Question

Guessing on a Test The designers of a true-false test would like to find out how a student would perform if the student just guessed all the answers at random. (Designers of multiple-choice tests like the SAT have to consider this issue as well.) There are 100 questions on the test. One point is awarded for each correct answer and 1 point is taken away for any other kind of answer (wrong, missing, ambiguous, or anything other than correct). The test designers have figured out that the number of questions the student gets right by guessing at random has expectation 50 and SD 5 a) Let W be the number of questions the student does not get right. Find E(W) and SD(W). b) Let S be the student's score on the test. Find E(S) and SD(S).

Answer 1

Answer:

a) E(W) = 50

SD(W) = 5

b) E(S) = 0

SD(S) = 10

Step-by-step explanation:

Let X be the amount of answers the student got right by guessing. We know that E(X) = 50 and sd(X) = 5.

a) Note that W = 100-X because there are 100 questions and X are the ones solved right.

Thus E(W) = E(100-X) = E(100) - E(X) = 100 - 50 = 50.

Remember also that sd(aX+ b) = |a|* sd(X) for any constants a and cm therefore sd(W) = sd(100-X) = |-1| sd(X) = 5 (here a = -1 and b = 100).

b) Observe that S = X-W

E(S) = E(X-W) = E(X)-E(W) = 50-50 = 0

Recall that S = X-W = X-(100-X) = 2X-100, thus

sd(S) = sd(2X -100) = |2| sd(X) = 10 (here a = 2, b = -100).