Question

(Can this be done on a Ti-84?) A certain flight arrives on time 83 percent of the time. Suppose140 flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that

?(a) exactly 109 flights are on time.

?(b) at least 109 flights are on time.

?(c) fewer than 114 flights are on time.

?(d) between 114 and 128 inclusive are on time.

Answer 1

Solution:-

p = 0.83

n = 140

Mean = n × p

Mean = 140 × 0.83

Mean = 116.2

a) The probability that exactly 109 flights are on time is 0.021.

x = 109

By applying normal distruibution:-

z = - 1.162

P(z = - 1.162) = 0.021

b) The probability that at least 109 flights are on time is 0.8776.

x = 109

By applying normal distruibution:-

z = - 1.162

P(z > -1.162) = 0.8776

c) The probability that a fewer than 114 flights are on time is 0.3101.

x = 114

By applying normal distruibution:-

z = - 0.495

P(z < - 0.495) = 0.3101

d) The probability that between 114 and 128 inclusive are on time is 0.6857.

x₁ = 114

x₂ = 128

By applying normal distruibution:-

z₁ = - 0.495

z₂ = 2.65

P(- 0.495 < z < 2.65) = P(z > - 0.495) - P(z > 2.65)

P(- 0.495 < z < 2.65) = 0.6897 - 0.004

P(- 0.495 < z < 2.65) = 0.6857