Suppose you are given a multiple choice test that has 100 questions with 4 answe

Question

Suppose you are given a multiple choice test that has 100 questions with 4 answer choices each (assume that each question has exactlyone correct answer). To pass the test, you must correctly answer at least 76 questions.Furthermore, suppose that you didn’t study at all and will have to guess randomly.
a1) What is your probability of answering any one particular problem correctly?

a2) If you guess randomly and consider this test a Bernoulli trial with “success” = “correct answer”, determine the values of p, q, and n, as well as E(X) and SD(X) (clearly box in your nal answers to this problem).

a3) Use you answers to part a2) to explain why you are not very likely to pass with this strategy. (Hint: your answer should make reference to the expected value and standard deviation).

4. (Continued) Now suppose that you studied for the test enough to have an 80%probability of answering any one question correctly.
b1) Once again treating the test as a Bernoulli trial with S = correct answer, compute the new values of p, q, n, E(X) and SD(X) (again, box in your answers).

b2) What is the probability that you answer 4 out of the rst 5 questions correctly?

b3) What is the probability that you pass the test? Remember that you must answer at least 76 questions correctly. (Hint: You will need to use the Normal Approximation.)

b4) What is the probability that the 4th question is the rst one you answer correctly?

4. (Continued) Continue assuming that you have an 80% probability of answering anyone question correctly. Now, suppose that each correct answer is worth 1 point, but each incorrect answer is worth 0.2 points.
c1) If you answer every question, what is your expected score on the test?

c2) If you still have to score 76 or higher to pass, what is the new probability that you will pass?

I really need a help to answer those question with enough explanations! thank you!

Explanation / Answer

Suppose you are given a multiple choice test that has 100 questions with 4 answer choices each (assume that each question has exactlyone correct answer). To pass the test, you must correctly answer at least 76 questions.Furthermore, suppose that you didn’t study at all and will have to guess randomly.

a1) What is your probability of answering any one particular problem correctly?

There are 4 options, and only 1 of them is correct. Hence,

p = 1/4 = 0.25 [ANSWER]

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a2) If you guess randomly and consider this test a Bernoulli trial with “success” = “correct answer”, determine the values of p, q, and n, as well as E(X) and SD(X) (clearly box in your nal answers to this problem).

p = 0.25
q = 1 - p = 0.75
n = 100
E(x) = mean = n p =    25
SD(x) = standard deviation = sqrt(np(1-p)) =    4.330127019

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a3) Use you answers to part a2) to explain why you are not very likely to pass with this strategy. (Hint: your answer should make reference to the expected value and standard deviation).

The usual values is only up to 2 standard deviations above the mean. That is

E(x) + 2*SD(x) = 25 + 2*4.33 = 33.66.

You need 76 to pass though. So this is not a good strategy.

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