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Chapter 2: Problem 4

Classify each of the following variables as either categorical or numerical. For those that are numerical, determine whether they are discrete or continuous. a. Brand of computer purchased by a customer b. State of birth for someone born in the United States c. Price of a textbook d. Concentration of a contaminant (micrograms per cubic centimeter) in a water sample e. Zip code (Think carefully about this one.) f. Actual weight of coffee in a 1 -pound can

Short Answer

Expert verified
a. Categorical \n b. Categorical \n c. Numerical - Continuous \n d. Numerical - Continuous \n e. Categorical \n f. Numerical - Continuous

Step by step solution

01

Classify the variable type for (a)

The brand of computer purchased by a customer is a qualitative variable and thus is a Categorical variable. This is because you cannot quantifiably measure the brand of a product; it is a matter of choosing one from different categories.
02

Classify the variable type for (b)

The state of birth for someone born in the United States refers to a specific category (one of the 50 states), thus, it is a Categorical variable.
03

Classify the variable type for (c)

The price of a textbook is a numerical value that can be measured quantitatively and thus it is a Numerical variable. It is also a continuous numerical variable since a price can potentially be any real number, even fractions of a cent.
04

Classify the variable type for (d)

The concentration of a contaminant (micrograms per cubic centimeter) in a water sample is a quantity that can be measured numerically. This would make it a Numerical variable. It is continuous since measurements could be any real number, based on the level of precision of the measurement.
05

Classify the variable type for (e)

A Zip code, although it is composed of numbers, is actually a Categorical variable. It may seem numerical but in fact, it is a representation of a category (specific geographical area). You cannot perform arithmetic operations on a Zip code in a meaningful way.
06

Classify the variable type for (f)

The actual weight of coffee in a 1-pound can is a Numerical variable that can be measured quantitatively. It is also classified as continuous because it can take any value within a certain range, considering that the weight is not always exactly 1 pound due to various factors like packaging and manufacturing differences.

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Key Concepts

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

Categorical Variables
Categorical variables are variables that describe categories or groups. They are not measured with numbers but rather represent different kinds of qualitative data.
  • For example, the brand of a computer is a categorical variable because different brands represent different categories. You can't measure or rank the brands numerically, but you can differentiate them by name.
  • Similarly, the state of birth is another categorical variable. The 50 states in the U.S. are distinct categories without any quantitative relationships between them.
Even if a variable seems numerical, like a zip code, it might still be categorical. Zip codes label geographical areas and performing numerical operations on them doesn't provide meaningful insights. Therefore, they're just categories tagged with numbers.
Numerical Variables
Numerical variables, unlike categorical ones, represent quantities that can be measured or counted. These variables take numbers and allow for meaningful arithmetic operations.
  • An example is the price of a textbook, which is quantifiable and varies. It can be calculated, analyzed, and averaged.
  • Another example is the actual weight of coffee in a can. This weight is expressed numerically, helping us compare it against the expected pound measure of the can.
Whether counting items or measuring aspects of them, numerical variables tell us the 'how much' or 'how many' and enable statistical analysis.
Discrete vs Continuous Data
When we talk about numerical data, we often classify it as discrete or continuous. This classification aids in understanding how the data behaves and how it can be represented.
Discrete data refers to numerical variables that assume distinct, separate values. These are usually whole numbers.
Continuous data, on the other hand, can take any value in a given range. Data such as the price of a textbook or weight of coffee is continuous. This is because these measurements can be infinitely detailed, capturing decimal and fractional values, constrained only by the precision of the measurement tools.
Data Classification
Data classification is the process of organizing data into categories or types that define its nature. This organization helps in deciding the kind of analysis or methodology needed to extract meaningful information.
  • Categorical variables belong to one classification where data is grouped, but not measured taken numerically.
  • Numerical variables, on the other hand, are data types where values are susceptible to arithmetic operations.
The distinction between these classifications is crucial for statistical analysis, helping researchers choose the right procedures to evaluate their data's characteristics.
Qualitative and Quantitative Data
Data can also be classified into qualitative and quantitative categories. Each plays a distinct role in data analysis and interpretation.
Qualitative data refers to descriptive information, often categorized under categorical variables.
Quantitative data is numerical, signifying quantities and powering calculations.
  • Categorical variables represent qualitative data types, like brands and zip codes.
  • Price and weight exemplify quantitative data, enabling measurement and analysis.
This classification is essential, determining the analytical approach and interpreting the results correctly.

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Most popular questions from this chapter

The Connecticut Agricultural Experiment Station conducted a study of the calorie content of different types of beer. The calorie contents (calories per \(100 \mathrm{ml}\) ) for 26 brands of light beer were (from the website brewery.org): $$ \begin{array}{lllllll} 29 & 28 & 33 & 31 & 30 & 33 & 30 \\ 28 & 27 & 41 & 39 & 31 & 29 & 23 \\ 32 & 31 & 32 & 19 & 40 & 22 & 34 \\ 31 & 42 & 35 & 29 & 43 & & \end{array} $$ Construct a stem-and-leaf display using stems \(1,2,3,\) and 4\. Write a sentence or two describing the calorie content of light beers.

The article "Ozzie and Harriet Don't Live Here Anymore" (San Luis Obispo Tribune, February 26,2002 ) looked at the changing makeup of America's suburbs. The article states that in the suburbs of the nation's largest cities, nonfamily households (for example, homes headed by a single professional or an elderly widow) now outnumber married couples with children. The article goes on to state: In the nation's 102 largest metropolitan areas, "nonfamilies" comprised 29 percent of households in 2000 , up from 27 percent in 1990 . While the number of married-with-children homes grew too, the share did not keep pace. It declined from 28 percent to 27 percent. Married couples without children at home live in another 29 percent of suburban households. The remaining 15 percent are single-parent homes. Use the given information on type of household in the year 2000 to construct a frequency distribution and a bar chart. (Make sure to only extract the year 2000 percentages from the given information.)

\((\mathrm{C} 1, \mathrm{C} 2)\) In a survey of 100 people who had recently purchased motorcycles, data on the following variables were recorded: Gender of purchaser Brand of motorcycle purchased Number of previous motorcycles owned by purchaser Telephone area code of purchaser Weight of motorcycle as equipped at purchase a. Which of these variables are categorical? b. Which of these variables are discrete numerical? c. Which type of graphical display would be an appropriate choice for summarizing the gender data, a bar chart or a dotplot? d. Which type of graphical display would be an appropriate choice for summarizing the weight data, a bar chart or a dotplot?

For the following numerical variables, state whether each is discrete or continuous. a. The number of insufficient-funds checks received by a grocery store during a given month b. The amount by which a 1 -pound package of ground beef decreases in weight (because of moisture loss) before purchase c. The number of New York Yankees during a given year who will not play for the Yankees the next year d. The number of students in a class of 35 who have purchased a used copy of the textbook

Heal the Bay is an environmental organization that releases an annual beach report card based on water quality (Heal the Bay Beach Report Card, May 2009). The 2009 ratings for 14 beaches in San Francisco County during wet weather were: $$ \mathrm{A}+\mathrm{C} \mathrm{B} \mathrm{A} \quad \mathrm{A}+\mathrm{A}+\mathrm{A} \mathrm{A}+\mathrm{B} \mathrm{D} \mathrm{C} \mathrm{D} \mathrm{F} \mathrm{F} $$ a. Summarize the wet weather ratings by constructing a relative frequency distribution and a bar chart. b. The dry weather ratings for these same beaches were: \(\mathrm{A} \mathrm{B} \mathrm{B} \quad \mathrm{A}+\mathrm{A} \mathrm{F} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{B} \quad \mathrm{A}\) Construct a bar chart for the dry weather ratings. c. Do the bar charts from Parts (a) and (b) support the statement that beach water quality tends to be better in dry weather conditions? Explain.
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