Here is a simple probability model for multiple-choice tests. Suppose that each
ID: 3132311 • Letter: H
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 = 0.79. (a) Use the normal approximation to find the probability that Jodi scores 74% or lower on a 100-question test. (Round your answer to four decimal places.) (b) If the test contains 250 questions, what is the probability that Jodi will score 74% or lower? (Use the normal approximation. Round your answer to four decimal places.) (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? questionsExplanation / Answer
a)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 0.74
u = mean = p = 0.79
s = standard deviation = sqrt(p(1-p)/n) = 0.040730824
Thus,
z = (x - u) / s = -1.22757154
Thus, using a table/technology, the left tailed area of this is
P(z < -1.22757154 ) = 0.109803925 [ANSWER]
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b)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 0.74
u = mean = p = 0.79
s = standard deviation = sqrt(p(1-p)/n) = 0.025760435
Thus,
z = (x - u) / s = -1.940961029
Thus, using a table/technology, the left tailed area of this is
P(z < -1.940961029 ) = 0.026131502 [ANSWER]
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c)
As the satndard deviation varies inversely as the square root of the sample size, then we need a sample size that is 4 times the original, 100, so
n = 4*100 = 400 questions [ANSWER]
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