Question

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.

Answer 1

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

Explanation:

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:

`V(x) = np(1-p)`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

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

Binomial

64 questions, so
`n = 64`

Each question is guessed, out of two possible answers. So
`p = (1)/(2) = 0.5`

`V(x) = np(1-p) = 64*0.5*0.5 = 16`

`√(V(X)) = √(np(1-p)) = 4`

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