Here is a simple probability model for multiplechoice tests

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.

4. 0/3 points | Previous Answers My Notes 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. Jodi is a good student for whom p = 0.84. Use 4 decimal places (a) Use the normal approximation to find the probability that Jodi scores 79% or lower on a 100-question test. (b) If the test contains 250 questions, what is the probability that Jodi will score 79% 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? Enter a number

Solution

p = 0.84

n = 100

SD = sqrt((0.84)(1 - 0.84) / 100) = .0366606

P[p <= 0.79) to be found

z = (x - u) / SD

z = (0.79 - 0.84) / .0366606 = -1.36386202080707898943279706278675198987468835752824558245

P(z < -1.36386202080707898943279706278675198987468835752824558245) = 0.0863 --> ANSWER

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n = 250 , p = 0.84

SD = sqrt(0.84 (1 - 0.84) / 250)
SD =0.0231862027939031017179498626514716347570242125996143569419134796

z = (x - u) / SD

z = (0.79 - 0.84) / 0.0231862027939031017179498626514716347570242125996143569419134796

z = -2.1564548729448569306128964519597874588006150111248471858178459619

P(z < -2.1564548729448569306128964519597874588006150111248471858178459619) is 0.0155

0.0155 ---> SECOND ANSWER

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On a 100 question test, her SD is :

SD = sqrt((0.84 * 0.16) / 100) = 0.0366606055596467

Half of that is : 0.01833030277982335

So, we get :

SD = sqrt(p(1 - p) / n)

0.01833030277982335 = sqrt(0.84 * 0.16 / n)

Squaring both sides :

0.000336 = 0.84 * 0.16 / n

n = (0.84 * 0.16) / 0.000336

n = 400 ----> THIRD ANSWER