SOLUTION: Here is a simple probability model for multiple-choice tests. Suppose that a student has probability p of correctly answering a question chosen at random from a universe of possibl

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Question 248978: Here is a simple probability model for multiple-choice tests. Suppose that a student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A good student has a higher p than a poor student.) The correctness of an answer to any specific question doesn't depend on other questions. A test contains n questions. Then the proportion of correct answers that a student gives is a sample proportion from an SRS of size n drawn from a population with population proportion p.
(a) Julie is a good student for whome p = 0.73. Find the probability that Julie scores 67% or lower on a 110 question test.
(c) How many questions must the test contain in order to reduce the standard deviation of Julie's proportion of correct answers to one-fourth its value for an 100-item test?
Answer by stanbon(75887) About Me  (Show Source):
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Here is a simple probability model for multiple-choice tests.
Suppose that a student has probability p of correctly answering a question chosen at random from a universe of possible questions.
(A good student has a higher p than a poor student.)
The correctness of an answer to any specific question doesn't depend on other questions.
A test contains n questions. Then the proportion of correct answers that a student gives is a sample proportion from an SRS of size n drawn from a population with population proportion p.

(a) Julie is a good student for whom p = 0.73. Find the probability that Julie scores 67% or lower on a 110 question test.
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mean = 0.73
std = sqrt{0.73*0.27/110] = 0.0391
z(0.67) = (0.67-0.73)/0.0391 = -1.5345
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P(0 <- phat <= 0.67) = P(-inf < z < -1.5345) = 0.0625
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(c) How many questions must the test contain in order to reduce the standard deviation of Julie's proportion of correct answers to one-fourth its value for an 100-item test?
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Since std = sqrt(pq/n)
(1/4)sqrt(pq/n) = sqrt(pq/16n)
16*110 = 1760
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Cheers,
Stan H.