Question

Use the appropriate normal distribution to approximate the resulting binomial distribution.
The manager of Madison Finance Company has estimated that, because of a recession year, 6% of its 300 loan accounts will be delinquent. If the manager's estimate is correct, what is the probability that 23 or more of the accounts will be delinquent? ROUND YOUR ANSWER TO FOUR DECIMAL PLACES

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 300 * 0.06 ) = 18
Variance = n * P * Q = ( 300 * 0.06 * 0.94 ) = 16.92
Standard deviation = √(variance) = √(16.92) = 4.1134

P ( X >= 23 ) = ?
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 23 - 0.5 ) =P ( X > 22.5 )

P ( X > 22.5 ) = 1 - P ( X < 22.5 )   
Standardizing the value   
Z = ( X - µ ) / σ  
Z = ( 22.5 - 18 ) / 4.1134  
Z = 1.09  
P ( ( X - µ ) / σ ) > ( 22.5 - 18 ) / 4.1134 )  
P ( Z > 1.09 )   
P ( X > 22.5 ) = 1 - P ( Z < 1.09 )   
P ( X > 22.5 ) = 1 - 0.8621  
P ( X > 22.5 ) = 0.1379