In: Statistics and Probability
Listed below are amounts (in millions of dollars) collected from parking meters by a security service company and other companies during similar time periods. Do the limited data listed here show evidence of stealing by the security service company's employees?
Security Servvice Company 1.5 1.6 1.4 1.3 1.7 1.4 1.6 1.7 1.5 1.8
Other Companies 1.8 1.9 1.6 1.7 1.6 1.9 1.5 1.7 1.7 1.7 1.6
The coefficient of variation for the amount collected by the security service company is %
The coefficient of variation for the amount collected by the other companies is %
Do the limited data listed here show evidence of stealing by the security service company's employees? Consider a difference of greater than 1% to be significant.
Solution:
Coefficient of variation for the amount collected by the security service company is
Mean = ( 1.5 + 1.6 + 1.4 + 1.3 + 1.7 + 1.4 + 1.6 + 1.7 + 1.5 +
1.8)/10
= 15.5/10
Mean = 1.55
Standard Deviation σ = √(1/10 - 1) x (( 1.5 - 1.55)2 +
(1.6 - 1.55)2 + (1.4 - 1.55)2 + (1.3 -
1.55)2 + (1.7 - 1.55)2 + (1.4 -
1.55)2 + (1.6 - 1.55)2 + (1.7 -
1.55)2 + (1.5 - 1.55)2 + (1.8 -
1.55)2)
= √(1/9) x ((-0.05)2 + (0.05)2 +
(-0.15)2 + (-0.25)2 + (0.15)2 +
(-0.15)2 + (0.05)2 + (0.15)2 +
(-0.05)2 + (0.25)2)
= √(0.1111) x ((0.0025) + (0.0025) + (0.0225) + (0.0625) + (0.0225)
+ (0.0225) + (0.0025) + (0.0225) + (0.0025) + (0.0625))
= √(0.1111) x (0.225)
= √(0.0249975)
= 0.1581
Cofficient of Varaiance =σ/ μ
=0.1581/1.55
Coefficient of Variance = 0.102
Coefficient of variation for the amount collected by the other companies is
Mean = ( 1.8 + 1.9 + 1.6 + 1.7 + 1.6 + 1.9 + 1.5 + 1.7 + 1.7 +
1.7 + 1.6)/11
= 18.7/11
Mean = 1.7
Standard Deviation σ = √(1/11 - 1) x (( 1.8 - 1.7)2 +
(1.9 - 1.7)2 + (1.6 - 1.7)2 + (1.7 -
1.7)2 + (1.6 - 1.7)2 + (1.9 -
1.7)2 + (1.5 - 1.7)2 + (1.7 -
1.7)2 + (1.7 - 1.7)2 + (1.7 -
1.7)2 + (1.6 - 1.7)2)
= √(1/10) x ((0.1)2 + (0.2)2 +
(-0.1)2 + (0)2 + (-0.1)2 +
(0.2)2 + (-0.2)2 + (0)2 +
(0)2 + (0)2 + (-0.1)2)
= √(0.1) x ((0.01) + (0.04) + (0.01) + (0) + (0.01) + (0.04) +
(0.04) + (0) + (0) + (0) + (0.01))
= √(0.1) x (0.16)
= √(0.016)
= 0.1265
Cofficient of Varaiance =σ/μ
=0.1265/1.7
Coefficient of Variance = 0.0744
Yes. There is a significant difference in the variation.