13.1 Chi-Square Goodness-of-Fit Test
2 min read•july 23, 2024
The helps determine if categorical data matches a specific distribution. It compares observed frequencies to expected ones, calculating a test statistic to assess the fit between sample data and hypothesized population distribution.
Interpreting results involves comparing the test statistic to a critical value or using the . This helps decide whether to reject or fail to reject the null hypothesis, indicating if the sample data significantly differs from the expected distribution.
Chi-Square Goodness-of-Fit Test
Purpose of chi-square goodness-of-fit test
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- Determines if a sample of categorical data (colors, types) comes from a population with a specific distribution
- Compares observed frequencies of categories in the sample to expected frequencies based on the hypothesized distribution
- Applicable when data is categorical or nominal, sample is randomly selected, and of each category is at least 5
Calculation of chi-square test statistic
- Calculate expected frequencies for each category by multiplying hypothesized probability of each category by total
- statistic calculated using formula
- represents for category (actual count)
- represents expected frequency for category (calculated based on hypothesized distribution)
- represents number of categories (options, choices)
Degrees of freedom for chi-square tests
- for chi-square goodness-of-fit test calculated as
- represents number of categories
- Critical value determined by significance level (), degrees of freedom
- Found using chi-square distribution table or statistical software (Excel, R)
Interpretation of chi-square test results
- Compare calculated chi-square test statistic to critical value
- If test statistic greater than critical value,
- Suggests sample data does not follow hypothesized distribution (significantly different)
- If test statistic less than or equal to critical value, fail to reject null hypothesis
- Suggests sample data consistent with hypothesized distribution (not significantly different)
- If test statistic greater than critical value,
- P-value also used to make decision
- If p-value less than significance level (), reject null hypothesis
- If p-value greater than or equal to significance level (), fail to reject null hypothesis
Key Terms to Review (18)
Calculate test statistic: To calculate a test statistic means to derive a numerical value that summarizes the difference between observed data and the expected outcomes under a null hypothesis. This value is crucial in hypothesis testing as it helps determine whether to reject or fail to reject the null hypothesis based on how far the observed data deviates from what was expected. The calculated test statistic is then compared against critical values from a statistical distribution to assess significance.
Categorical Distribution: A categorical distribution is a probability distribution that describes the likelihood of different outcomes in a categorical variable, where the data can fall into distinct categories rather than continuous values. This type of distribution is used to model situations where the outcomes are mutually exclusive and exhaustive, such as survey responses or preferences among different options. It serves as the foundation for various statistical tests that assess how well observed data aligns with expected distributions.
Cell Frequency: Cell frequency refers to the number of observations or data points that fall into a specific category or cell within a contingency table. This term is crucial when analyzing categorical data, as it helps to determine how well the observed data aligns with expected values in statistical tests, particularly in evaluating the goodness-of-fit for a given distribution.
Chi-Square Goodness-of-Fit Test: The chi-square goodness-of-fit test is a statistical method used to determine how well observed data fits a specific distribution. It helps assess whether the differences between observed and expected frequencies in categorical data are due to chance or if there is a significant deviation from the expected distribution.
Chi-Square Test: The chi-square test is a statistical method used to determine if there is a significant association between categorical variables or if the observed frequencies in a dataset differ from the expected frequencies. This test is often applied in different contexts to assess goodness-of-fit, independence, and relationships within contingency tables, making it an essential tool for analyzing data and making inferences about populations.
Consumer Preferences: Consumer preferences refer to the individual tastes, likes, and dislikes that influence a person's choice among various products or services. Understanding consumer preferences is essential for businesses as it helps them tailor their offerings to meet the demands and desires of their target audience, impacting product development, marketing strategies, and overall success in the marketplace.
Contingency Table: A contingency table is a type of data display that shows the frequency distribution of variables and helps to analyze the relationship between two categorical variables. It organizes data into rows and columns, allowing for a clear comparison and understanding of how the different categories intersect. This table is particularly useful in statistical analysis to determine if there is a significant association between the variables, which can be tested using methods like the Chi-Square Goodness-of-Fit Test.
Degrees of Freedom: Degrees of freedom refers to the number of independent values or quantities that can vary in an analysis without breaking any constraints. This concept is crucial in statistical tests because it affects the distribution of the test statistic, influencing how we determine significance. When conducting various statistical tests, understanding degrees of freedom helps in accurately interpreting results and making valid conclusions.
Df: In statistics, 'df' stands for degrees of freedom, which refers to the number of independent values or quantities that can vary in an analysis without breaking any constraints. It is crucial in various statistical tests, as it helps determine the distribution of the test statistic under the null hypothesis. Specifically, in the context of the Chi-Square Goodness-of-Fit Test, degrees of freedom are used to interpret the test results by identifying the appropriate critical value from the Chi-Square distribution table.
Expected Frequency: Expected frequency refers to the theoretical frequency of an event occurring in a statistical experiment, based on the assumption of a specific distribution of data. This concept is crucial for analyzing categorical data and helps in determining whether the observed frequencies deviate significantly from what is expected under a given hypothesis, thereby allowing for statistical testing and inference.
Independence of Observations: Independence of observations refers to the condition where the data collected from one observation does not influence or affect the data collected from another observation. This concept is crucial in statistical analyses as it ensures that each data point contributes uniquely to the overall results, allowing for valid inferences and conclusions. When observations are independent, it means that the occurrence or value of one observation does not provide any information about another, which is important for the validity of various statistical tests.
Market Research: Market research is the process of gathering, analyzing, and interpreting information about a market, including information about the target audience, competitors, and overall industry trends. It helps businesses understand their customers' needs and preferences, enabling them to make informed decisions regarding product development, marketing strategies, and sales approaches.
Observed Frequency: Observed frequency refers to the actual count or number of occurrences of a specific event or category in a dataset. This term is crucial in statistical analysis, especially in tests that compare observed data to expected outcomes, such as evaluating how well a particular distribution fits the observed data. It provides a basis for determining if the differences between observed and expected frequencies are significant.
P-value: A p-value is a statistical measure that helps determine the significance of results from a hypothesis test. It represents the probability of obtaining results at least as extreme as the observed data, given that the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis, leading to its rejection in favor of an alternative hypothesis.
Reject null hypothesis: To reject the null hypothesis means to determine, based on statistical analysis, that there is enough evidence to conclude that the null hypothesis is not true. This decision typically arises from hypothesis testing where a p-value is compared to a predetermined significance level. If the p-value is less than this significance level, we conclude that the observed data is unlikely under the assumption of the null hypothesis, thus leading to its rejection.
Sample size: Sample size refers to the number of observations or data points included in a statistical sample, which is crucial for ensuring the reliability and validity of the results. A larger sample size can lead to more accurate estimates and stronger statistical power, while a smaller sample size may result in less reliable outcomes. Understanding the appropriate sample size is essential for various analyses, as it affects the confidence intervals, error rates, and the ability to detect significant differences or relationships within data.
State Hypotheses: State hypotheses refers to the specific, testable statements about a population parameter that researchers formulate before conducting statistical tests. These hypotheses are critical in determining the direction and significance of a statistical analysis, helping to guide the research process and allowing for clear conclusions based on collected data.
Uniform Distribution: Uniform distribution is a type of probability distribution where all outcomes are equally likely to occur within a specified range. This means that the probability of each outcome is the same, leading to a rectangular shape when graphed. It’s important in statistics as it provides a simple model for scenarios where each outcome has an equal chance, making it useful for testing and simulations.