Question

Here is a simple model for multiple-choice tests. Suppose that each student has a probability of 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= 0.76. (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? (d) Laura is a weaker student for whom p= 0.71. Does the answer you gave in (c) for the standard deviation of Jodi's score apply to Laura's standard deviation also?

Answer 1

(c) This experiment can be modeled as Binomial Experiment, in which "Success" is defined as "A question answered correctly by a student" and "Number of trials" corresponds to total number of questions.

With this model, Jodi's proportion of correct answers is a Binomial Random variable with N = 100; and p = 0.76;

(d) Yes, answer to part (c) will STILL APPLY TO LAURA also because the factor of p(1-p) will get cancelled while writing new sigma in terms of old sigma value. So regardless of what value p takes, answer to (c) applies here too.

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