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 = 0.81.

(a) Use the Normal approximation to find the probability that Jodi scores 77% 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 77% or lower? (Use the normal approximation. Round your answer to four decimal places.)

Solution

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.

            standard error of P=sqrt(0.81*(1-0.81)/100)) =0.0392

z value for 77%, z=(0.77-0.81)/0.0392 =-1.02

P( z < -1.02) = 0.1539

standard error of P=sqrt(0.81*(1-0.81)/250)) =0.0248

z value for 77%, z=(0.77-0.81)/0.0248 = -1.61

P( z < -1.61) = 0.0537