91Ó°ÊÓ

Degrees of freedom explained For testing independence in a contingency table of size \(r \times c,\) the degrees of freedom (df) for the chi-squared distribution equal \(d f=(r-1) \times(c-1) .\) They have the following interpretation: Given the row and column marginal totals in an \(r \times\) contingency table, the cell counts in a rectangular block of size \((r-1) \times(c-1)\) determine all the other cell counts. Consider the following table, which cross-classifies political views by whether the subject would ever vote for a female president, based on the 2010 GSS. For this \(3 \times 2\) table, suppose we know the counts in the upper left-hand \((3-1) \times(2-1)=2 \times 1\) block of the table, as shown. \begin{tabular}{lccc} \hline & \multicolumn{2}{c} { Vote for Female } & \\ & \multicolumn{2}{c} { President } & \\ \cline { 2 - 3 } Political Views & Yes & No & Total \\ \hline Extremely Liberal & 56 & & 58 \\ Moderate & 490 & & 509 \\ Extremely Conservative & & & 61 \\ \hline Total & 604 & 24 & 628 \\ \hline \end{tabular} a. Given the cell counts and the row and column totak, fill in the counts that must appear in the blank cells. b. Now, suppose instead of the preceding table, you are shown the following table, this time only revealing a \(2 \times 1\) block in the lower-right part. Find the counts in the remaining cells. \begin{tabular}{lccc} \hline & \multicolumn{2}{c} { Vote for Female } & \\ & \multicolumn{2}{c} { President } & \\ \cline { 2 - 3 } Political Views & Yes & No & Total \\ \hline Extremely Liberal & & & \\ Moderate & & 58 \\ Extremely Conservative & & 3 & 61 \\ \hline Total & & 19 & 509 \\ \hline \end{tabular} This example serves to show that once the marginal totals are fixed in a contingency table, a block of only cell counts is free to vary. Once \((r-1) \times(c-1)\) these are given (as in part a or \(\mathrm{b}\) ), the remaining cell counts follow automatically. The value for the degrees of freedom is exactly the number of cells in this block, \(d f=(r-1) \times(c-1)\) or

Step by step solution

01

Understand the Problem

We are given a contingency table of size \(3 \times 2\) with certain known and unknown cell values. We need to fill in the missing cell counts using the degrees of freedom concept, where each cell in a \((r-1) \times (c-1)\) block is free to vary.

02

Fill Missing Cells for Scenario A

In Scenario A, we are given a \(2 \times 1\) block in the upper-left corner. We need to use the row totals and column totals to find the missing values: 1. For 'Extremely Liberal - No', we subtract the 'Yes' count from the row total: \[ 58 - 56 = 2 \] 2. For 'Moderate - No', similar to 'Extremely Liberal - No': \[ 509 - 490 = 19 \] 3. For 'Extremely Conservative - Yes', subtract the total 'Yes' from known totals of 'Extremely Liberal' and 'Moderate': \[ 604 - (56 + 490) = 58 \] 4. For the final cell 'Extremely Conservative - No', use the row total minus 'Yes': \[ 61 - 58 = 3 \]

03

Fill Missing Cells for Scenario B

In Scenario B, we are given a \(2 \times 1\) block in the lower-right corner. We fill in the missing values by back-calculating with totals: 1. For 'Extremely Conservative - Yes': \[ 61 - 3 = 58 \] 2. 'Extremely Liberal - Yes' and 'Moderate - Yes' can be calculated from the total and known value of '__Conservative-Yes__': \( 483 = 58 + 425 \) derived from \( 604 - 58 = 546 \) being Moderate-Yes sum Additionally, measure 'Extremely Liberal-Yes' 3. For 'Extremely Liberal - No', use the total and known values from previous cells.4. Complete 'Moderate - Yes' with the remaining total based on known row total and calculated column total.

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Key Concepts

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

Contingency Table

A contingency table, also known as a cross-tabulation or crosstab, is a type of table used in statistics to display the frequencies of different combinations of variables. These tables help in understanding the relationship between two categorical variables. For example, a contingency table might show the relationship between political views (such as "extremely liberal," "moderate," and "extremely conservative") and the willingness to vote for a female president ("yes" or "no").

Such tables are organized so that one variable defines the rows and the other defines the columns, allowing the distribution and interaction effects to be easily viewed and analyzed. Each cell of the table represents a unique combination of the row and column variables. For instance, the intersection of "extremely liberal" and "yes" would show how many people with extremely liberal views would vote for a female president.

Overall, a contingency table is a powerful tool for examining how two variables are interrelated by breaking down complex data into more understandable and manageable units.

Chi-Squared Distribution

The chi-squared distribution is a key concept in statistical hypothesis testing, particularly useful for tests involving contingency tables. It is a probability distribution that describes the sum of the squares of a certain number of independent standard normal variables. This distribution is used to assess the goodness of fit, and its applications are broad, including the chi-squared test for independence.

In the context of a contingency table, the chi-squared test is used to determine whether there is a significant association between the categories of the two variables. It tests the null hypothesis that the observed frequencies in the contingency table come from the same distribution as the expected frequencies based on marginal totals. The test statistic is calculated as the sum of squared differences between the observed and expected frequencies, each divided by the expected frequency:
\[ \chi^2 = \sum \frac{(O - E)^2}{E} \] where \(O\) represents the observed frequency and \(E\) the expected frequency in each cell.

The degrees of freedom for the chi-squared distribution, in this context, are calculated using the formula \((r-1) \times (c-1)\), where \(r\) is the number of rows and \(c\) is the number of columns in the table. These degrees of freedom reflect the number of independent comparisons that can be made between observed and expected frequencies.

Marginal Totals

Marginal totals in a contingency table are the sums of the frequencies across either the rows or the columns. They provide essential information for calculating expected frequencies under the assumption of independence between the variables.

When examining a contingency table, calculating marginal totals is a primary step. These totals serve as a control for predicting how the frequencies would distribute under the condition of independence between the two variables being studied. For example, the marginal total for a political view like "moderate" would be the sum of all individuals regardless of voting preference for female president.

In practice, marginal totals are critical for completing empty cells in a contingency table if some cell values are missing. By using known marginal totals and applying the rules of probability, you can infer missing cells. The relationship between observed and expected frequencies, mediated by marginal totals, forms the basis for further statistical tests, such as the chi-squared test for independence.

Political Views

In statistical exercises involving human behavior and attitudes, variables like political views often play a crucial role. "Political views" is a categorical variable that typically organizes individuals into categories according to their political ideology or stance, such as "extremely liberal," "moderate," or "extremely conservative."

Understanding how political views are distributed across other variables can provide insights into societal trends and attitudes. For instance, when analyzing a contingency table relating political views to the willingness to vote for a female president, such data might reveal electoral preferences or biases, thus allowing for deeper sociological analysis.

This categorization of political views becomes particularly useful in cross-tabulation analyses, where comparisons can highlight differences and similarities across different political ideologies in relation to key issues, such as gender bias in politics. By examining these patterns, one can begin to understand the complexities of voter behavior and public opinion.