Question

For students who guess at all answers, find the mean for the random variable X, the number of correct answers?

Answer 1

For each question there are four possible answers out of which only one is correct, thus the probability that a question is answered correctly is given by:

P(correctly answering a question), p = (Number of correct answers) / (Number of possible answers) = 1/4

Now, we are given that X is the random variable representing the number of correct answers (out of the 16 questions on the test) for a student who guesses at all answers.

Since, there is a fixed number of questions on the test (equal to 16), each question has two outcomes (correctly answered or incorrectly answered) and each question is correctly answered with probability 1/4 independent of other questions, thus we can conclude that:

X ~ Binomial(n = 16, p = 1/4)

Thus, the mean for the random variable X is given by:

E(X) = np [Using the formula for mean of a Binomial Distribution]

= 16*(1/4)

= 4 [ANSWER]