Here is a simple probability model for multiple-choice tests. Suppose that each
ID: 3363899 • 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 = 0.77.
Yes, the smaller p for Laura has no effect on the relationship between the number of questions and the standard deviation.No, the smaller p for Laura alters the relationship between the number of questions and the standard deviation.
Explanation / Answer
Solution:- p = 0.77
a) Mean = np = 100(.77) = 77
Standard deviation = sqrt [np(1-p)] = sqrt[100(.77)(.23)] = 4.2083
P( x <= 72)
= 77
= 4.2083
standardize x to z = (x - ) /
P(x < 72) = P( z < (72-77) / 4.2083)
= P(z < -1.1881)
= 1 P(Z < 1.1881)
= 1 0.883
= 0.1170
b) Mean = np = 250(.77) = 192.5
Standard deviation = sqrt [np(1-p)] = sqrt[250(.77)(.23)] = 6.6539
72% of 250 = 180
P( x <= 180) =
= 192.5
= 6.6539
standardize x to z = (x - ) /
P(x < 180) = P( z < (180-192.5) / 6.6539)
= P(z < -1.8786)
= 1 P(Z < 1.8786)
= 1 0.9699
= 0.0301
c) s100 /sn = 2/1 => sqrt(n/100) => n = 400
d) Yes, the smaller p for Laura has no effect on the relationship between the number of questions and the standard deviation.