Answer to Question #129101 in Statistics and Probability for Manpreet Kaur

Question

4. 30% of the customers at XYZ Stores use a debit card to pay for their purchases. In a recent market research study, 12 of the Store’s over 20,000 customers on record was randomly selected for two focus groups on new payment technologies. (12 Marks)
a. Is the selection of 12 customers a binomial experiment? Please explain.
b. What is the probability that 5 of the 12 customers selected use a debit card to pay for their purchases?
c. What is the probability that at least 4 of the 12 customers selected use a debit card to pay for their purchases?
d. What is the probability that at most 4 of the 12 customers selected use a debit card to pay for their purchases

Answer 1

a)  Yes, it is Binomial experiment.

1. The experiment consists of repeated trials.

2. Each trial results in an outcome: a success or a failure.

3. The probability of success, p, remains constant from trial to trial.

4. The repeated trials are independent.

Let $X=$ the number of customers using a debit card to pay for their purchases:

$X\sim Bin (n, p)$

Given $p=0.3, n=12$

b)

$$P(X=5)=\dbinom{12}{5}(0.3)^5(1-0.3)^{12-5}\approx0.15849579$$

c)

$$P(X\geq4)=1-P(X=0)-P(X=1)-$$

$$-P(X=2)-P(X=3)\approx0.507484$$

d)

$$P(X\leq4)=P(X=0)+P(X=1)+P(X=2)+$$

$$+P(X=3)+P(X=4)\approx0.723655$$