Question

The HR/Payroll office at a given firm is responsible for processing all requests for reimbursement of business related expenses. For all expenses less than $200, employees of the company use their own money for the initial charge. They then collect itemized receipts for the purchase, complete a reimbursement form, obtain their supervisor’s signature on the form, and then submit the receipts with completed form to the HR/Payroll Office. If the expenses are approved, employees receive a direct deposit of the same amount into their bank account within 14 business days. In an audit of reimbursement requests received in early January, the following receipts were submitted from six different employees who were seeking reimbursement: $157, $132, $109, $145, $125, and $139. Each was from a local restaurant, presumably for dining charges incurred for business purposes.

a. Compute the z-scores associated with each receipt. NOTE: To do this, you will first have to compute the sample mean and standard deviation.

b. Compute the 1, 2, and 3 sigma limits for the receipts in this problem.

c. In the context of the Empirical Rule and your answer to Part B above, what do you notice about the six receipts?

d. Would a receipt of $190 be considered an outlier? Please provide the rationale for your answer.

Answer 1

a)

sample mean = 134.5
std. dev. = 16.6343
n = 6

x	xbar	z = (x - xbar)/(s/sqrt(n))
157	134.5	3.3132
132	134.5	-0.3681
109	134.5	-3.7550
145	134.5	1.5462
125	134.5	-1.3989
139	134.5	0.6626

b)

sigma	1-sigma	2-sigma	3-sigma
16.6343	16.6343	33.2686	49.9029

1-sigma limits = (134.5 - 16.6343, 134.5 + 16.6343) = (117.8657, 151.1343)

2-sigma limits = (134.5 - 2*16.6343, 134.5 + 2*16.6343) = (101.2314, 167.7686)

3-sigma limits = (134.5 - 3*16.6343, 134.5 + 3*16.6343) = (84.5971, 184.4029)

c)

Around 68% of receipts's amount would lie within (117.8657, 151.1343)

Around 95% of receipts's amount would lie within (101.2314, 167.7686)

Around 99% of receipts's amount would lie within (84.5971, 184.4029)

d)

Yes, $190 would be considered an outlier. Because this is lie beyond 3-sigma level.