Correct Answer - Option 3 : 23.33 / 7.62
ANOVA (Analysis of variance)
- It is a statistical technique that is used to check whether the difference of means of two or more groups is significant.
- It helps to find out if whether the null hypothesis rejected or accepted.
- ANOVA checks the impact of one or more factors by comparing the means of different samples.
- ANOVA uses the F-test to determine whether the variability between group means is larger than the variability of the observations within the groups. If that ratio is sufficiently large, you can conclude that not all the means are equal.
- For example - Different teaching methods: Demonstration, lecture, and experiment methods are used to teach a particular topic. ANOVA is used to check which method is better than the others.
- The F-ratio is defined as the ratio of the between-group variance (MSB) to the within-group variance (MSW).
Steps to Calculate the F-ratio:
- Step 1: Calculate all the means for all the groups. Then calculate overall means with all the data combined as one single group.
-
Step 2: Form the null and alternate hypothesis
- The null hypothesis assumes that there is no variance in different groups. In other words, the means are the same
- The alternate hypothesis states the means are different.
-
Step 3: Calculate the Sum of Squares Total:
- The SStotal is the external Sum of Squares(SS) of the mean of the entire set in all the groups. \(SS_{total} = \displaystyle\sum_{j=1}^{n} ({\bar{X_{j}} - {\bar{X}}})^{2}\)
- Calculate the Sum of Squares within the group: This is the internal sum of squares within each group. \(SS_{within} = \displaystyle\sum_{j=1}^{k} \displaystyle\sum_{j=1}^{n} ({{X} - {\bar{X}}}_{j})^{2}\)
- Calculate the Sum of Squares between the group: This is the sum of squares with the groups taken as single units. \(SS_{between} = \displaystyle\sum_{j=1}^{k} ({\bar{X_{j}} - {\bar{X}}})^{2}\)
- Step 4: Calculate the Degrees of Freedom (df)
- Calculate the Degrees of Freedom Total (dft): \(df_{total} = n - 1\)
- Calculate the Degrees of Freedom within groups (dfw): \(df_{within} = k - 1\)
- Calculate the Degrees of Freedom between groups (dfb): \(df_{between} = n - k\)
- Step 5: Calculate the Mean Squares
- Calculate the Mean Squares between groups (MSB): \(MS_{between} = \frac{SS_{between}}{df_{between}}\)
- Calculate the Mean Squares within groups (MSW): \(MS_{within} = \frac{SS_{within}}{df_{within}}\)
- Step 6: Calculate the F statistic Value:
- \(F = \frac{between\ group\ variance}{within\ group\ variance} = \frac{MSB}{MSW} = \frac{23.33}{7.62}\)
Ans: Option 3
