Chapter 6.1, Problem 5E

As an example of a situation in which several different statistics could reasonably be used to calculate a point estimate, consider a population of N invoices. Associated with each invoice is its “book value" the recorded amount of that invoice. Let T denote the total book value, a known amount. Some of these book values are erroneous. An audit will be carried out by randomly selecting n invoices and determining the audited (correct) value for each one. Suppose that the sample gives the following results (in dollars).

			Invoice
	1	2	3	4	5
Book value	300	720	526	200	127
Audited value	300	520	526	200	157
Error	0	200	0	0	-30

Let

$\bar{Y}$ = sample mean book value

$\bar{X}$ = sample mean audited value

$\bar{D}$ = sample mean error

Propose three different statistics for estimating the total audited (i.e., correct) value—one involving just N and $\bar{X}$, another involving T, N, and $\bar{D}$, and the last involving T and $\bar{X}$/ $\bar{Y}$. If N = 5000 and T = 1,761,300, calculate the three corresponding point estimates. (The article “Statistical Models and Analysis in Auditing.” Statistical Science, 1989: 2-33 discusses properties of these estimators.)