1. A student takes an exam with 25 multiple choice questions. Each question has

Question

1. A student takes an exam with 25 multiple choice questions. Each question has 4 possible answers, one of which is correct. The student gets 3 points for each correct answer and loses 1 point for each wrong answer, so his/her maximum (resp. minimum) final score is 75 (resp. -25). Consider a (hypothetical) student who answers each question at random. (a) Is the final score (number of points) of the student a Binomial random variable? (b) what is approximately the probability that the student’s final score (number of points) is larger than 12?

Explanation / Answer

Answer:

Part a

A final score of the student is a binomial random variable because each question has the probability of correct answer as 0.25. All trials of selecting answer to questions are independent from each other. It is given that student answers each question randomly.

Part b

We are given

n = 25,

p = 0.25,

q = 1 – p = 1 – 0.25 = 0.75,

n*p = 25*0.25 = 6.25,

n*q = 25*0.75 = 18.75,

n*p & n*q > 5, so normal approximation applicable.

Mean = n*p = 25*0.25 = 6.25

SD = sqrt(n*p*q) = sqrt(25*0.25*0.75) = 2.165064

We have to find P(X>12)

P(X>12) = 1 – P(X<12)

Z = (X – mean) / SD

Z = (12 - 6.25) / 2.165064

Z = 2.655811

P(Z<2.655811) = P(X<12) = 0.996044

P(X>12) = 1 - 0.996044

P(X>12) = 0.003956

Required probability = 0.003956

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