Question

As an example of a situation in which several different statis tics 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

\({\rm{T}}\)- sample mean book value

\(\bar X\)- sample mean advised value

\({\rm{\bar D}}\)- sample mean errors

Propose three different statistics for estimating the total audited (i.e., correct) value-one involving just N and,another involving T, N, and \({\rm{\bar D,}}\) and the last involving T and \({\rm{\bar X/\bar Y}}{\rm{.}}\)If \({\rm{N = 5000}}\)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.)

Short answer

a.The estimated value is\({\rm{1,704,000}}\).

b.The estimated value is\({\rm{1,591,300}}\).

c.The estimated value is\({\rm{1,601,438}}{\rm{.281}}\).

Step-by-step solution

Concept introduction

The standard deviation (SD) is a measure of the variability, or dispersion, between individual data values and the mean, whereas the standard error of the mean (SEM) is a measure of how far the sample mean (average) of the data is expected to differ from the genuine population mean. Always, the SEM is smaller than the SD.

Calculating the total audited values

The total audited values are denoted by\({\theta _T}\).

First, using simply\(N\)and\(\bar X\)the following statistic can be employed.

\({{\rm{\hat \theta }}_{{{\rm{T}}_{\rm{1}}}}}{\rm{ = N \times \bar X}}\)

Which is simply the \(N\) invoice multiplied by the average.

Second, using\({\rm{N \times \bar X}}\), and\(\bar D\), the following statistic can be employed.

\({{\rm{\hat \theta }}_{{{\rm{T}}_{\rm{2}}}}}{\rm{ = T - N \times \bar D}}\)

When the total sample mean error is deducted from the total book value, the total book value is the result.

Third, using \(T\)and \(\bar X/\bar Y\),the following statistic can be employed.

This is the quotient of the sample mean audited value and the sample mean book average multiplied by the total book value: \({{\rm{\hat \theta }}_{{{\rm{T}}_{\rm{3}}}}}{\rm{ = T \times }}\frac{{{\rm{\bar X}}}}{{{\rm{\bar Y}}}}\)

Estimating the point using the invoice table

Consider the given,

\(\begin{array}{l}{\rm{N = 5000 }}\\{\rm{T = 1,761,300}}\end{array}\)

The point estimations can be determined using the invoice table and the invoice table

\(\begin{array}{c}{\rm{\bar y = }}\frac{{\rm{1}}}{{\rm{5}}}{\rm{(300 + 720 + 526 + 200 + 127)}}\\{\rm{ = 374}}{\rm{.6}}\end{array}\)

The average audited value in the sample is

\(\begin{array}{c}{\rm{\bar x = }}\frac{{\rm{1}}}{{\rm{5}}}{\rm{(300 + 520 + 526 + 200 + 157)}}\\{\rm{ = 340}}{\rm{.6,}}\end{array}\)

Where the standard deviation of the sample mean error is

\(\begin{array}{c}{\rm{\bar d = }}\frac{{\rm{1}}}{{\rm{5}}}{\rm{(0 + 200 + 0 + 0 - 30)}}\\{\rm{ = 34}}\end{array}\)

As a result, the point estimations

\(\begin{array}{c}{\rm{\& }}{{{\rm{\hat \theta }}}_{{{\rm{T}}_{\rm{1}}}}}{\rm{ = N \times \bar x = 5000 \times 340}}{\rm{.6}}\\{\rm{ = 1,704,000,}}\\{\rm{n\& }}{{{\rm{\hat \theta }}}_{{{\rm{T}}_{\rm{2}}}}}{\rm{ = T - N \times \bar d = 1,761,300 - 5,000 \times 34}}{\rm{.0}}\\{\rm{ = 1,591,300,}}\\{{{\rm{\hat \theta }}}_{{{\rm{T}}_{\rm{3}}}}}{\rm{ = T \times }}\frac{{{\rm{\bar x}}}}{{{\rm{\bar y}}}}\\{\rm{ = 1,761,300 \times }}\frac{{{\rm{340}}{\rm{.6}}}}{{{\rm{374}}{\rm{.6}}}}\\{\rm{ = 1,601,438}}{\rm{.281}}\end{array}\)

Therefore, the required estimated value is\({\rm{1,601,438}}{\rm{.281}}\).