Question

Question 3     

Two examiners mark every script. Examiner A gives scores according to a Normal distribution with mean 65 and standard deviation of 10. Examiner B gives scores according to a Normal distribution with mean 55 and standard deviation of 15.

1. What is the threshold mark, given by examiner A, that is achieved by the top 20% of the students? What is the equivalent threshold mark given by examiner B?
2. If the pass mark in the exams is 50, what percentage of students pass by each of the examiners?
3. Is the probability that a student gets a mark between 55 and 65 higher by examiner A or by examiner B?

Answer 1

Let X be the marks given by examiner A and Y be the marks given by examiner B. then

a.

for finding the threshold mark, given by examiner A and B, that is achieved by the top 20% of the students we have to find the value of x at which the probability is

for we get the value for probability 0.2 as

x=73.4162

for examiner B threshold value of Y is

for we get value of y for probability 0.2 as

y=67.6243

we used normal distribution table here

or excel function of NORM.INV can also be used.

b.

if the pass mark in the exams is 50, percentage of students pass by each of examiners is given by

examiner A

BY STANDARD NORMAL DISTRIBUTION TABLE WE GET

so 93% of students pass by examiner A.

EXAMINER B

BY STANDARD NORMAL DISTRIBUTION TABLE WE GET

SO 63% of students pass by examiner B.

c.

the probability that a student gets a mark between 55 and 65 by examiner A is

for the mean and standard deviation value of X we get this value by standard normal distribution

the probability that a student gets a mark between 55 and 65 by examiner B is

so we conclude that Probability that a student gets a mark between 55 and 65 is higher by examiner A.

REMARK:- we can use excel function NORMSDIST or standard distribution table.