Question

To increase business, the manager of a sporting goods store is running a promotion in which a customer's bill can be selected at random to receive a discount. When a customer's bill is printed, a program in the cash register determines randomly whether the customer will receive a discount on the bill. The program was written to generate a discount with a probability of 0.25; that is, 25% of the bills get a discount in the long run. However, the owner is concerned the program is incorrect and is not generating the intended long-run proportion of 0.25.

The owner selects a random sample of bills and finds that only 22% of them received a discount. A confidence interval for p, the proportion of bills that will receive a discount in the long run, is 0.22 ± 0.04, and all conditions for inference are met.

Consider the confidence interval 0.22 ± 0.04.

Part A: Does the confidence interval provide convincing statistical evidence that the program is NOT working as intended? Justify your answer. (3 points)

Part B: Does the confidence interval provide convincing statistical evidence that the program generates the discount with a probability of 0.25? Justify your answer. (2 points)

Part C: A second random sample of bills is taken that is four times the size of the original sample. In the second sample, 22% of the bills received the discount. Determine the value of the margin of error based on the second sample of bills used to compute an interval for p with the same confidence level as that of the original interval. (2 points)

Part D: Based on the margin of error in part C obtained from the second sample, is the program working as intended? Justify your answer. (3 points) (10 points)

Answer 1

Ho :   p =    0.25
H1 :   p ╪   0.25

part A:

NO,confidence interval DOES NOT provide convincing statistical evidence that the program is NOT working as intended.

As CI is (0.18,0.26) and it contains 0.25, null hypothesis will not be rejected

..............

part B:

confidence interval IS providing convincing statistical evidence that the program generates the discount with a probability of 0.25

As CI is (0.18,0.26) and it contains 0.25, null hypothesis will not be rejected

............

part c:

if the number of sample becomes four times then,

value of margin of error = 1/2 of original value

so,

margin of error = 1/2 * 0.04 = 0.02

.............

part d:

now,

CI = 0.22+-0.02

(0.20 , 0.24)

now, CI does not contain 0.25, so null hypothesis will be rejected.

so,

the program is not working as intended

thanks . revert back for doubt