You are to take a multiple-choice exam consisting of 64 questions with two possible responses to each question. Suppose that you have not studied and so must guess (select one of the two answers in a completely random fashion) on each question. Let x represent the number of correct responses on the test.(a) What kind of probability distribution does x haveb) Compare the variance and standard deviation of x.

For each question, there are only two possible outcomes. Either the person answer it correctly, or the person answers it wrong. The probability of answering a question correctly is independent of other questions. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The variance of the binomial distribution is:

[tex]V(x) = np(1-p)[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

a) What kind of probability distribution does x haveb) Compare the variance and standard deviation of x.

Binomial

64 questions, so [tex]n = 64[/tex]

Each question is guessed, out of two possible answers. So [tex]p = \frac{1}{2} = 0.5[/tex]

[tex]V(x) = np(1-p) = 64*0.5*0.5 = 16[/tex]

[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = 4[/tex]

x has a binomial distribution. The variance of x is 16 and the standard deviation is 4.