Chapter 3.2, Problem 32E

a.

To determine

Find the mean back-to-school expenditure for each group.

Check whether the data are consistent with the National Retail Federation’s report.

Answer to Problem 32E

The mean back-to-school expenditure for Freshman is 1,285 and the mean back-to-school expenditure for Seniors is 433.

Yes, the data are consistent with the National Retail Federation’s report.

Explanation of Solution

Calculation:

The given information is a sample data comparing the back-to-school expenditures for 25 freshmen and 20 seniors.

Software Procedure:

Step by step procedure to obtain the mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Freshman and Seniors.
• In Statistics select mean.
• Click OK.

Output using the MINITAB software is given below:

Thus, the mean back-to-school expenditure for Freshman is 1,285 and for Seniors is 433.

The data are consistent with the National Retail Federation’s report because the mean back-to-school expenditure for Freshman is higher than Seniors.

b.

To determine

Find the range for the expenditures in each group.

Answer to Problem 32E

The range for the expenditures in Freshman is 1,720 and for Seniors is 352.

Explanation of Solution

Calculation:

The range is calculated as follows:

Software Procedure:

Step by step procedure to obtain the range using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Freshman and Seniors.
• In Statistics select Range.
• Click OK.

Output using the MINITAB software is given below:

Thus, the range for the expenditures in Freshman is 1,720 and for Seniors is 352.

c.

To determine

Find the interquartile range for the expenditures in each group.

Answer to Problem 32E

The interquartile range for Freshman is 404 and for Seniors is 131.5.

Explanation of Solution

Calculation:

The formula for the location of $p^{th}$ percentile is given below:

$i=\left(\frac{p}{100}\right)\left(n\right)$

where n is the sample size.

For Freshmen:

The 25^th percentile is calculated as follows:

$\begin{array}{c}i=\left(\frac{25}{100}\right)\left(25\right)\\ =\frac{25\left(25\right)}{100}\\ =\frac{625}{100}\\ =6.25\end{array}$

Here, i is not an integer. Therefore, the 25^th percentile is the next integer greater than i. That is, the position of the 25^th percentile is 7^th position.

Thus, the first quartile is 1,079.

The 75^th percentile is calculated as follows:

$\begin{array}{c}i=\left(\frac{75}{100}\right)\left(25\right)\\ =\frac{75\left(25\right)}{100}\\ =\frac{1,875}{100}\\ =18.75\end{array}$

That is, the position of the 75^th percentile is 19^th position.

Thus, the third quartile is 1,475.

The IQR can be obtained as follows:

$IQR=Q_3-Q_1$

Substitute $Q_1=1,075$ and $Q_3=1,479$ in the formula.

$\begin{array}{c}IQR=1,479-1,075\\ =404\end{array}$

Thus, the IQR is 404.

Thus, the interquartile range for Freshman is 404.

For Seniors:

The 25^th percentile is calculated as follows:

$\begin{array}{c}i=\left(\frac{25}{100}\right)\left(20\right)\\ =\frac{25\left(20\right)}{100}\\ =\frac{500}{100}\\ =5\end{array}$

Here, i is an integer. Therefore, the 25^th percentile is the average of the values in the positions i and i+1.

The 25^th percentile is obtained below:

$\begin{array}{c}{25}^{\text{th}}\text{percentile = }\frac{368+373}{2}\\ =\frac{741}{2}\\ =370.5\end{array}$

Thus, the first quartile is 370.5.

The 75^th percentile is calculated as follows:

$\begin{array}{c}i=\left(\frac{75}{100}\right)\left(20\right)\\ =\frac{75\left(20\right)}{100}\\ =\frac{1,500}{100}\\ =15\end{array}$

Here, i is an integer. Therefore, the 75^th percentile is the average of the values in the positions i and i+1.

The 75^th percentile is obtained below:

$\begin{array}{c}{75}^{\text{th}}\text{percentile = }\frac{489+515}{2}\\ =\frac{1,004}{2}\\ =502\end{array}$

Thus, the third quartile is 502.

The IQR can be obtained as follows:

$IQR=Q_3-Q_1$

Substitute $Q_1=370.5$ and $Q_3=502$ in the formula.

$\begin{array}{c}IQR=502-370.5\\ =131.5\end{array}$

Thus, the IQR is 131.5.

Thus, the interquartile range for Seniors is 131.5.

d.

To determine

Find the standard deviation for the expenditures in each group.

Answer to Problem 32E

The standard deviation for the expenditures in Freshman is 367 and Seniors is 97.

Explanation of Solution

Calculation:

The standard deviation is calculated as follows:

Software Procedure:

Step by step procedure to obtain the standard deviation using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Freshman and Seniors.
• In Statistics select Standard deviation.
• Click OK.

Output using the MINITAB software is given below:

Thus, the standard deviation for the expenditures in Freshman is 367 and Seniors is 97.

e.

To determine

Explain whether freshmen or seniors have more variation in back-to-school expenditures.

Explanation of Solution

From parts (b) to (d), all measures of variability suggest that freshmen have more variation in back-to-school expenditures.