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Cases are classified according to one variable, with categories \(\mathrm{A}\) and \(\mathrm{B},\) and also classified according to a second variable with categories \(1,2,\) and 3 . The cases are shown, with the first digit indicating the value of the first variable and the second digit indicating the value of the second variable. (So "A1" represents a case in category \(\mathrm{A}\) for the first variable and category 1 for the second variable.) Construct a two- way table of the data. Thirty cases: \(\begin{array}{llllllllll}\text { A1 } & \text { A1 } & \text { A2 } & \text { A2 } & \text { A2 } & \text { A2 } & \text { A2 } & \text { A2 } & \text { A3 } & \text { A3 } \\ \text { A3 } & \text { A3 } & \text { B1 } & \text { B1 } & \text { B1 } & \text { B1 } & \text { B1 } & \text { B2 } & \text { B2 } & \text { B3 } \\ \text { B3 } & \text { B3 } & \text { B3 } & \text { B3 } & \text { B3 } & \text { B3 } & \text { B3 } & \text { B3 } & \text { B3 } & \text { B3 }\end{array}\)

Step by step solution

01

Find the frequencies of each case

Count each instance of Ai and Bj (where i can be either 1, 2 or 3 and j can be either A or B). The result will be: \[ A1: 2, A2: 6, A3: 4, B1: 5, B2: 2, B3: 11 \]

02

Create a Two-way Table

Create a two-way (also known as a contingency) table with A and B as the row variable identified in rows, and 1, 2 and 3 as the column variables identified in columns. The intersection of each row and column will represent the count of instances for each case (Ai, Bj)

03

Populate the Two-way Table

Fill in the table with the frequencies of each case derived from step 1. Create a total row for summing up the frequencies under each column and likewise, a total column for summing up the frequencies for each row.

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Key Concepts

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

Categorical Data Analysis

In the world of statistics, when we talk about categorical data, we refer to data that can be sorted into categories or groups. Unlike numerical data, which symbolizes quantities and can be measured, categorical data represents characteristics or attributes and is often descriptive in nature.

For instance, categorical variables can be something like 'type of fruit' with categories being apples, bananas, and oranges, or 'shirt size' with categories like small, medium, large. In the exercise provided, the categories are represented by two categorical variables (A and B), and each has a further classification (1, 2, 3).

To analyze such data, it's crucial to organize it in a way that allows us to see patterns or relationships. This is particularly useful when we want to investigate how often different combinations of categories occur together, which leads to the need for tools like two-way tables and contingency tables, assisting in visually representing and analyzing categorical data.

Frequency Distribution

The frequency distribution is a basic concept in statistics that describes how often each value in a data set occurs. It is typically used for summarizing categorical variables, just like in the given exercise.

The process begins by counting the occurrences (frequencies) of each category or case. In the context of the exercise, we would tally how many times A1 appears, how many times A2 appears, and so on. These counts provide us with a basic but powerful understanding of the dataset's structure: what's common, what's rare, or perhaps, what's absent.

With the use of frequency tables, pie charts, or bar graphs, frequency distribution can be visualized to make it easier to identify trends or patterns. In educational settings, breaking down this process into step-by-step solutions can significantly help students to grasp the underlying patterns within the data.

Contingency Table

The contingency table, also known as a cross-tabulation or two-way table, is a valuable tool for categorical data analysis. It is a type of frequency distribution table that helps us understand the relationship between two categorical variables by displaying their frequencies.

In the exercise, we see that a contingency table is created by using two variables: A and B, which are the row categories, and 1, 2, 3, which are the column categories. Each cell in the table represents the intersection of these variables—essentially answering the question, 'How many cases are there for each possible combination of the two variables?'

By summarizing the data in this manner, a contingency table provides a foundation for various statistical methods aimed at testing hypotheses, such as evaluating whether there is an association between the two variables. It is an essential instrument for researchers and statisticians in almost any field, from medicine to marketing, and of course, education.