Question

A student takes a multiple choice test with 40 questions. The probability that a student answers a given question correctly is 0.5, independent of all other questions. The probability that the student answers more than N questions correctly is greater than 0.10. The probability that the student answers more than N+1 questions correctly is less than 0.10. Calculate N using normal approximation with continuity correction. Answer is 23. Please explain your work.

Answer 1

Explanation:

X no of questions student answers is binomial with n =40 and p =0.5

If approximated to normal, X is Normal with

mean = np = 20 and variance = npq = 10

P(X>N) >0.10

We use std normal distribution table to get z value first then convert to x value

`z>-1.28`

So
`x>20-1.28(10) = 20-12.8 = 7.2`

This is with continuity correction.

Hence without continuity correction this equals 7.2-0.5 = 6.7

x>6.7

n = 7