Question

Assume a binomial probability distribution with n=55 and π=0.34. Compute the following: (Round all your z values to 2 decimal places.)

a. The mean and standard deviation of the random variable. (Round your "σ" to 4 decimal places and mean to 1 decimal place.)

b. The probability that X is 23 or more. (Use the rounded values found above. Round your answer to 4 decimal places.)

c. The probability that X is 13 or less. (Use the rounded values found above. Round your answer to 4 decimal places.)

Answer 1

Condition check for Normal Approximation to Binomial
n * P >= 10 = 55 * 0.34 = 18.7
n * (1 - P ) >= 10 = 55 * ( 1 - 0.34 ) = 36.3

part a)

Using Normal Approximation to Binomial
Mean = n * P = ( 55 * 0.34 ) = 18.7
Variance = n * P * Q = ( 55 * 0.34 * 0.66 ) = 12.342
Standard deviation = √(variance) = √(12.342) = 3.5131

Part b)

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

X ~ N ( µ = 18.7 , σ = 3.5131 )
P ( X > 22.5 ) = 1 - P ( X < 22.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 22.5 - 18.7 ) / 3.5131
Z = 1.08
P ( ( X - µ ) / σ ) > ( 22.5 - 18.7 ) / 3.5131 )
P ( Z > 1.08 )
P ( X > 22.5 ) = 1 - P ( Z < 1.08 )
P ( X > 22.5 ) = 1 - 0.8599
P ( X > 22.5 ) = 0.1401

Part c)

P ( X <= 13 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 13 + 0.5 ) = P ( X < 13.5 )

X ~ N ( µ = 18.7 , σ = 3.5131 )
P ( X < 13.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 13.5 - 18.7 ) / 3.5131
Z = -1.48
P ( ( X - µ ) / σ ) < ( 13.5 - 18.7 ) / 3.5131 )
P ( X < 13.5 ) = P ( Z < -1.48 )
P ( X < 13.5 ) = 0.0694