Problem
In statistics, the phrase “f on a test” often means the F-statistic used in an F-test. Consider a one-way ANOVA with three groups (each with \(n=5\)). The sample means are \(72\), \(78\), \(84\) and the sample variances are \(16\), \(20\), \(18\). Compute the ANOVA test statistic \(F\). At significance level \(\alpha=0.05\), decide using the critical value \(F_{0.95}(2,12)=3.89\).
Step 1: state the one-way ANOVA F-test structure
\[ F=\frac{\text{MSB}}{\text{MSW}} \]
\[ \text{MSB}=\frac{\text{SSB}}{k-1},\qquad \text{MSW}=\frac{\text{SSW}}{N-k} \]
Here \(k\) is the number of groups and \(N\) is the total sample size. For this problem, \(k=3\) and \(N=5+5+5=15\).
Step 2: compute the overall mean
With equal group sizes, the overall mean \(\bar{y}\) is the average of the group means:
\[ \bar{y}=\frac{72+78+84}{3}=\frac{234}{3}=78 \]
Step 3: compute SSB (between-groups sum of squares)
For equal group size \(n=5\),
\[ \text{SSB}=\sum_{j=1}^{k} n\left(\bar{y}_j-\bar{y}\right)^2 \]
\[ \text{SSB}=5\left[(72-78)^2+(78-78)^2+(84-78)^2\right] \]
\[ \text{SSB}=5\left[36+0+36\right]=5\times 72=360 \]
Step 4: compute SSW (within-groups sum of squares)
Using sample variances \(s_j^2\),
\[ \text{SSW}=\sum_{j=1}^{k}(n-1)s_j^2 \]
\[ \text{SSW}=4\times 16+4\times 20+4\times 18=4\times (16+20+18)=4\times 54=216 \]
Step 5: compute degrees of freedom, mean squares, and \(F\)
\[ \text{df}_{B}=k-1=3-1=2,\qquad \text{df}_{W}=N-k=15-3=12 \]
\[ \text{MSB}=\frac{360}{2}=180,\qquad \text{MSW}=\frac{216}{12}=18 \]
\[ F=\frac{180}{18}=10 \]
The F-statistic in a one-way ANOVA is always nonnegative because it is a ratio of mean squares; a negative “F on a test” value indicates a computation or transcription error.
Step 6: decision using the critical value
The rejection rule for a right-tailed ANOVA F-test is: reject \(H_0\) if \(F \ge F_{0.95}(2,12)\).
\[ 10 \ge 3.89 \quad \Rightarrow \quad \text{Reject } H_0 \text{ at } \alpha=0.05 \]
Conclusion: the data provide sufficient evidence that at least one population mean differs among the three groups.
ANOVA summary table
| Source | SS | df | MS | F |
|---|---|---|---|---|
| Between (Treatments) | \(360\) | \(2\) | \(180\) | \(10\) |
| Within (Error) | \(216\) | \(12\) | \(18\) | |
| Total | \(576\) | \(14\) |