1. Esteemed Insurance Company
In order to determine the likelihood of fraud committed by the wholesale furniture retailer store
against your company I was provided with legally obtained information regarding the selling
price, profit, profit margin, and month sold for the 3,005 furniture items that sold in the year
prior to the fire. By using this data I found that the retailer very likely committed fraud by
overestimating the amount lost in the fire. In my analysis is a discussion of my methods and the
limitations and weaknesses of my findings.
To analyze the data I used several statistical tests which calculated the odds that the furniture
retailer obtained the results that they did. My statistical tests show a more accurate way to
calculate how much lost profit the retailer company suffered as a result of the fire. These tests
will help your company determine how much the retailer overestimated the amount and
ultimately help you determine whether you should file a lawsuit and how much you should
demand in punitive damages.
Using these different methods I found that the furniture retailer very likely committed fraud. The
furniture retailer first claimed that they drew a random sample of 253 items from their 1991
inventory of 3,005 items. I found that their claim that a random sample of 253 items yielded the
lost profit that it did was most likely not true. My tests revealed that there is a 98.6% chance that
they would not have achieved these results. In terms of statistics anything higher than 95% is
considered very accurate, making my findings significantly condemning for the retailer.
After additional inquiry the retailer revealed that their random sample was in fact an average of
two random samples that they made from their 1991 inventory of 3,005 furniture items. Running
tests I found that their average was nearly 1.9% higher than the average that I calculated. In other
words, their estimate was nearly $30,400 more than my estimation. Now what is more significant
is the likelihood that they even reached the conclusion that they did. The chance that they drew
their first sample then drew what is equivalent to their second result is extremely small. In fact so
small that if they were to repeat this process 1,000 times they would only achieve their
approximate results 3 times out a 1,000. In other words, it is 99.7% likely that they did not
randomly draw these results.
In light of this overwhelming evidence a lawsuit seeking punitive damages is justified. As a
result of the unlikelihood of the retailer’s data, I conclude that the furniture retailer manipulated
the data or very likely chose a sample from a number of high profit items to demand more money
from their insurance agreement.
Although, my data shows an extremely high possibility that the retailer intentionally
overestimated lost profits, it is important to realize that there is a 0.3% chance that the retailer
obtained the results that they did. Additionally, there is the possibility that because of calculation
errors, my math, and as a result my conclusions could be wrong. Also important to consider is
the reality that the profit loss of the year before the fire does not take into account different
influencing factors such as inflation or an increased or decreased price in the product for that
industry. I recommend a more in-depth study analyzing changes in furniture prices and inflation
for the year 1992 to achieve more accurate results. Despite these limitations, I am confident that
these results will be of great assistance in your lawsuit.

2. Appendix
As I started my analysis of the data I assumed that the furniture retailer company was honest and
presented accurate data that they found through random sampling. I also assumed that their data
was accurate as they had large enough sample sizes allowing the central value theorem to apply.
First I collected the population mean and standard deviation using stata then found the
probability that the furniture company 1) had a random sample of 253 items and achieved a mean
of 50.8% for GPF, 2) had a random sample of 119 and a mean of 119 for GPF, 3) had a random
sample of 134 and had a mean of 50.6%. I then found the probability that the furniture retailer
would draw a random sample and first get a mean of 50.6% and then a second sample with a
mean of 51%. I found this by using the equation for the joint distribution for random variables. Finally
I calculated the amount the retailer overestimated assuming that an 1% increase in GPF results in an
additional $16,000 as stated in the data.
Equations Z-scores Probabilities Percentages
Pr(Y>50.8%) if n = 253 2.19 1 - 0.986 = 0.014 1.4%
Pr(Y>51%) n=119 1.65 1 - 0.95 = 0.05 5%
Pr(Y>50.6%) n=134 1.43 1 - 0.924 = 0.076 7.6%
Pr( 𝑋 = 𝑥) × Pr⁡( 𝑌 =) - 0.0038 0.38%
I first obtained the population mean by using the command sum margin in stata
Find population mean - $48.90
Standard deviation – 13.83
I then found the following probabilities/z scores given a certain sample size and Y value.
Pr(Y>50.8%) ifn = 253
Z=
𝑌−𝜇
(
𝜎
√ 𝑛
)
Y=50.8
𝜇 = 48.9
𝜎 = 13.83⁡(Standard deviation)
n = 253
Z=
50.8−48.9
(
13.83
15.9
)
Z=
1.9
0.869

3. Z=2.19
= 1-0.986
= 0.014
=1.4%
Pr(Y>51%) n=119
Z=
𝑌−𝜇
(
𝜎
√ 𝑛
)
Y=51
𝜇 = 48.9
𝜎 = 13.83⁡(Standard deviation)
n = 119
Z=
51−48.9
(
13.83
√119
)
Z=
2.1
(1.27)
Z=1.65
=0.95
=0.05
Pr(Y>50.6%) n=134
Z=
𝑌−𝜇
( 𝜎
√ 𝑛
)
Z =
50.6−48.9
(
13.83
√134
)
Z=
1.7
(
13.83
11.58
)
Z=
1.7
1.19
Z=1.43
= 0.924
=1-0.924
= 0.076

4. The equation for the joint distribution for random variables is
𝐏𝐫( 𝑿 = 𝒙,𝒀 = 𝒚) = 𝐏𝐫( 𝑿 = 𝒙) × 𝐏𝐫⁡( 𝒀 =)
=0.076× 0.05
=0.0038
The probability that the furniture retailer would draw a random sample and first get a mean of
50.6% and then second get a mean of 51% for a second sample is very small.
In fact if they were to draw this sample 1,000 times, they would get these results only 3 times out
of a 1,000.
50.8% - 48.9%
= 1.9%
Amount overestimated
= 1.9 × 16,000
=$30,400
(Equation for joint distribution on Pg. 31)
(Equation for the z score Slide 30, January 13th
)
(Received help from TA Mckenna)
(Checked answers with Noelle Goodine, Tatiana Flexman, Garrette Gibbson)

5. Grader’s Appendix
Stata Output
Do-file