Question

It is estimated that 16% of those taking the quantitative methods portion of the certified public accountant (CPA) examination fail that section. Seventy students are taking the examination this Saturday.

  

a-1. How many would you expect to fail? (Round the final answer to 2 decimal places.)

Number of students           

a-2. What is the standard deviation? (Round the final answer to 2 decimal places.)

Standard deviation           

b. What is the probability that exactly eight students will fail? (Round the final answer to 4 decimal places.)

Probability           

c. What is the probability at least eight students will fail? (Round the final answer to 4 decimal places.)

Probability           

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 70 * 0.16 ) = 11.2
Variance = n * P * Q = ( 70 * 0.16 * 0.84 ) = 9.408
Standard deviation = √(variance) = √(9.408) = 3.0672

Part a-1)

Mean = n * P = ( 70 * 0.16 ) = 11.2

part a-2)

Standard deviation = √(variance) = √(9.408) = 3.07

Part c)

P ( X = 8 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 8 - 0.5 < X < 8 + 0.5 ) = P ( 7.5 < X < 8.5 )

X ~ N ( µ = 11.2 , σ = 3.0672 )
P ( 7.5 < X < 8.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 7.5 - 11.2 ) / 3.0672
Z = -1.2063
Z = ( 8.5 - 11.2 ) / 3.0672
Z = -0.8803
P ( -1.21 < Z < -0.88 )
P ( 7.5 < X < 8.5 ) = P ( Z < -0.88 ) - P ( Z < -1.21 )
P ( 7.5 < X < 8.5 ) = 0.1893 - 0.1139
P ( 7.5 < X < 8.5 ) = 0.0755

Part d)

P ( X >= 8 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 8 - 0.5 ) =P ( X > 7.5 )

X ~ N ( µ = 11.2 , σ = 3.0672 )
P ( X > 7.5 ) = 1 - P ( X < 7.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 7.5 - 11.2 ) / 3.0672
Z = -1.2063
P ( ( X - µ ) / σ ) > ( 7.5 - 11.2 ) / 3.0672 )
P ( Z > -1.2063 )
P ( X > 7.5 ) = 1 - P ( Z < -1.2063 )
P ( X > 7.5 ) = 1 - 0.1139
P ( X > 7.5 ) = 0.8861