Question

Suppose a random sample of 30 customers is taken to test a company’s claim that 85% of customers are satisfied with their dog food. Assume trials are independent. What is the probability that at least 22 of the customers are satisfied? Also, what is the probability 8 customers are not satisfied?

Answer 1

According to company's claim 85% of customers are satisfied which implies probability of a customer to be satisfied is 0.85

Let E be Experiment that whether a person is satisfied or not

assume a discrete random variable X = no. of customers satisfied out of 30 people in sample

Experiment E can have binomial distribution as:

i)no of total trials are constant i.e. 30

ii) Satisfaction of each customer is independent of other customers

iii) there are only two outcomes for each customer - satisfied or not satisfied with constant probability for each outcome

for this binomial distribution n=30 and probability of success( we take a customer being satisfied to be success) =p  

= 0.85

possible values of X =0,1,2,3,4, ------------ 27,28,29,30

Now , we need to calculate probability that atleast 22 Customers are satisfied i.e P(X>=22)

= P(X=22)+ P(X=23) -------- P(X=29) + P(X=30)

in binomial theorem (n,p) n= no of trials , p = probability of sucess

so

required probability    P(X>=22) = 0.9722

(b) also if 8 people are not satisfied means 22 people are satisfied i.e. we need to calculate P(X=22)

P(X=22) = 0.042