Question

onsider an exam which has n multiple choice questions. Each question has k possible answers, among which only one answer is correct. (a) Consider a student who chooses at random the answers for all questions of the exam. Let X be the number of correct answers of this student. What is distribution of X? What is the expected value of X? (b) To eliminate the effect of guessing, the instructor decides to mark the according to the following rule: for each correct answer he gives 1 exam mark, but for each incorrect answer he subtracts x marks with 0< x< 1. Find the value of x such that the average grade of a student who chooses at random the answers for all questions is 0. Hint: Denote by Y the grade of a student who chooses at random all answers, under the new rule. Express E(Y) in the equation E(Y) = 0. (c) Find the value x for a k 5 as a function of x. Solve 25 and multiple choice exam as above with n

Answer 1

accon ding to the given up bin (n,b) X E (x)= (n b) 6r-1)-(x-1) n-1 M-1- a1- m-1 ニの Linlnt,b) i (n-4) n-2)-(-2)m(n-1) -2 (n-2)- n(n-)p2 M-2 D-2 m(n-2) p2 Vr (x)Ex(X- E(x) -E() (4u MEMSM

a) X binomial (n,p) E (x) p= 1/k-Probabli ty ob Selechng Conect debine given E(Y)0-E X.1 (n-x * っ0 X + (n-x) E(x) E(x) +nn- (n- n) 1/A 4-4/K K-1 K-1 exhraion Reguned ince anegetve no. 1 4 O- 25 K-1