Reference: Chapter 5.1 Summary Here is a simple probability model for multiple-c

Question

Reference: Chapter 5.1 Summary 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. Jod is a good student for whom p 0.8 Use 4 decimal places. (a) Use the normal approximation to find the probability that Jodi scores 75% or lower on a 100-question test. (b) If the test contains 250 questions, what is the probability that Jodi will score 75% 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

Solution :-

(a) Mean () = 100(0.80) = 80

Standard deviation () = sqrt(np(1-p)) = sqrt(100*0.8*0.2) = sqrt(16) = 4

We want to find: P( x <= 75)

z = (x - ) /

P(x < 75) = P( z < (75-80) / 4)

z = -1.25

Therefore, the p-value = 0.1057

(b)
Mean = np = 250(0.80) = 200
Standard deviation = sqrt [np(1-p)] = sqrt[250(.80)(.20)] = 6.32455

75% of 250 = 187.5
P( x <= 187.5) =

= 200
= 6.32455
standardize x to z = (x - ) /
P(x < 187.5) = P( z < (187.5 - 200) / 6.32455)
= P(z < -1.9764) = 0.0241
(From Normal probability table)

(c) The standard deviation for proportions (not counts) in the 100 question test is 0.043301 (I used the formula for proportions).

Half of that is 0.021651.

0.021651 = sqrt[(0.20(0.80) / n] solve for n.

n = 341

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