Question

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 companys employees?
security service company: 1.3 1.7 1.5 1.7 1.4 1.4 1.6 1.8 1.4 1.7
other companies: 1.5 1.9 1.6 1.8 1.7 1.9 1.6 1.7 1.8 1.6
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.)

Explanation:

Step1: Calculate the mean for Security Service Company

Let \(x_1,x_2,\cdots,x_n\) be the data - points. The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\).
For Security Service Company data: \(n = 10\), \(\sum_{i=1}^{10}x_i=1.3 + 1.7+1.5+1.7+1.4+1.4+1.6+1.8+1.4+1.7 = 15.5\), \(\bar{x}_1=\frac{15.5}{10}=1.55\).

Step2: Calculate the standard deviation for Security Service Company

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\).
\(\sum_{i = 1}^{10}(x_i - 1.55)^2=(1.3 - 1.55)^2+(1.7 - 1.55)^2+\cdots+(1.7 - 1.55)^2\)
\(=( - 0.25)^2+0.15^2+\cdots+0.15^2=0.385\)
\(s_1=\sqrt{\frac{0.385}{9}}\approx0.206\).

Step3: Calculate the coefficient of variation for Security Service Company

The coefficient of variation \(CV=\frac{s}{\bar{x}}\times100\%\).
\(CV_1=\frac{0.206}{1.55}\times100\%\approx13.3\%\).

Step4: Calculate the mean for Other Companies

For Other Companies data: \(n = 10\), \(\sum_{i=1}^{10}x_i=1.5+1.9+1.6+1.8+1.7+1.9+1.6+1.7+1.8+1.6 = 17.1\), \(\bar{x}_2=\frac{17.1}{10}=1.71\).

Step5: Calculate the standard deviation for Other Companies

\(\sum_{i = 1}^{10}(x_i - 1.71)^2=(1.5 - 1.71)^2+(1.9 - 1.71)^2+\cdots+(1.6 - 1.71)^2\)
\(=( - 0.21)^2+0.19^2+\cdots+( - 0.11)^2=0.289\)
\(s_2=\sqrt{\frac{0.289}{9}}\approx0.179\).

Step6: Calculate the coefficient of variation for Other Companies

\(CV_2=\frac{0.179}{1.71}\times100\%\approx10.5\%\).

Step7: Compare the coefficients of variation

The coefficient of variation for the security service company is approximately \(13.3\%\) and for other companies is approximately \(10.5\%\). The security - service company has a higher coefficient of variation, indicating more relative variability in the amounts collected.

Answer:

13.3