Question

Suppose that 19% of all sales are for amounts greater than $1,000. In a random sample of 30 invoices, what is the probability that more than six of the invoices are for over $1,000? Select one:

a. 0.0560

b. 0.9440

c. 0.5557

d. None of the suggested answers are correct

e. 0.4443

Answer 1

Solution:

Given,

p = 19% = 0.19

1 - p = 1 - 0.19 = 0.81

n = 30

X follows the Binomial(30 , 0.19)

Using binomial probability formula ,

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

Find P(More than 6)

= 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) + P(X = 6) }

= 1 - { (₃₀ C ₀) * 0.19⁰ * 0.81^{30 - 0} + (₃₀ C ₁) * 0.19¹ * 0.81^{30 - 2} + (₃₀ C ₂) * 0.19² * 0.81^{30 - 2} + (₃₀ C ₃) * 0.19³ * 0.81^{30 - 3} + (₃₀ C ₄) * 0.19⁴ * 0.81^{30 - 4}+ (₃₀ C ₅) * 0.19⁵ * 0.81^{30 - 5} + (₃₀ C ₆) * 0.19⁶ * 0.81^{30 - 6} }

= 1 - { 0.0017970103+0.01264562804+0.04301074721+0.09416344657+0.14909212373+0.18185557808+0.17773950533}

= 1 - 0.66030403926

= 0.33969596074

= 0.3397

P(More than 6) = 0.3397

None of the suggested answers are correct

Option d is correct.