In Exercises 21–24, find the coefficient of variation for each of the two samples; then compare the variation. (The same data were used in Section 3-1.) 21.

Bank Queues Waiting times (in seconds) of customers at the Madison Savings Bank are recorded with two configurations: single customer line; individual customer lines.

Single Line 390 396 402 408 426 438 444 462 462 462

Individual Lines 252 324 348 372 402 462 462 510 558 600

The waiting times of customers are given under two categories: single customer line and individual customer line.

The coefficient of variation is a percentage of change in the standard deviation measure over the mean of data values.

$C.V.=\frac{s}{\overline{x}}\times 100$

, where

$s$is the sample standard deviation;

$\overline{x}$is the sample mean.

The mean waiting time for a single customer line is calculated as shown below:

$$\begin{gathered} {\overline{x}}_1=\frac{{\sum }_{i=1}^{n_1}{x}_{1i}}{n_1} \\ =\frac{\left(390+396+...+462\right)}{10} \\ =429.0 \end{gathered}$$

Thus, the mean waiting time for a single customer line is 429.0 seconds.

The mean waiting time for individual customer lines is calculated as follows:

$$\begin{gathered} {\overline{x}}_2=\frac{{\sum }_{i=1}^{n_2}{x}_{2i}}{n_2} \\ =\frac{\left(252+324+...+600\right)}{10} \\ =429.0 \end{gathered}$$

Thus, the mean waiting time for individual customer lines is 429.0 seconds.

The standard deviation for waiting time for a single customer line is given as follows:

$$\begin{gathered} s_1=\sqrt{\frac{{\sum }_{i=1}^n{\left(x_{1i}-{\overline{x}}_1\right)}^2}{n_1-1}} \\ =\sqrt{\frac{{\left(390-429.0\right)}^2+{\left(396-429.0\right)}^2+...+{\left(462-429.0\right)}^2}{10-1}} \\ =28.6 \end{gathered}$$

Thus, the standard deviation for waiting time for a single customer line is 28.6 seconds.

The standard deviation for waiting time for individual customer lines is given as follows:

$$\begin{gathered} s_2=\sqrt{\frac{{\sum }_{i=1}^{n_2}{\left(x_{2i}-{\overline{x}}_2\right)}^2}{n_2-1}} \\ =\sqrt{\frac{{\left(252-429.0\right)}^2+{\left(324-429.0\right)}^2+...+{\left(600-429.0\right)}^2}{10-1}} \\ =109.3 \end{gathered}$$

Thus, the standard deviation for waiting time for individual customer lines is 109.3 seconds.

The coefficient of variation for waiting time for a single customer line is computed as follows:

$$\begin{gathered} C{V}_1=\frac{s_1}{{\overline{x}}_1}\times 100 \\ =\frac{28.6}{429}\times 100 \\ =6.7\% \end{gathered}$$

Therefore, the coefficient of variation for waiting time for a single customer line is equal to 6.7%.

The coefficient of variation for waiting time for individual customer lines is computed as follows:

$$\begin{gathered} C{V}_2=\frac{s_2}{{\overline{x}}_2}\times 100 \\ =\frac{109.3}{429}\times 100 \\ =25.5\% \end{gathered}$$

Therefore, the coefficient of variation for waiting time for individual customer lines is equal to 25.5%.

The coefficient of variation for thewaiting time of individual customer lines isgreater than the coefficient of variation forwaiting time for individual customer lines by a large margin.

Hence, the waiting time for individual customer lines varies more than for a single customer line.