独立性检验的假设检验

The null hypothesis states that the two variables being tested are independent events with no relationship between them. For example, in testing alcohol consumption and accident fatality, the null hypothesis claims these variables are unrelated. This hypothesis is rejected only if statistical evidence strongly suggests otherwise.

Expected frequency is calculated by multiplying the row total by the column total, then dividing by the sum of all frequencies in the contingency table. This formula generates the expected count for each table entry under the assumption that the variables are independent, providing a baseline for comparison with observed frequencies.

The chi-square test statistic measures the difference between observed frequencies and expected frequencies in a contingency table. A larger test statistic indicates greater deviation from independence. If this statistic exceeds the critical value from a chi-square table, the null hypothesis of independence is rejected.

Compare the calculated chi-square test statistic to the critical value obtained from a chi-square table at your chosen significance level. If the test statistic is larger than the critical value and falls in the critical region, reject the null hypothesis. Otherwise, accept it and conclude the variables are independent.

Rejecting the null hypothesis indicates that the two variables are dependent, meaning a statistically significant relationship exists between them. In the alcohol and accident fatality example, rejection at a 5% significance level provides sufficient evidence that these variables are related, not independent.

A contingency table organizes observed frequencies for two qualitative variables, allowing you to calculate expected frequencies and compare them systematically. The table structure enables computation of row and column totals needed for the chi-square formula and provides the framework for organizing and analyzing the relationship between variables.

Degrees of freedom determine which chi-square distribution to use when finding the critical value. For a test of independence, degrees of freedom are calculated based on the contingency table dimensions. This value ensures the critical value threshold is appropriate for your specific table size and significance level.