Here is a simple probability model for multiple-choice tests. Suppose that each

Question

Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) The correctness of answers to different questions are independent. Jodi is a good student for whom p = 0.85.
(a) Use the normal approximation to find the probability that Jodi scores 80% or lower on a 100-question test.

(b) If the test contains 250 questions, what is the probability that Jodi will score 80% or lower?

(c) How many questions must the test contain in order to reduce the standard deviation of Jodi's proportion of correct answers to half its value for a 100-item test?

Explanation / Answer

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 = 0.85. Find the probability that Julie scores 80% or lower on a 100 question test.
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mean = 0.85
std = sqrt{0.85*0.15/100] = 0.0357
z(0.80) = (0.80-0.85)/0.0357 = -1.40056
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P(0 <- phat <= 0.80) = P(-inf < z < -1.40056) = 0.0808

b)mean = 0.85
std = sqrt{0.85*0.15/250] = 0.02258
z(0.80) = (0.80-0.85)/0.02258 = -2.2143
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P(0 <- phat <= 0.80) = P(-inf < z < -2.2143) = 0.0136

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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) = 0.85*0.15/100 = 0.0357
(1/2)sqrt(pq/n) = sqrt(pq/4n)

0.01785 = sqrt 0.85*0.15/4n

0.0003186 =0.85*0.15 /4n

0.001274n = 0.1275

n=100.39 or 100

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