Question 1 - A-Level Maths

Exam Board	OCR MEI
Module	S3 (Statistics 3)
Year	2014
Session	June
Topic	Linear combinations of normal random variables
Type	Expectation and variance with context application

Let \(X\) be a random variable with variance \(\sigma ^ { 2 }\). The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are both distributed as \(X\). Write down the variances of \(X _ { 1 } + X _ { 2 }\) and \(2 X\); explain why they are different. A large company has produced an aptitude test which consists of three parts. The parts are called mathematical ability, spatial awareness and communication. The scores obtained by candidates in the three parts are continuous random variables \(X , Y\) and \(W\) which have been found to have independent Normal distributions with means and standard deviations as shown in the table.
Mean Standard deviation
Mathematical ability, \(X\) 30.1 5.1
Spatial awareness, \(Y\) 25.4 4.2
Communication, \(W\) 28.2 3.9
Find the probability that a randomly selected candidate obtains a score of less than 22 in the mathematical ability part of the test.
Find the probability that a randomly selected candidate obtains a total score of at least 100 in the whole test.
For a particular role in the company, the score \(2 X + Y\) is calculated. Find the score that is exceeded by only \(2 \%\) of candidates.
For a different role, a candidate must achieve a score in communication which is at least \(60 \%\) of the score obtained in mathematical ability. What proportion of candidates do not achieve this?