Glossary

11.3 Test of Independence

2 min read•june 25, 2024

The helps us figure out if two are related. We use observed and expected frequencies to calculate a , which we compare to a to make a decision.

This test is crucial for understanding relationships between variables in real-world scenarios. By analyzing contingency tables and interpreting results, we can draw meaningful conclusions about associations in our data.

Chi-Square Test of Independence

Test statistic calculation for independence

Top images from around the web for Test statistic calculation for independence

  • Test of Independence (3 of 3) | Concepts in Statistics View original

    Is this image relevant?

  • Chi square calculator - wikidoc View original

    Is this image relevant?

  • Test of Independence (3 of 3) | Concepts in Statistics View original

    Is this image relevant?

1 of 3

Top images from around the web for Test statistic calculation for independence

  • Test of Independence (3 of 3) | Concepts in Statistics View original

    Is this image relevant?

  • Chi square calculator - wikidoc View original

    Is this image relevant?

  • Test of Independence (3 of 3) | Concepts in Statistics View original

    Is this image relevant?

1 of 3
  • Calculate chi-square using formula χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}
    • OO in each cell of (actual counts)
    • EE in each cell of (theoretical counts assuming )
  • Determine expected frequency for each cell
    • Multiply by divide by
    • Formula E=(row total)×(column total)grand totalE = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}
  • Calculate (r−1)(c−1)(r - 1)(c - 1)
    • rr number of rows in contingency table (categories of one variable)
    • cc number of columns in contingency table (categories of other variable)

Interpretation of independence test results

  • Test of independence assesses relationship between two
  • Null hypothesis (H0H_0) variables are independent (no )
  • (HaH_a) variables are not independent (associated)
  • Compare calculated chi-square test statistic to from table
    • Use degrees of freedom (0.05) to find critical value
  • If calculated chi-square test statistic > critical value reject null hypothesis
    • Sufficient evidence to suggest variables are not independent (associated)
  • If calculated chi-square test statistic < critical value fail to reject null hypothesis
    • Insufficient evidence to suggest variables are associated (independent)

Chi-square analysis in real-world scenarios

  • Identify categorical variables of interest in real-world scenario (gender, age group)
  • Construct contingency table with observed frequencies for each category combination
  • Calculate expected frequencies for each cell in contingency table
  • Calculate chi-square test statistic using observed expected frequencies
  • Determine degrees of freedom based on number of rows columns in contingency table
  • Choose significance level (0.05) find critical value from chi-square distribution table
  • Compare calculated chi-square test statistic to critical value make conclusion about variable relationship
    • If test statistic > critical value conclude variables are associated (preference, behavior)
    • If test statistic < critical value conclude insufficient evidence to suggest association between variables (independence)

Statistical Inference and Hypothesis Testing

  • Chi-square test of independence is a form of
  • Uses sample data to draw conclusions about population parameters
  • Follows framework to make decisions about independence or association
  • Relies on (chi-square distribution) to determine critical values
  • Chi-square test is a , making fewer assumptions about the underlying population distribution

Key Terms to Review (26)

Alternative Hypothesis: The alternative hypothesis is a statement that suggests a potential outcome or relationship exists in a statistical test, opposing the null hypothesis. It indicates that there is a significant effect or difference that can be detected in the data, which researchers aim to support through evidence gathered during hypothesis testing.
Association: Association is a statistical concept that describes the relationship or connection between two or more variables. It measures the degree to which changes in one variable are accompanied by changes in another variable, without necessarily implying a causal relationship.
Binomial probability distribution: A binomial probability distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by two parameters: the number of trials (n) and the probability of success (p).
Categorical variables: Categorical variables are variables that represent categories or groups and have a limited number of distinct values. These values are usually qualitative and describe characteristics or attributes.
Categorical Variables: Categorical variables are variables that represent a set of categories or groups, rather than numerical values. They are used to classify or group data based on qualitative characteristics or attributes, and are commonly used in statistical analysis and data visualization.
Chi-Square Distribution: The chi-square distribution is a continuous probability distribution that arises when independent standard normal random variables are squared and summed. It is widely used in statistical hypothesis testing, particularly in evaluating the goodness-of-fit of observed data to a theoretical distribution and in testing the independence of two categorical variables.
Chi-Square Test of Independence: The chi-square test of independence is a statistical test used to determine whether there is a significant relationship or association between two categorical variables. It examines the differences between the observed frequencies and the expected frequencies in each category to assess whether the variables are independent or related.
Column Total: The column total is the sum of all the values in a particular column of a data table or contingency table. It represents the total count or frequency for that column and is a crucial component in the analysis of the relationship between two categorical variables.
Contingency table: A contingency table, also known as a cross-tabulation or crosstab, is a type of table in a matrix format that displays the frequency distribution of variables. It is commonly used to analyze the relationship between two categorical variables.
Contingency Table: A contingency table, also known as a cross-tabulation or a two-way table, is a type of table that displays the frequency distribution of two or more categorical variables. It is used to analyze the relationship between these variables and determine if they are independent or associated with each other.
Critical value: A critical value is a point on the scale of the standard normal distribution that is compared to a test statistic to determine whether to reject the null hypothesis. It separates the region where the null hypothesis is not rejected from the region where it is rejected.
Critical Value: The critical value is a threshold value in statistical analysis that is used to determine whether to reject or fail to reject a null hypothesis. It serves as a benchmark for evaluating the statistical significance of a test statistic and is a crucial concept across various statistical methods and hypothesis testing procedures.
Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation without breaking any constraints. It plays a crucial role in determining the appropriate statistical tests and distributions used for hypothesis testing, estimation, and data analysis across various contexts.
Expected Frequency: The expected frequency is the anticipated or predicted frequency of an outcome in a statistical analysis, particularly in the context of contingency tables, goodness-of-fit tests, and tests of independence. It represents the expected number of observations in a particular cell or category under the null hypothesis.
Grand Total: The grand total is the overall sum or final result obtained by adding up all the individual values or subtotals in a data set or table. It represents the comprehensive total that encompasses all the components or parts of a larger whole.
Hypothesis Testing: Hypothesis testing is a statistical method used to determine whether a claim or hypothesis about a population parameter is likely to be true or false based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, collecting and analyzing sample data, and making a decision to either reject or fail to reject the null hypothesis.
Independence: Independence is a fundamental concept in statistics that describes the relationship between events or variables. When events or variables are independent, the occurrence or value of one does not depend on or influence the occurrence or value of the other. This concept is crucial in understanding probability, statistical inference, and the analysis of relationships between different factors.
Nonparametric Test: A nonparametric test is a statistical hypothesis test that does not rely on the data following a specific probability distribution, such as the normal distribution. These tests are often used when the assumptions for parametric tests, like normality, are not met.
Observed Frequency: Observed frequency refers to the actual count or number of occurrences of a particular event or outcome in a dataset or sample. It represents the empirical or observed data, as opposed to the expected or theoretical frequency. This term is crucial in understanding and interpreting various statistical analyses, including contingency tables, goodness-of-fit tests, and tests of independence.
P-value: The p-value is the probability of obtaining a test statistic at least as extreme as the one actually observed, assuming the null hypothesis is true. It is a crucial concept in hypothesis testing that helps determine the statistical significance of a result.
Probability Distribution: A probability distribution is a mathematical function that describes the likelihood or probability of different possible outcomes or events occurring in a given situation or experiment. It provides a comprehensive representation of the possible values a random variable can take on and their corresponding probabilities.
Row Total: The row total is the sum of all the values in a specific row of a data table or contingency table. It represents the total count or frequency for that particular row, providing information about the marginal distribution of the row variable.
Significance Level: The significance level, denoted as α (alpha), is the probability of rejecting the null hypothesis when it is true. It represents the maximum acceptable probability of making a Type I error, which is the error of rejecting the null hypothesis when it is actually true. The significance level is a crucial concept in hypothesis testing and statistical inference, as it helps determine the strength of evidence required to draw conclusions about a population parameter or the relationship between variables.
Statistical Inference: Statistical inference is the process of using data analysis and probability theory to draw conclusions about a population from a sample. It allows researchers to make educated guesses or estimates about unknown parameters or characteristics of a larger group based on the information gathered from a smaller, representative subset.
Test statistic: A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to determine whether to reject the null hypothesis.
Test Statistic: A test statistic is a numerical value calculated from sample data that is used to determine whether to reject or fail to reject a null hypothesis in a hypothesis test. It serves as the basis for decision-making in statistical inference, providing a quantitative measure to evaluate the strength of evidence against the null hypothesis.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide