Question

Problem 5-4: In a binomial distribution, n = 12 and π = .60. Find the following probabilities.

1. x = 5.
2. x ≤ 5.
3. x ≥ 6.

Assume a binomial distribution where n = 5 and π = .30.

1. Determine the mean and standard deviation of the distribution from the general definitions of the formulas or template.

Answer 1

Solution:

Problem 5-4: In a binomial distribution, n = 12 and π = .60.

Using binomial probability formula ,

P(X = x) = (_n C _x) * π^x * (1 - π)^{n - x} ; x = 0 ,1 , 2 , ....., n

a)

P(X = 5) = (₁₂C ₅) * 0.60⁵ * (1 - 0.60)^{12 - 5} = 0.10090237133

P(X = 5) = 0.10090237133

b)

P(X 5 )

=   P(X=0) + P(X=1) + P(X=2) + P(X=3) +  P(X = 4) + P(X = 5)

= (₁₂C ₀) * 0.60⁰ * (1 - 0.60)^{12 - 0} + (₁₂C ₁) * 0.60¹ * (1 - 0.60)^{12 - 1} +(₁₂C ₂) * 0.60² * (1 - 0.60)^{12 - 2} +(₁₂C ₃) * 0.60³ * (1 - 0.60)^{12 - 3} +(₁₂C ₄) * 0.60⁴ * (1 - 0.60)^{12 - 4} +(₁₂C ₅) * 0.60⁵ * (1 - 0.60)^{12 -5}

=  0.00001677722+0.00030198989+0.00249141658+0.01245708288+0.04204265472+0.10090237

=  0.15821229261

P(X 5 ) = 0.15821229261

c)

P(X    6)

= 1 - P(X < 6)

= 1 - { P(X=0) + P(X=1) + P(X=2) + P(X=3) +  P(X = 4) + P(X = 5) }

= 1- {(₁₂C ₀) * 0.60⁰ * (1 - 0.60)^{12 - 0} + (₁₂C ₁) * 0.60¹ * (1 - 0.60)^{12 - 1} +(₁₂C ₂) * 0.60² * (1 - 0.60)^{12 - 2} +(₁₂C ₃) * 0.60³ * (1 - 0.60)^{12 - 3} +(₁₂C ₄) * 0.60⁴ * (1 - 0.60)^{12 - 4} +(₁₂C ₅) * 0.60⁵ * (1 - 0.60)^{12 -5} _}

= 1 - {0.15821229261}

= 0.84178770739

P(X    6) =  0.84178770739

Question :

Assume a binomial distribution where n = 5 and π = .30.

Determine the mean and standard deviation of the distribution from the general definitions of the formulas or template.

Mean = = n * π = 5 * 0.30 = 1.5

Mean = 1.5

Standard deviation = = [n *π * (1 - π )] = [5 * 0.30 (1 - 0.30) = 1.05 = 1.0247

Standard deviation = 1.0247