Applied statisticians should know
conditional probability inside and out; this is the source of a great deal of misunderstandings about p-values and type I assertion probability $\alpha$ as well as holding back more usage of the Bayesian paradigm
experimental design, sources of bias and variability
measurement properties and how to optimize them
how to ...
aov/anova ouputs are not adjusted for multiple comparisons. This is actually a good thing. Comparison adjustments are not always required, and different adjustments are favored in different situations. So the software is leaving the analyst in charge and not making any assumptions on her behalf.
Simple effects are defined as the effect of one variable, but restricted only to some level of some other variable. This meaning is used in for instance the R package (on CRAN) phia and discussed in its vignette. Another term which seems to be used with the same meaning is Simple slope.
For a simple illustration, let there be two factors, a and b with ...
A statistician, as in a Mathematical that is specialized in Statistics, as in my experience, has a basic set of theorical knowledge in:
Probability Theory (most importat one)
Exploratory Data Research
Keep in mind that, even if it's not as complex as other ...