Question

According to Marsha Bertrand’s book “Fraud!: How to Protect Yourself from Schemes, Scams, and Swindles”, the most common hook in franchise fraud is earnings misrepresentation. In the early 1990s, Tower Cleaning Systems, a commercial cleaning franchisor, promised potential franchisees that they'd have revenues normally distributed with a mean of $9,500 and a standard deviation $1200 per month. The company, however, provided no documentation to back that up. You, as a forensic accountant, wanted to justify this claim and collected some data from their current franchisees. You created a histogram to better see how data looks. The data indicated the following:

Revenue Amount (bins) #	# of Franchisees Less than
Less than $8,000	22
Between $8,000 and $9,000	25
Between $9,000 and $10,000	34
More than $10,000	28

Do you think what Tower telling to their potential franchisees is correct? Perform an analysis and make a suggestion to potential franchisees.
(Use Excel).

Answer 1

for x=8000, z=(8000-9500)/1200=-1.25

for x=9000, z=(9000-9500)/1200=-0.4167

for x=10000, z=(10000-9500)/1200=0.4167

P(X<8000)=P(Z<-1.25)=0.1057

P(8000<X<9000)=P(-1.25<Z<-0.4167)=P(Z<-0.4167)-P(Z<-1.25)=0.3384-0.1057=0.2327

P(9000<X<10000)=P(-0.4167<Z<0.4167)=P(Z<0.4167)-P(Z<-0.4167)=0.6616-0.3384=0.3232

P(X>10000)=P(Z>0.4167)=1-P(Z<0.4167)=1-0.6616=0.3384

now we use chi-square test and chi-square=sum((O-E)²/E)=13.58 with k-1=4-1=3 df

with null hypothesis H0: obeserved frequency=expected frequency

alternate hypothesis Ha: observed frequency is not equal to expected frequecny

critical chi-square(0.05,3)=7.81 is less than calculated chi-square=13.58, so reject null hypothesis and conclude that potential franchisees is not correct.

Revenue Amount (bins) #	Observed(O)	Probability	expected(E)	(O-E)	(O-E)2/E
Less than $8,000	22	0.1057	10.57	11.43	12.36
Between $8,000 and $9,000	25	0.2327	23.27	1.73	0.13
Between $9,000 and $10,000	34	0.3232	32.32	1.68	0.09
More than $10,000	28	0.3384	33.84	-5.84	1.01
		1	100	9	13.58