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 **

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