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 Service Company:Security Service Company:
1.5
1.8
1.4
1.7
1.7
1.5
1.8
1.6
1.4
1.7
Other Companies:Other Companies:
1.8
1.9
1.6
1.7
1.6
1.9
1.6
1.5
1.8
1.7
Find the coefficient of variation for each of the two samples, then compare the variation.
The coefficient of variation for the amount collected by the security service company is _.
(Round to one decimal place as needed.)
The coefficient of variation for the amount collected by the other companies is _. (Round to one decimal place as needed.)
To find the coefficient of variation, you first need to calculate the standard deviation and mean for each set of data.
For the security service company's data:
1. Calculate the mean: add up all the values and divide by the total number of values. In this case, the mean is (1.5 + 1.8 + 1.4 + 1.7 + 1.7 + 1.5 + 1.8 + 1.6 + 1.4 + 1.7) / 10 = 1.64 million dollars.
2. Calculate the standard deviation: subtract the mean from each value and square the result. Add up all the squared results and divide by the total number of values. Finally, take the square root of this value. In this case, the standard deviation is √[((1.5-1.64)^2 + (1.8-1.64)^2 + (1.4-1.64)^2 + (1.7-1.64)^2 + (1.7-1.64)^2 + (1.5-1.64)^2 + (1.8-1.64)^2 + (1.6-1.64)^2 + (1.4-1.64)^2 + (1.7-1.64)^2) / 10] ≈ 0.13 million dollars.
3. Calculate the coefficient of variation: divide the standard deviation by the mean, and multiply by 100 to express it as a percentage. In this case, the coefficient of variation for the security service company is (0.13 / 1.64) * 100 ≈ 7.93%.
Now repeat the same steps for the other companies' data:
1. Calculate the mean: add up all the values and divide by the total number of values. In this case, the mean is (1.8 + 1.9 + 1.6 + 1.7 + 1.6 + 1.9 + 1.6 + 1.5 + 1.8 + 1.7) / 10 = 1.72 million dollars.
2. Calculate the standard deviation: subtract the mean from each value and square the result. Add up all the squared results and divide by the total number of values. Finally, take the square root of this value. In this case, the standard deviation is √[((1.8-1.72)^2 + (1.9-1.72)^2 + (1.6-1.72)^2 + (1.7-1.72)^2 + (1.6-1.72)^2 + (1.9-1.72)^2 + (1.6-1.72)^2 + (1.5-1.72)^2 + (1.8-1.72)^2 + (1.7-1.72)^2) / 10] ≈ 0.14 million dollars.
3. Calculate the coefficient of variation: divide the standard deviation by the mean, and multiply by 100 to express it as a percentage. In this case, the coefficient of variation for the other companies is (0.14 / 1.72) * 100 ≈ 8.14%.
Comparing the coefficients of variation, we can see that both the security service company and the other companies have similar variation in the amount collected from parking meters, but there is no evidence of stealing based solely on these limited data.
To calculate the coefficient of variation for each sample, you need to find the standard deviation and mean for each sample. Here are the steps to calculate the coefficient of variation for both the security service company and the other companies:
1. Calculate the mean for each sample:
Security Service Company:
Sum all values: 1.5 + 1.8 + 1.4 + 1.7 + 1.7 + 1.5 + 1.8 + 1.6 + 1.4 + 1.7 = 16.3
Mean = Sum of values / Number of values = 16.3 / 10 = 1.63 million dollars
Other Companies:
Sum all values: 1.8 + 1.9 + 1.6 + 1.7 + 1.6 + 1.9 + 1.6 + 1.5 + 1.8 + 1.7 = 17.7
Mean = Sum of values / Number of values = 17.7 / 10 = 1.77 million dollars
2. Calculate the standard deviation for each sample:
Security Service Company:
Subtract the mean from each value, square the result, and sum them all:
(1.5 - 1.63)^2 + (1.8 - 1.63)^2 + (1.4 - 1.63)^2 + (1.7 - 1.63)^2 + (1.7 - 1.63)^2 + (1.5 - 1.63)^2 + (1.8 - 1.63)^2 + (1.6 - 1.63)^2 + (1.4 - 1.63)^2 + (1.7 - 1.63)^2 = 0.467
Divide the sum by the number of values minus 1, and take the square root:
Standard Deviation = √(0.467 / (10 - 1)) = √(0.467 / 9) ≈ √0.0519 ≈ 0.228
Other Companies:
Subtract the mean from each value, square the result, and sum them all:
(1.8 - 1.77)^2 + (1.9 - 1.77)^2 + (1.6 - 1.77)^2 + (1.7 - 1.77)^2 + (1.6 - 1.77)^2 + (1.9 - 1.77)^2 + (1.6 - 1.77)^2 + (1.5 - 1.77)^2 + (1.8 - 1.77)^2 + (1.7 - 1.77)^2 = 0.311
Divide the sum by the number of values minus 1, and take the square root:
Standard Deviation = √(0.311 / (10 - 1)) = √(0.311 / 9) ≈ √0.0346 ≈ 0.186
3. Calculate the coefficient of variation for each sample:
Coefficient of Variation (Security Service Company) = (Standard Deviation / Mean) * 100
= (0.228 / 1.63) * 100 ≈ 13.98
Coefficient of Variation (Other Companies) = (Standard Deviation / Mean) * 100
= (0.186 / 1.77) * 100 ≈ 10.51
Therefore, the coefficient of variation for the amount collected by the security service company is approximately 13.98, and for the other companies, it is approximately 10.51.