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This report was created by R version 3.6.3 (2020-02-29) and its package jfa (version 0.5.0)1. jfa provides Bayesian and classical audit sampling analyses and is available on CRAN.

jfa is free software and you can redistribute it and or modify it under the terms of the GNU GPL-3 as published by the Free Software Foundation. The package is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability of fitness for a particular purpose.

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Summary

The objective of this sampling procedure is to determine with a confidence of 95% whether the percentage of misstatement in the population is lower than the performance materiality of 17.425% ( = $2,000,000)

Table 1: Summary of the sampling results.
Sample size Deviations Total taint
130 6 NaN
a The total taint could not be calculated because
some ist values are zero.


The table below summarizes the estimated misstatement in the population, including the most likely error, the 95% upper bound, and the obtained precision. As a justification of these results, appendix A contains the derivations upon which they are based. Appendix B lists the input data.

Table 2: Summary of the inferential results.
Most likely error Lower bound Upper bound Precision
x 100 = % 0.000 -0.012 0.012 0.012
$ -1,383.038 -142,457.836 139,691.760 141,074.798
a The top row shows the misstatement as a fraction of the total value.

Appendix A: Derivations

These results have been calculated using the difference estimator (Touw & Hoogduin, 2011). The units of inference are individual transactions.

Notation:

\[\begin{equation}\begin{split}\text{Ist position: }& ist = \text{ist_euro} \\ \text{Soll position: }& soll = \text{soll_euro} \\ \text{Sample size: }& n = 130 \\ \text{Population size: }& N = 50,152\\ \text{Population value: }& B = 11,477,820 \\ \text{Sampling risk: }& \alpha = 0.05\end{split} \hspace{4cm} \begin{split} \text{Estimated population misstatement: }& \hat{E} \\ \text{Lower bound population misstatement: }& E_{0.025} \\ \text{Upper bound population misstatement: }& E_{0.975} \\ \text{Estimated unit misstatement: }& \hat{\theta} \end{split}\end{equation}\]

Calculation:

\(\begin{aligned} \text{The estimated population misstatemen}&\text{t } \hat{E}\text{ is a generalization of the estimated unit misstatement }\hat{\theta}\text{:} \\ \\ \hat{E} &= N \times \hat{\theta} \\ &= 50,152 \times -0.02757692 = -1,383.038 \\ \\ \text{The confidence interval for the populat}&\text{ion misstatement } [E_{0.025}; E_{0.975}] \text{ is a generalization of the unit upper and lower bounds:} \\ \\ [E_{0.025}; E_{0.975}] &= N \times \left( \hat{\theta} \pm t_{1 - \frac{\alpha}{2}} \times \frac{s_e}{\sqrt{n}} \right) \\ &= 50,152 \times \left(-0.02757692 \pm 2.812945\right) = [-142,457.84;139,691.76 ] \\ \\ \text{The other relevant quantities can be de}&\text{rived using the following formulas:} \\ \\ \hat{\theta} &= \frac{\sum_{i=1}^n ist_i - soll_i}{n} = \frac{-3.585}{130} = -0.02757692\\ t_{1 - \frac{\alpha}{2}} &= t_{0.975} = 1.979\\ s_e &= \sqrt{\frac{\sum_{i=1}^n ([ist_i - soll_i] - \hat{\theta})^2}{n - 1}} = \sqrt{\frac{33,897.88}{129}} =16.21031\\ t_{0.975} \times \frac{s_e}{\sqrt{n}} &= 1.979 \times \frac{16.21031}{\sqrt{130}} = 2.812945\end{aligned}\)

Evaluation:

✔️\(\hspace{0.5cm} \text{Upper bound lower than performance materiality} \rightarrow E_{0.975} < E_{max} \rightarrow 139,691.76<2,000,000\)

References:

Appendix B: Data

Table 3: Relevant data obtained from the input sample.
Row ist_euro soll_euro Difference Taint
96 0.000 91.546 -91.546 -Inf
97 63.770 63.770 0.000 0.000
98 0.000 0.000 0.000 NaN
99 1,451.754 1,451.754 0.000 0.000
100 106.608 106.608 0.000 0.000
101 162.621 162.621 0.000 0.000
102 1,795.500 1,795.500 0.000 0.000
103 129.715 129.715 0.000 0.000
104 138.892 0.000 138.892 1.000
105 78.000 78.000 0.000 0.000
106 102.451 102.451 0.000 0.000
107 36.457 36.457 0.000 0.000
108 139.936 139.936 0.000 0.000
109 22.000 22.000 0.000 0.000
110 0.000 0.000 0.000 NaN
111 1,807.029 1,807.029 0.000 0.000
112 0.000 0.000 0.000 NaN
113 433.680 433.680 0.000 0.000
114 0.000 0.000 0.000 NaN
115 0.000 0.000 0.000 NaN
116 0.000 0.000 0.000 NaN
117 0.000 0.000 0.000 NaN
118 34.603 34.603 0.000 0.000
119 37.800 37.800 0.000 0.000
120 28.256 28.256 0.000 0.000
121 35.568 35.568 0.000 0.000
122 72.146 72.146 0.000 0.000
123 11.684 11.684 0.000 0.000
124 0.000 0.000 0.000 NaN
125 50.494 50.494 0.000 0.000
126 54.032 54.032 0.000 0.000
127 114.840 114.840 0.000 0.000
128 12.937 12.937 0.000 0.000
129 36.994 36.994 0.000 0.000
130 43.738 43.738 0.000 0.000
131 49.603 49.603 0.000 0.000
132 765.191 765.191 0.000 0.000
133 68.560 68.560 0.000 0.000
134 71.040 71.040 0.000 0.000
135 36.648 36.648 0.000 0.000
136 17.201 22.934 -5.734 -0.333
137 40.219 40.219 0.000 0.000
138 72.944 72.944 0.000 0.000
139 1,060.050 1,060.050 0.000 0.000
140 44.160 44.160 0.000 0.000
141 0.000 0.000 0.000 NaN
142 28.026 28.026 0.000 0.000
143 420.350 420.350 0.000 0.000
144 304.560 304.560 0.000 0.000
145 0.000 72.248 -72.248 -Inf
146 20.176 20.176 0.000 0.000
147 54.828 54.828 0.000 0.000
148 50.286 50.286 0.000 0.000
149 343.992 347.901 -3.909 -0.011
150 76.045 76.045 0.000 0.000
151 30.745 30.745 0.000 0.000
152 51.177 51.177 0.000 0.000
153 157.489 157.489 0.000 0.000
154 46.211 46.211 0.000 0.000
155 502.632 502.632 0.000 0.000
156 195.200 195.200 0.000 0.000
157 55.958 55.958 0.000 0.000
158 146.364 146.364 0.000 0.000
159 84.806 84.806 0.000 0.000
160 226.044 226.044 0.000 0.000
161 115.306 115.306 0.000 0.000
162 268.320 237.360 30.960 0.115
163 55.094 55.094 0.000 0.000
164 17.080 17.080 0.000 0.000
165 0.000 0.000 0.000 NaN
166 68.560 68.560 0.000 0.000
167 2,059.260 2,059.260 0.000 0.000
168 64.500 64.500 0.000 0.000
169 84.395 84.395 0.000 0.000
170 17.286 17.286 0.000 0.000
171 85.160 85.160 0.000 0.000
172 21.600 21.600 0.000 0.000
173 13.032 13.032 0.000 0.000
174 31.500 31.500 0.000 0.000
175 40.952 40.952 0.000 0.000
176 47.760 47.760 0.000 0.000
177 38.271 38.271 0.000 0.000
178 17.219 17.219 0.000 0.000
179 51.480 51.480 0.000 0.000
180 221.707 221.707 0.000 0.000
181 491.882 491.882 0.000 0.000
182 32.191 32.191 0.000 0.000
183 59.944 59.944 0.000 0.000
184 49.024 49.024 0.000 0.000
185 37.357 37.357 0.000 0.000
186 730.850 730.850 0.000 0.000
187 95.856 95.856 0.000 0.000
188 1,178.573 1,178.573 0.000 0.000
189 39.040 39.040 0.000 0.000
190 34.596 34.596 0.000 0.000
191 38.550 38.550 0.000 0.000
192 26.046 26.046 0.000 0.000
193 2,418.150 2,418.150 0.000 0.000
194 519.360 519.360 0.000 0.000
195 0.000 0.000 0.000 NaN
196 23.680 23.680 0.000 0.000
197 53.060 53.060 0.000 0.000
198 19.667 19.667 0.000 0.000
199 51.341 51.341 0.000 0.000
200 0.000 0.000 0.000 NaN
201 28.346 28.346 0.000 0.000
202 0.000 0.000 0.000 NaN
203 22.374 22.374 0.000 0.000
204 3.494 3.494 0.000 0.000
205 31.120 31.120 0.000 0.000
206 72.674 72.674 0.000 0.000
207 13.183 13.183 0.000 0.000
208 17.862 17.862 0.000 0.000
209 70.442 70.442 0.000 0.000
210 52.104 52.104 0.000 0.000
211 10.104 10.104 0.000 0.000
212 25.027 25.027 0.000 0.000
213 224.100 224.100 0.000 0.000
214 43.738 43.738 0.000 0.000
215 812.818 812.818 0.000 0.000
216 170.395 170.395 0.000 0.000
217 75.900 75.900 0.000 0.000
218 72.240 72.240 0.000 0.000
219 18.609 18.609 0.000 0.000
220 35.830 35.830 0.000 0.000
221 1,066.464 1,066.464 0.000 0.000
222 758.880 758.880 0.000 0.000
223 795.648 795.648 0.000 0.000
224 37.440 37.440 0.000 0.000
225 39.200 39.200 0.000 0.000

  1. jfa’s source code can be found on its GitHub page↩︎