Question

If the sample means for each of the k treatment groups were identical, what would be the observed value of the ANOVA test statistic?

• A. 0.0
• B. 1.0
• C. 0.1
• D. Very small number

Hint:

The F-statistic in ANOVA is a ratio of between-group variation to within-group variation. If there is absolutely no variation between the group means, the numerator becomes zero, making $F = 0$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In Analysis of Variance (ANOVA), the F-test statistic is used to test the null hypothesis that all treatment group means are equal.
Key Formula or Approach:
The F-statistic is computed as: \[ F = \frac{\text{Mean Square for Treatments (MST)}}{\text{Mean Square for Error (MSE)}} \]

Step 2: Detailed Explanation:
The Sum of Squares for Treatments (SST), which measures the variation between different group means, is given by: \[ \text{SST} = \sum_{i=1}^k n_i (\bar{Y}_{i\cdot} - \bar{Y}_{\cdot\cdot})^2 \] where $\bar{Y}_{i\cdot}$ is the mean of the $i$-th treatment group and $\bar{Y}_{\cdot\cdot}$ is the grand mean of all observations.
If the sample means for each of the $k$ treatment groups are identical: \[ \bar{Y}_{1\cdot} = \bar{Y}_{2\cdot} = \dots = \bar{Y}_{k\cdot} = \bar{Y}_{\cdot\cdot} \]
This perfect uniformity means that: \[ \bar{Y}_{i\cdot} - \bar{Y}_{\cdot\cdot} = 0 \text{ for all } i \] \[ \text{SST} = 0 \implies \text{MST} = \frac{\text{SST}}{k-1} = 0 \]
Substituting $\text{MST} = 0$ into the F-statistic formula: \[ F = \frac{0}{\text{MSE}} = 0 \]
Thus, the observed F-statistic is exactly zero.

Step 3: Final Answer:
The observed value of the ANOVA test statistic is 0.0.