Chi-square goodness-of-fit test

Definition

The chi-square goodness-of-fit test is a statistical method used to determine if a sample data set matches a population with a specific distribution. It compares the observed frequencies of events in the sample to the expected frequencies under the assumed distribution, allowing researchers to assess how well the sample fits the theoretical model.

5 Must Know Facts For Your Next Test

1. The chi-square goodness-of-fit test is applicable for categorical data, making it suitable for determining if sample distributions align with theoretical distributions.
2. To perform the test, you need a sufficient sample size; typically, each expected frequency should be 5 or more to ensure the validity of the results.
3. The test statistic is calculated by summing the squared differences between observed and expected frequencies divided by the expected frequencies.
4. A high chi-square statistic indicates a poor fit between observed and expected data, leading to potential rejection of the null hypothesis.
5. The significance level (alpha) determines how likely you are to reject the null hypothesis; common levels are 0.05 and 0.01.

Review Questions

• How does the chi-square goodness-of-fit test help in determining whether a sample fits a given distribution?
  • The chi-square goodness-of-fit test assesses whether the observed frequencies in your sample data align with the expected frequencies under a specific theoretical distribution. By comparing these frequencies using a chi-square statistic, you can evaluate how closely your sample matches what is expected. If there are significant discrepancies, it suggests that your sample may not fit the proposed distribution.
• What assumptions must be met before applying the chi-square goodness-of-fit test?
  • Before using the chi-square goodness-of-fit test, certain assumptions must be satisfied. The data should be categorical, and the sample size needs to be adequate; specifically, each category's expected frequency should typically be at least 5. Additionally, observations should be independent of one another, ensuring that each data point contributes uniquely to the analysis without overlap.
• Evaluate the implications of rejecting the null hypothesis in a chi-square goodness-of-fit test and how it affects decision-making.
  • Rejecting the null hypothesis in a chi-square goodness-of-fit test indicates that there is a significant difference between observed and expected frequencies, suggesting that the sample does not fit the proposed distribution well. This outcome can guide decision-making by prompting further investigation into potential underlying factors affecting data collection or suggesting modifications to hypotheses about population characteristics. Consequently, it informs researchers about necessary adjustments to their models or methods based on empirical evidence.

Related terms

Chi-square test: A statistical test that measures how expectations compare to actual observed data to assess if there are significant differences between categorical variables.

Degrees of freedom: The number of independent values or quantities that can be assigned to a statistical distribution; in the context of the chi-square test, it is calculated as the number of categories minus one.

Expected frequency: The anticipated number of occurrences of each category based on the null hypothesis, used to compare against observed frequencies in the chi-square test.