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ANOVA (Analysis of Variance)

ANOVA (Analysis of Variance) is a statistical method used to test differences in means among three or more groups. This is achieved by partitioning the total variance in the data into between-group variance and within-group variance.

How ANOVA Works:

1. Formulation of Hypotheses: Null hypothesis (\(H_0\)) and alternative hypothesis (\(H_A\)) are stated. The null hypothesis asserts that all group means are equal.
2. Calculation of Variances: Total variance is divided into two parts:
   • Between-Group Variance: Measure of differences between group means.
   • Within-Group Variance: Measure of variance within each group.
3. F-Test: An F-test is conducted by calculating the ratio of between-group variance to within-group variance.
4. Decision Making: Based on the F-test, a decision is made whether to reject the null hypothesis. A significant F-value indicates differences between the groups.

Applications of ANOVA:

• Experimental Design: Testing differences in means under different experimental conditions.
• Quality Control: Comparing product quality across different production lines.
• Educational Research: Investigating performance differences between different schools or classes.
• Medical Studies: Examining efficacy differences between different treatment groups.

Example:

Suppose we want to know if there is a significant difference in average test scores among three different teaching methods. ANOVA could be used to answer this question.