Answer to Question #195784 in Statistics and Probability for Erick

Question

 The marks of 500 candidates in an examination are normally distributed 

with 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 obtained 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)

$$P(Y\geq x)=1-P(Y<x)$$

$$=1-P(Z<\dfrac{x-45}{20})=0.05$$

$$P(Z<\dfrac{x-45}{20})=0.95$$

$$\dfrac{x-45}{20}=1.6449$$

$$x=78$$