Question

The manager of a large company that sells pet supplies online wants to increase sales by encouraging repeat purchases. The manager believes that if past customers are offered $10 off their next purchase, more than 40 percent of them will place an order. To investigate the belief, 90 customers who placed an order in the past year are selected at random. Each of the selected customers is sent an e-mail with a coupon for $10 off the next purchase if the order is placed within 30 days. Of those who receive the coupon, 38 place an order.
Required:
a. Is there convincing statistical evidence, at the significance level of a = 0.05, that the managerâs belief is correct? Complete the appropriate inference procedure to support your answer.
b. Based on your conclusion from part (a), which of the two errors, Type I or Type II, could have been made? Interpret the consequence of the error in context.

Answer 1

Answer:

There is not a convincing statistical evidence, at the significance level of a = 0.05, that the manager's belief is correct.

As the null hypothesis maybe false accepting it makes a type II error.

Step-by-step explanation:

Let the null and alternate hypotheses be

H0: p ≤ 0.4  vs Ha : p>0.4

q= 1-p= 1-0.4= 0.6

The significance level alpha is 0.05

The critical region for one tailed test is Z> ± 1.645

The sample proportion is p^= x/n= 38/90=0.42222

Using the z statistic

z= p^- p/ √pq

z= 0.422-0.4/ √0.4*0.6

z= 0.04536

Since the calculated value of z= 0.04536 does not lie in the critical region  Z> ± 1.645 we fail to reject null hypothesis.

There is not sufficient evidence to support the manager's claim.

Type I error is when we reject the true null hypothesis .

Type II error is when we accept the false null hypothesis .

As the null hypothesis maybe false accepting it makes a type II error.