184. One-Way ANOVA

Compares the means of three or more groups based on one independent variable

• ${𝐻}_0$: ${𝜇}_1={𝜇}_2=\dots ={𝜇}_{𝑘}$
• ${𝐻}_1$: At least one ${𝜇}_{𝑖}$ differs from the others

Step 1: Calculate Between-Group Variation ($𝑆{𝑆}_{\text{between}}$)

• ${𝑛}_1$: Number of observations in group $𝑖$
• ${\overline{𝑋}}_{𝑖}$: Mean of group $𝑖$
• ${\overline{𝑋}}_{\text{overall}}$: Overall mean of all groups

Step 2: Calculate Within-Group Variation ($𝑆{𝑆}_{\text{within}}$)

• ${𝑋}_{𝑖𝑗}$: Observation $𝑗$ in group $𝑖$

Step 3: Calculate Total Variation ($𝑆{𝑆}_{\text{total}}$)

Step 4: Calculate Mean Squares

• Mean Square Between ($𝑀{𝑆}_{\text{between}}$)

• Mean Square Within ($𝑀{𝑆}_{\text{within}}$)

• $𝑁$: total number of observations
• $𝑘$: number of groups

Step 5: Calculate the F-statistic

Step 6: Decision Rule

• Compare the F-statistic to the critical value from the F-distribution table (based on chosen significance level $𝛼$).
• Alternatively, compare the p-value to the significance level $𝛼$.
• Reject ${𝐻}_0$ if the F-statistic is greater than the critical value or if the p-value is less than $𝛼$, indicating that at least one group mean is significantly different.