Question

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

Short answer

The coefficient of variation for waiting times corresponding to the single customer line is equal to 6.7%.

The coefficient of variation for waiting times corresponding to the individual customer lines is equal to 25.5%.

There is a significantly large difference in the variation of the waiting times of the single customer line and the individual customer lines.

Step-by-step solution

Given information

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

Formula for the coefficient of variation

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.

Calculation of sample means 

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.

Calculations for sample standard deviations

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.

Calculations for sample coefficients of variation

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%.

Comparison of the two coefficient of variation measure

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.