Question

The marks of 500 candidates in an examination are normally distributed with a mean of 45

marks and a standard deviation of 20 marks.

(a) Given that the pass mark is 41, estimate the number of candidates who passed the

examination

(b) If 5% of the candidates obtain a distinction by scoring 𝑥 marks or more, estimate the value

of 𝑥.

Answer 1

Let $Y=$ the mark in an examination: $Y\sim N(\mu, \sigma^2)$

Given $\mu=45, \sigma=20.$

a)

$$P(Y\geq41)=1-P(Y<41)$$ $$=1-P(Z<\dfrac{41-45}{20})=1-P(Z<-0.2)$$ $$\approx0.579260$$ $$500(0.579260)=289.63$$

289 candidates passed the examination.

b)

$$P(Y\geq x)=1-P(Y<x)$$ $$=1-P(Z<\dfrac{x-45}{20})=0.05$$ $$P(Z<\dfrac{x-45}{20})=0.95$$ $$\dfrac{x-45}{20}=1.6449$$ $$x=78$$