Here is a simple probability model for multiplechoice tests

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.

Use 4 decimal places.



(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?

Solution

a)

Proportion ( P ) =0.85
Standard Deviation ( sd )= Sqrt (P*Q /n) = Sqrt(0.85*0.15/100)
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
                  
P(X > 0.8) = (0.8-0.85)/0.0357
= -0.05/0.0357 = -1.4006
= P ( Z >-1.401) From Standard Normal Table
= 0.9193                  
              
P(X<=0.80) = 1 - 0.9193 = 0.0807
b)
P(X > 0.8) = (0.8-0.85)/0.0226
= -0.05/0.0226 = -2.2124
= P ( Z >-2.212) From Standard Normal Table
= 0.9865                  
P(X<=0.80) = 1 - 0.9865 = 0.0135