.

[ 5 ]

Let the heights be normally distributed with mean $\mu$ and standard deviation $\sigma$ . The two given percentages give
$\frac{1.56-\mu}{\sigma}=-0.44, \qquad \frac{1.86-\mu}{\sigma}=0.674.$
Solving these simultaneous equations gives
$\mu=1.68\text{ m}, \qquad \sigma=0.269\text{ m}.$

CAIE A-Level Mathematics AS 5.5.2 Problems Concerning a Variable X Questions

Exam points

CAIE A-Level Mathematics AS 5.5.2 Problems Concerning a Variable X Questions question 1

Question (a)

Question (b)