Question

Question 4 Essay Worth 10 points) (04.01 MC) To increase sales, an online dothing store began giving a 50% off coupon to random customers. Customers didn't know whether they would receive the coupon until after the final sale The website claimed that one in five customers received the coupon. Six customers each made purchases from the website. Let X = the number of customers that received the 50% off coupon Part A: Is X a binomial random variable? Explain (3 points) Part B: What is the mean and standard deviation of X? Provide an interpretation for each value in context. (4 points) Part C: Two of the six customers receive the coupon with their purchase. Is the store's daim accurate? Compute P(x 2 2) and use the result to justify your answer (3 points)

Answer 1

a) As one in five customers received, the coupon, therefore
P(received coupon) = 1/5 = 0.2

Therefore out of 6 customers, the number of customers who received the coupon here is modelled as:

$X \sim Bin(n = 6, p = 0.2)$

Therefore this is a binomial distribution here. This is because the probability of any customer receiving the coupon is same and equal to 0.2. Also the number of customers here is constant n = 6 here.

b) The mean and standard deviation of X here is computed as:

$\mu = np = 6*0.2 = 1.2$

o= np(1 - p) = 6 * 0.2 + (1 -0.2) = 0.9798

c) P(X >= 2) = 1 - P(X = 0) - P(X = 1) = 1 - 0.8⁶ - 6*0.8⁵*0.2 = 0.34464

As the probability here is 0.34464 > 0.05, therefore the given that two of the six customers received the coupon the event is not unusual here. Therefore we cannot reject the claims here.