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 = 0.79.

(a) Use the normal approximation to find the probability that Jodi scores 74% or lower on a 100-question test.(Round the probablity to the 4th decimal place,)

I got .1131 for this


(b) If the test contains 250 questions, what is the probability that Jodi will score 74% or lower? (Round the probablity to the 4th decimal place,)

I got .0268 for this


(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?

Stumped on how to do this

Explanation / Answer

Ans:

sampling distribution of sample proportions:

mean=p

standard deviation=sqrt(p(1-p)/n)

a)mean=0.79

standard deviation=sqrt(0.79*(1-0.79)/100)=0.0407

z=(0.74-0.79)/0.0407=-1.23

P(z<=-1.23)=0.1093

b)standard deviation=sqrt(0.79*(1-0.79)/250)=0.0258

z=(0.74-0.79)/0.0258=-1.94

P(z<=-1.94)=0.0262

c)As,standard deviation is inversely proportional to sqrt(n),so for reducing the standard deviation to half,number of questions must be increased 4 times i.e. 400

standard deviation=sqrt(0.79*(1-0.79)/400) i.e. just half of the standard deviation in part a)

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