Question

A multiple choice exam has 40 questions, each with 5 possible answers. A well prepared student believes she has a probability of 0.5 of getting any particular question correct. She uses the normal approximation to determine the probability of getting less than 25 correct. In her first attempt she forgets to use the continuity correction and then corrects herself. Determine the probability both with and without the continuity correction. with continuity correction: 0.9386; without continuity correction: 0.9512 with continuity correction: 0.9334, without continuity correction: 0.9512 with continuity correction: 0.9334; without continuity correction: 0.9429 with continuity correction: 0.9222; without continuity correction: 0.9429 with continuity correction: 0.9222; without continuity correction: 0.9358

Answer 1

n = 40

P = 0.5

$\mu$ = np = 40 * 0.5 = 20

$\sigma$ = sqrt(np(1 - p))

= sqrt(40 * 0.5 * 0.5)

= 3.1623

with continuity correction

P(X < 25)

= P(X < 24)

= P((X - $\mu$ )/$\sigma$ < (24.5 - $\mu$ )/$\sigma$)

= P(Z < (24.5 - 20)/3.1623)

= P(Z < 1.42)

= 0.9222

without continuity correction

P(X < 25)

= P((X - $\mu$ )/$\sigma$ < (25 - $\mu$ )/$\sigma$)

= P(Z < (25 - 20)/3.1623)

= P(Z < 1.58)

= 0.9429

Option - D) with continuity correction: 0.9222; without continuity correction: 0.9429