P-value vs Critical F
Comparing the observed F to a critical value and comparing the p-value to alpha are two routes to the same decision. They differ only in how the answer is communicated. This page explains why the two are equivalent and when each is the right thing to report.
Two equivalent decisions
If the observed F exceeds the critical F at level alpha, the upper-tail probability beyond the observed F is less than alpha by definition. That upper-tail probability is the p-value. So F > F_alpha if and only if p < alpha. The two procedures cannot disagree.
Reporting p-value the right way
Report exact p-values rather than thresholded statements like p less than 0.05. Most readers want to see whether the result was borderline (p approximately equal to 0.04) or strongly significant (p approximately equal to 1e-6). Avoid p equal to 0; if your software returns zero it really means below the precision floor, conventionally written p less than 0.0001. Use scientific notation for very small p-values.
When the critical F still matters
Two situations still favour reporting the critical F: when you are showing the calculation by hand for teaching, and when the test does not have a closed-form p (rare for F but common for some specialised distributions). For routine ANOVA work the p-value plus F(df1, df2) is enough.