Mistake Master
A Large p-Value Fails to Convict. It Does Not Acquit
You'll learnhow to carry a two-proportion z-test from statistic to sentence: the gap divided by the pooled standard error, the p-value as a tail area of the null's own distribution, a conclusion that carries the direction and both populations, why failing to reject never becomes a finding of equality, and how the test lines up with the confidence interval built from the same data.
The arithmetic left after Topic 3.12's setup is one division and a tail area: the observed gap of 0.10 sits 2.37 pooled standard errors from where the null expects it, and two populations with a shared proportion would produce a gap that far out about 9 times in a thousand. The picture on every step is the null's own sampling distribution — the world the null describes, drawn as a curve — with the observed difference planted on it. A gap in the far tail is evidence; the shaded area is exactly how much surprise the data carry. Then the sentence, which is where this topic is actually lost: a small p-value earns a direction and two populations, a large one earns precisely "not convincing evidence" — and a pilot study whose p-value is 0.28 has not shown the systems perform equally, because at its sample sizes only a gap past 11 points was ever going to register.