Here is a simple probability model for multiple-choice tests. Suppose that each

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 = 0.85.

(a) Use the Normal approximation to find the probability that Jodi scores 80% 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 80% 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?

Explanation / Answer

a)here p=0.85

hence for sample size n=100 ; std errror =(p(1-p)/n)1/2 =0.0357

therefore P(X<0.8)=P(Z<(0.8-0.85)/0.0357)=P(Z<-1.4)=0.0807

b) for sample size n=250 ; std errorr =0.0226

therefore P(X<0.8)=P(Z<(0.8-0.85)/0.0226)=P(Z<-2.2140)=0.0134

c)for std error is inversely proportional to square root of sample size.

hence to reduce the standard deviation of Jodi's proportion of correct answers to half its value : sample size required =100*22 =400

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