Null distribution

The null distribution is the sampling distribution of a statistic in a world where $H_0$ holds: centered at the value $H_0$ names, with a spread computed under that same assumption. It is not the distribution of the population and not the distribution of the values in your sample. It describes one summary number across every sample of size $n$ you might have drawn.

With $H_0: \mu = 100$, $\sigma$ (sigma) known to be 15 and $n = 25$, the null distribution of $\bar{x}$ (x-bar) is normal, centered at 100, with standard error $15/\sqrt{25} = 3$. An observed $\bar{x} = 106.2$ sits $\frac{106.2 - 100}{3} = 2.07$ standard errors above the center, and the area beyond it, 0.0194, is the one-sided p-value. Change the hypothesized mean and the whole curve slides sideways, while the sample mean you actually observed leaves it exactly where it is.

That last point has a consequence students meet without recognizing it. Testing $H_0: p = 0.50$ with $n = 100$, the standard deviation is $\sqrt{0.50(0.50)/100} = 0.05$, built from the hypothesized 0.50 even when the sample gave $\hat{p}$ (p-hat) of 0.60. A confidence interval has no hypothesis to borrow from, so it uses $\hat{p}$ instead and gets $\sqrt{0.60(0.40)/100} = 0.049$. Same data, two denominators, which is one reason an interval and a test can disagree.

A simulated null distribution, the kind a randomization test builds, is a finite pile of reshuffled outcomes rather than a curve, so its p-value moves in steps of 1 divided by the number of trials and shifts a little when you rerun it. A theory-based curve earns its shape only when the conditions hold; without them the p-value is an area under the wrong curve, correct arithmetic and all.

The area this distribution hands back is AP Statistics topic 3.6, p-Values.