R (Stats) - Read anova table and prove Ha Don't down-vote me for not including more context--You don't need it (unless you're curious for more info).
I have a dataset and have the following output in the anova table:
(the stats are based on this study: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC164340/ )

I understand that the F value is the test statistic and Pr(>F) denotes the associated p-value. Using this data, I've been asked to "report the conclusion at the 0.05 significance level for Ho : no difference..."
Do we reject Ho because of the significantly small p-value? After doing a little research, I've found that statisticians look down on making decisions as to the significance of studies/results solely based on the p-value.
Is there some other comparison that can/should be made, or should our decision solely be based on the p-value?
 A: It is a standard practice to reject the null hypothesis because the p-value is small, and if you're asked to use a significance level of $0.05$, then that means you are being told to reject the null hypothesis if the p-value is less than $0.05$.
The choice of $0.05$ is an often subjective economic decision; it is the highest probability of Type I error that you are willing to tolerate.
I suspect (but only suspect) that your statement that "statisticians look down on making decisions as to the significance of studies/results solely based on the p-value" is about proposed Bayesian alternatives to the use of p-values.  In Bayesian as opposed to frequentist statistics, one assigns probabilities to uncertain propositions even when the uncertainty does not arise from randomness and does not correspond to any proportion of a population.  Thus in Bayesian statistics one can actually assign a probability to the null hypothesis, even though doing the experiment over again -- taking another independent random sample -- will not randomly change whether the null hypothesis is true or not.  In that context, one would find the conditional probability of the null hypothesis given the data.
Here's one simple criticism of the use of p-values: If the p-value is $0.01$, then one gets results at least as extreme as what appeared with probability $0.01$, given that the null hypothesis is true.  But what if the result is equally improbable given the alternative hypothesis?
