91Ó°ÊÓ

A 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 as well as 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 finding the median.) $$\text { Fresh Men}$$ \(\begin{array}{clccccl} \text { Student } & \text { Region of Birth } & \text { Legalization } & \text {Out-ofPocket Expenses } & \text { Movies } & \text { Food } & \text { Religion } \\ \hline \text { A } & \text { North } & 7 & 33 & 0 & 10 & \text { Protestant } \\\ \text { B } & \text { North } & 4 & 39 & 14 & 7 & \text { Protestant } \\ \text { C } & \text { South } & 3 & 45 & 10 & 2 & \text { Catholic } \\ \text { D } & \text { Midwest } & 2 & 47 & 7 & 1 & \text { None } \\ \text { E } & \text { North } & 3 & 62 & 5 & 8 & \text { Protestant } \\ \text { F } & \text { North } & 5 & 48 & 1 & 6 & \text { Jew } \\ \text { G } & \text { South } & 1 & 52 & 0 & 10 & \text { Protestant } \\ \text { H } & \text { South } & 4 & 65 & 14 & 0 & \text { Other } \\ \text { ? } & \text { Midwest } & 1 & 51 & 3 & 5 & \text { Other } \\ \text { J } & \text { West } & 2 & 43 & 4 & 6 & \text { Catholic } \end{array}\) \(\text { SENIORS }\) \(\begin{array}{clccccl} \text { Student } & \text {Region of Birth } & \text { Legalization } & \text {Out-ofPocket Expenses } & \text { Movies } & \text { Cafeteria Food } & \text { Religion } \\ \hline \mathrm{K} & \text { North } & 7 & 65 & 0 & 1 & \text { None } \\ \mathrm{L} & \text { Midwest } & 6 & 62 & 5 & 2 & \text { Protestant } \\ \mathrm{M} & \text { North } & 7 & 60 & 11 & 8 & \text { Protestant } \\ \mathrm{N} & \text { North } & 5 & 90 & 3 & 4 & \text { Catholic } \\ \mathrm{O} & \text { South } & 1 & 62 & 4 & 3 & \text { Protestant } \\ \mathrm{P} & \text { South } & 5 & 57 & 14 & 6 & \text { Protestant } \\ \mathrm{Q} & \text { West } & 6 & 40 & 0 & 2 & \text { Catholic } \\ \mathrm{R} & \text { West } & 7 & 49 & 7 & 9 & \text { None } \\ \mathrm{S} & \text { North } & 3 & 45 & 5 & 4 & \text { None } \\ \mathrm{T} & \text { West } & 5 & 85 & 3 & 7 & \text { Other } \\ \mathrm{U} & \text { North } & 4 & 78 & 5 & 4 & \text { None } \end{array}\)

Step by step solution

01

Identify Level of Measurement

Determine the level of measurement for each variable: nominal, ordinal, interval, or ratio.

02

Region of Birth

Region of Birth is a nominal variable. The most appropriate measure of central tendency is the mode.

03

Legalization of Marijuana

Legalization of Marijuana is an ordinal variable. The most appropriate measure of central tendency is the median.

04

Out-of-Pocket Expenses

Out-of-Pocket Expenses are a ratio variable. The most appropriate measure of central tendency is the mean.

05

Number of Movies

Number of Movies is a ratio variable. The most appropriate measure of central tendency is the mean.

06

Cafeteria Food Rating

Cafeteria Food Rating is an interval variable. The most appropriate measure of central tendency is the mean.

07

Religion

Religion is a nominal variable. The most appropriate measure of central tendency is the mode.

08

Calculate Measures for Freshmen

For freshmen:Region of Birth: Mode = NorthLegalization: Median = 3.5Out-of-Pocket Expenses: Mean = 48.5Number of Movies: Mean = 5.8Cafeteria Food Rating: Mean = 5.5Religion: Mode = Protestant

09

Calculate Measures for Seniors

For seniors:Region of Birth: Mode = NorthLegalization: Median = 5Out-of-Pocket Expenses: Mean = 61.8Number of Movies: Mean = 5.2Cafeteria Food Rating: Mean = 4.0Religion: Mode = Protestant

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nominal Variables

Nominal variables classify data into distinct categories with no mathematical meaning. Each category is unique and used for labeling or identifying groups.
Examples in the exercise include 'Region of Birth' and 'Religion.'

For Region of Birth, the most common category is 'North,' making the mode the best measure of central tendency.
Similarly, for Religion, 'Protestant' is the most frequent category among both freshmen and seniors, making mode the appropriate choice for this variable.

Ordinal Variables

Ordinal variables order data in a meaningful sequence, but the differences between data points can't be measured precisely.
In the exercise, 'Legalization of Marijuana' and 'Cafeteria Food Rating' are ordinal variables.

For 'Legalization of Marijuana,' responses range from 1 (strong opposition) to 7 (strong support). The median is the best measure of central tendency here, as it shows the middle value when data is ordered.

Similarly, 'Cafeteria Food Rating' also uses a scale, making the median a suitable measure.

Ratio Variables

Ratio variables have all the characteristics of interval variables, with the addition of a true zero point, allowing for meaningful comparisons using multiplication or division.
'Out-of-Pocket Expenses' and 'Number of Movies' are ratio variables in the exercise.

'Out-of-Pocket Expenses' can be averaged (mean) because they are measured in consistent units, and zero represents no expenses.
'Number of Movies' watched is similar; the mean provides the best measure for summarizing this variable.

Interval Variables

Interval variables represent data with meaningful intervals but no true zero point. Temperature scales are typical examples.
In the exercise, 'Cafeteria Food Rating' could also be considered an interval variable because it measures opinions on a fixed scale.

The best measure of central tendency for interval variables like 'Cafeteria Food Rating' is the mean. This measure helps understand the average perception of cafeteria food among students.

Mean

The mean is calculated by summing all values and dividing by the number of values. It provides an average that represents the central point of the data.
For ratio and interval variables, like 'Out-of-Pocket Expenses,' 'Number of Movies,' and 'Cafeteria Food Rating,' the mean is typically the most suitable measure of central tendency.

In our exercise:

• Freshmen: Out-of-Pocket Expenses Mean = 48.5; Number of Movies Mean = 5.8; Cafeteria Food Rating Mean = 5.5
• Seniors: Out-of-Pocket Expenses Mean = 61.8; Number of Movies Mean = 5.2; Cafeteria Food Rating Mean = 4.0

Median

The median is the middle value in a dataset when the values are arranged in ascending or descending order. It is less affected by extreme values, making it useful for ordinal data.
In this exercise, 'Legalization of Marijuana' is an ordinal variable, so the median is appropriate.

For Legalization:

• Freshmen: Median = 3.5
• Seniors: Median = 5

The median helps show the central tendency without being skewed by strong opposition or support values.

Mode

The mode is the value that appears most frequently in a dataset. It's the best measure for nominal data but can also apply to other types of variables when identifying the most common category or score.
In our exercise, nominal variables such as 'Region of Birth' and 'Religion' use the mode as the measure of central tendency.

For Region of Birth:

• Freshmen: Mode = North
• Seniors: Mode = North

For Religion:

• Freshmen: Mode = Protestant
• Seniors: Mode = Protestant

The mode helps in identifying the most frequent categories among the students.