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 = 0.75.

Use 4 decimal places.



(a) Use the normal approximation to find the probability that Jodi scores 70% or lower on a 100-question test.

(b) If the test contains 250 questions, what is the probability that Jodi will score 70% 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?

Explanation / Answer

here p=0.75

mean =np=75

hence std error=(np(1-p))1/2 =4.33

also X=100*0.7=70

hence P(P<70)= P(Z<(70.5-75)/4.33)=P(Z<-1.0392)=0.1493

b) for n=250

mean =187.5

and std deviation =6.8465

X=nphat=250*0.7=175

hence P(X<=175)=P(Z<(175.5-187.5)/6.8465)=P(Z<-1.7527)=0.0398

c)for std deviation is inversely proportion to square root of sample size

hence sample size =4*100=400

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