Question

It is estimated that 40% of those taking the quantitative methods portion of the CPA examination fail that section. Ninety students are taking the exam this fall. Using a normal approximation of the binomial probabilities,

1. How many would you expect to fail?
2. What is the standard deviation?
3. What is the probability that exactly 34 students will fail?
4. What is the probability that at least 36 will fail?

Answer 1

n = 90

p = 0.4

q = 1-p = 0.6

np = 90*0.4 = 36 >5

nq = 90* 0.6 = 54 >5

A) number of students expected to fail

= np

= 90 * 0.4

= 36 (mean)

B) standard deviation

= √(npq)

= √(90 * 0.4 * 0.6)

= 4.65

C) probablity that exact 34 students will fail

p( x = 34)

= p( 33.5 < X < 34.5)

Z = X - mean / standard deviation

= p( 33.5 - 36/4.65 < Z < 34.5 -36/4.65)

= p( -0.54 < Z < -0.32)

= 0.2054 - 0.1255 [standard normal distribution table]

= 0.0799

D) probabality that atleast 36 will fail

p(X >= 36)

= p(x > 35.5)

= p( z > 35.5 - 36/4.65)

= p( z > -0.11)

= 0.0438 +0.5. [standard normal distribution table]

= 0.5438