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 = 0.8.

(a) Use the Normal approximation to find the probability that Jodi scores 76% or lower on a 100-question test. (Round your answer to four decimal places.)
1

(b) If the test contains 250 questions, what is the probability that Jodi will score 76% or lower? (Use the normal approximation. Round your answer to four decimal places.)
2

(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?
3 questions

(d) Laura is a weaker student for whom p = 0.75. Does the answer you gave in (c) for standard deviation of Jodi's score apply to Laura's standard deviation also?

4

Yes, the smaller p for Laura has no effect on the relationship between the number of questions and the standard deviation. No, the smaller p for Laura alters the relationship between the number of questions and the standard deviation.    

Answer 1

(a)

Given,

p = 0.8

n = 100

Using Normal approximation, the score will follow normal distribution with mean = np = 100 * 0.8 = 80 and

variance = np(1-p) = 100 * 0.8 * (1 - 0.8) = 16

standard deviation = = 4

Probability that Jodi scores 76% or lower on a 100-question test = P(X 76)

= P[Z (76 - 80) / 4]

= P{Z -1]

= 0.1587

(b)

76% of 250 = 0.76 * 250 = 190

Using Normal approximation, the score will follow normal distribution with mean = np = 250 * 0.8 = 200 and

variance = np(1-p) = 250 * 0.8 * (1 - 0.8) = 40

standard deviation = = 6.324555

Probability that Jodi scores 76% or lower on a 250-question test = P(X 190)

= P[Z (190 - 200)/ 6.324555]

= P{Z -1.58]

= 0.0571

(c)

Let N be the required questions.

Standard deviation of Jodi's proportion of correct answers for a 100-item test = 4

To reduce the standard deviation of Jodi's proportion of correct answers to half its value for a 100-item test,

Np(1-p) = (4/2)²

=> N 0.8 * (1 - 0.8) = 4

=> N = 4 / 0.16

=> N = 25

(d)

No, since the standard deviation depends on probability p.

For p = 0.75

=> N 0.75* (1 - 0.75) = 4

=> N = 4 / 0.1875

=> N = 21.33

The answer is,

No, the smaller p for Laura alters the relationship between the number of questions and the standard deviation.