Here is a simple probability model for multiple-choice tests. Suppose that each
ID: 3360351 • 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 = 0.81. (a) Use the normal approximation to find the probability that Jodi scores 76% or lower on a 100-question test.(Round the probablity to the 4th decimal place,) (b) If the test contains 250 questions, what is the probability that Jodi will score 76% or lower? (Round the probablity to the 4th decimal place,) (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
Mean = np = 100(.81) = 81
Standard deviation = sqrt [np(1-p)] = sqrt[100(.81)(.19)] = 3.923
We want to find: P( x <= 76)
= 81
= 3.923
standardize x to z = (x - ) /
P(x < 76) = P( z < (76-81) / 3.923)
= P(z < -1.274) = .102
(From Normal probability table)
b)
Mean = np = 250(.81) =202.5
Standard deviation = sqrt [np(1-p)] = sqrt[250(.81)(.19)] = 6.202
76% of 250 = 190
P( x <= 190) =
= 202.5
= 6.202
standardize x to z = (x - ) /
P(x < 190) = P( z < (190-202.5) / 6.202)
= P(z < -2.015) = 0.0217
c) sd = p(1- p) /n
if reduce 1/2 of sd =1/2 p(1-p) / n = p(1-p) / 2^2(n)
4*n =4*100 =400