Chapter 3, Problem 3.1P

SOCA variety of information has been gathered from a sample of college freshmen and seniors, including

• Their region of birth;

• The extent to which they support legalization of marijuana (measured on a scale on which 7 = strong support, 4 = neutral, and 1 = strong opposition);

• The amount of money they spend each week out-of-pocket for food, drinks, and entertainment;

• How many movies they watched in their dorm rooms last week;

• Their opinion of cafeteria food (10 = excellent, 0 = very bad); and

• Their religious affiliation.

Some results are presented here. Find the most appropriate measure of central tendency for each variable for freshmen and then for seniors. Report both the measure you selected and its value for each variable (e, g., “Mode = 3” or ‘Median = 3.5”). (HINT: Determine the level of measurement for each variable first. In general, this will tell you which measure of central tendency is appropriate. See the section “Choosing a Measure of Central Tendency” to review the relationship between measure of central tendency and level of measurement. Also, remember that the mode is the most common score, and especially remember to array scores from high to low before find the median.)

Freshmen
Student	Region of Birth	Legalization	Out-of-Pocket Expenses	Movies	Cafeteria Food	Religion
A	North	7	43	0	10	Protestant
B	North	4	49	14	7	Protestant
C	South	3	55	10	2	Catholic
D	Midwest	2	57	7	1	None
E	North	3	72	5	8	Protestant
F	North	5	58	1	6	Jew
G	South	1	62	0	10	Protestant
H	South	4	75	14	0	Other
I	Midwest	1	61	3	5	Other
J	West	2	53	4	6	Catholic

Seniors
Student	Region of Birth	Legalization	Out-of-Pocket Expenses	Movies	Cafeteria Food	Religion
K	North	7	75	0	1	None
L	Midwest	6	72	5	2	Protestant
M	North	7	70	11	8	Protestant
N	North	5	95	3	4	Catholic
O	South	1	72	4	3	Protestant
P	South	5	67	14	6	Protestant
Q	West	6	50	0	2	Catholic
R	West	7	59	7	9	None
T	West	5	95	3	7	Other
U	North	4	88	5	4	None

To determine

a. To find:

The percentages of the respondents in each complex are married

Answer to Problem 3.1P

Solution:

Variable	Measure	Freshmen	Seniors
Region of birth	Mode	North	North
Legalization	Median	3	5
Expenses	Mean	58.5	72.55
Movies	Mean	5.80	5.18
Food	Median	6	4
Religion	Mode	Protestant	Protestant, None

Explanation of Solution

Description:

When the variables measured have non-numerical scores or categories, they are said to have a nominal level of measurement.

When the variables measured have non-numerical scores or categories that could be ranked from high to low, they are said to have an ordinal level of measurement.

When the variables measured have numerical scores and can be used for further statistical analysis, they are said to have an interval ratio level of measurement.

From the given information, there are six variables Region of Birth, Legalization, % College Bound, Out-of-Pocket Expenses, Movies, Cafeteria Food, and Religion. Here, the variables Out-of-Pocket Expenses and Movies have a numeric value; therefore, the level of measurement of these variables is interval-ratio. The variables Region of Birth and Religion can be classified into categories; therefore, the level of measurement of these variables is Nominal. Further, the variables legalization and Cafeteria Food can be classified into categories and can be ranked; therefore, the level of measurement is Ordinal.

Choosing a Measures of Central Tendency

When the data is Nominal type — Mode

When the data is Ordinal type — Median

When the data is Interval ratio type — Mean

Freshmen:

For the variable Out-of-Pocket Expenses, the measure of central tendency is: Mean

Formula Used:

The formula to calculate mean is given by,

$\overline{X}=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Substitute 10 for $N$, 43 for $X_1$, 49 for $X_2$ and so on in the above mentioned formula,

$\begin{array}{rll}\overline{X}& =& \frac{43+49+\mathrm{...}+61+53}{10}\\ & =& \frac{585}{10}\\ & =& 58.5\end{array}$

For the variable Movies, the measure of central tendency is: Mean

Formula Used:

The formula to calculate mean is given by,

$\overline{X}=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Substitute 10 for $N$, 0 for $X_1$, 14 for $X_2$ and so on in the above mentioned formula,

$\begin{array}{rll}\overline{X}& =& \frac{0+14+\mathrm{...}+3+4}{10}\\ & =& \frac{58}{10}\\ & =& 5.8\end{array}$

For the variable Region of Birth, the measure of central tendency is: Mode.

Formula used:

The Mode:

The mode of any distribution of scores is the value that occurs most frequently.

In the data: “North” occurs for most frequently. Therefore, the mode is “North”.

For the variable, Religion, the measure of central tendency is: Mode.

Formula used:

The Mode:

The mode of any distribution of scores is the value that occurs most frequently.

In the data: “Protestant” occurs for most frequently. Therefore, the mode is “Protestant”.

For the variable Legalization, the measure of central tendency is: Median.

Arrange the data in the increasing order.

The data in increasing order is given by,

1

1

2

2

3

3

4

4

5

7

The formula to calculate median for even number of terms is given by,

$\text{Median}=\frac{\left({\left(\frac{N}{2}\right)}^{th}+{\left(\frac{N}{2}+1\right)}^{th}\right)}{2}$

Substitute 10 for $N$ in the above mentioned formula,

$\begin{array}{rll}\text{Median}& =& \frac{\left({\left(\frac{10}{2}\right)}^{th}+{\left(\frac{10}{2}+1\right)}^{th}\right)}{2}\\ & =& \frac{{\left(5\right)}^{th}+{\left(5+1\right)}^{th}}{2}\\ & =& \frac{{\left(5\right)}^{th}+{\left(6\right)}^{th}}{2}\end{array}$

The value corresponding to $5^{th}$ and $6^{th}$ is 3 and 3.

The required median is,

$\begin{array}{rll}\text{Median}& =& \frac{3+3}{2}\\ & =& \frac{6}{2}\\ & =& 3\end{array}$

For the variable Cafeteria Food, the measure of central tendency is: Median.

Arrange the data in the increasing order.

The data in increasing order is given by,

0

1

2

5

6

6

7

8

10

10

The formula to calculate median for even number of terms is given by,

$\text{Median}=\frac{\left({\left(\frac{N}{2}\right)}^{th}+{\left(\frac{N}{2}+1\right)}^{th}\right)}{2}$

Substitute 10 for $N$ in the above mentioned formula,

$\begin{array}{rll}\text{Median}& =& \frac{\left({\left(\frac{10}{2}\right)}^{th}+{\left(\frac{10}{2}+1\right)}^{th}\right)}{2}\\ & =& \frac{{\left(5\right)}^{th}+{\left(5+1\right)}^{th}}{2}\\ & =& \frac{{\left(5\right)}^{th}+{\left(6\right)}^{th}}{2}\end{array}$

The value corresponding to $5^{th}$ and $6^{th}$ is 6 and 6.

The required median is,

$\begin{array}{rll}\text{Median}& =& \frac{6+6}{2}\\ & =& \frac{12}{2}\\ & =& 6\end{array}$

Seniors:

For the variable Out-of-Pocket Expenses, the measure of central tendency is: Mean

Formula Used:

The formula to calculate mean is given by,

$\overline{X}=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Substitute 11 for $N$, 95 for $X_1$, 88 for $X_2$ and so on in the above mentioned formula,

$\begin{array}{rll}\overline{X}& =& \frac{75+72+\mathrm{...}+95+88}{11}\\ & =& \frac{615}{11}\\ & =& 72.55\end{array}$

For the variable Movies, the measure of central tendency is: Mean

Formula Used:

The formula to calculate mean is given by,

$\overline{X}=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Substitute 11 for $N$, 0 for $X_1$, 5 for $X_2$ and so on in the above mentioned formula,

$\begin{array}{rll}\overline{X}& =& \frac{0+5+\mathrm{...}+3+5}{11}\\ & =& \frac{57}{10}\\ & =& 5.18\end{array}$

For the variable Region of Birth, the measure of central tendency is: Mode.

Formula used:

The Mode:

The mode of any distribution of scores is the value that occurs most frequently.

In the data: “North” occurs for most frequently. Therefore, the mode is “North”.

For the variable, Religion, the measure of central tendency is: Mode.

Formula used:

The Mode:

The mode of any distribution of scores is the value that occurs most frequently.

In the data: “Protestant” and “None” occurs for most frequently. Therefore, the mode is “Protestant” and “None”.

For the variable Legalization, the measure of central tendency is: Median.

The ordered data is:

1

3

4

5

5

5

6

6

7

7

7

The number of terms is 11, which is odd.

The median for the odd number of terms is given by,

$\text{Median}=\frac{\left(\text{Number of terms}+1\right)}{2}$

Substitute 11 for number of terms in the above mentioned formula,

$\begin{array}{rll}\text{Median}& =& \frac{\left(11+1\right)}{2}\\ & =& \frac{12}{2}\\ & =& 6\end{array}$

The median corresponding to $6^{th}$ number is 5.

Therefore, the median is 5.

For the variable Cafeteria Food, the measure of central tendency is: Median.

The ordered data is:

1

2

2

3

4

4

4

6

7

8

9

The number of terms is 11, which is odd.

The median for the odd number of terms is given by,

$\text{Median}=\frac{\left(\text{Number of terms}+1\right)}{2}$

Substitute 11 for number of terms in the above mentioned formula,

$\begin{array}{rll}\text{Median}& =& \frac{\left(11+1\right)}{2}\\ & =& \frac{12}{2}\\ & =& 6\end{array}$

The median corresponding to $6^{th}$ number is 4.

Therefore, the median is 5.

Therefore,

Variable	Measure	Freshmen	Seniors
Region of birth	Mode	North	North
Legalization	Median	3	5
Expenses	Mean	58.5	72.55
Movies	Mean	5.80	5.18
Food	Median	6	4
Religion	Mode	Protestant	Protestant, None