Question

(Normal approximation to Binomial) Suppose that each student has probability p of correctly answering a question chosen at random from a large inventory of questions. The correctness of answer to a question is independent of the correctness of answers to other questions. An exam consisting of n questions can be considered as a simple random sample of size n from the population of all questions. Therefore, the number of correct answers X in an exam of n questions is a Binomial(n, p) random variable. This is true for every student writing the test. However, the success probability p varies from student to student.

Assume that each question is worth 1 point, that is, the maximum marks is n. In the following questions, use Normal approximation to compute the Binomial probabilities. Use continuity correction,

1. Roxi is a good student for whom p = 0.85. Compute the probability that Roxi will score 90% or better on a 100 question exam.
2. If the exam contains 200 questions, what is the probability that Roxi will score 90% or better? Note that 90% of 200 is 180.
3. Zulekha is a weaker student for whom p = 0.65. Suppose the passing grade for the exam is 70%. What is the probability that Zulekha will pass a 200 questions exam?
4. Suppose Zulekha is given a choice to write a 100 questions exam instead of 200 questions exam. Should she write the shorter exam? Show your calculations to justify your answer.

Answer 1