Null hypothesis

The null hypothesis, written $H_0$ and read H-naught (or H-zero), is the skeptical claim that nothing is going on: no effect, no difference, no association. It is a statement about the population, never about the sample in hand. When it concerns a parameter such as $p$ or $\mu$ (mu) it pins that parameter to a single number, which is why it carries an equals sign while the alternative $H_a$ carries $<$, $>$ or $\ne$.

That single number is what makes a test possible. Suppose you flip a coin 100 times to test whether it is fair. Setting $H_0: p = 0.5$ fixes the sampling distribution of $\hat{p}$ (p hat) completely: it is centered at 0.5 with standard deviation $\sqrt{\frac{0.5(0.5)}{100}} = 0.05$. An alternative such as $H_a: p \ne 0.5$ names no particular value, so no distribution can be built from it. Every test compares the observed data against a distribution built from the null, which is why the null has to be specific enough to determine one, and that is the whole content of the phrase assume the null is true.

Stating the hypothesis about the wrong quantity is the most common setup error. $H_0: \hat{p} = 0.5$ is not a hypothesis, because $\hat{p}$ is something you measured rather than something unknown. You already know what it equals; there is nothing left to test.

The second error lives in the conclusion. A test either rejects $H_0$ or fails to reject it, and failing to reject is not the same as accepting. A large p-value says the data are consistent with $H_0$, but they are also consistent with a whole range of nearby parameter values, which is precisely what a confidence interval puts on display. Rejection is not proof either: a Type I error is the case where a true null gets rejected, and its long-run rate is the significance level $\alpha$ (alpha) chosen before the data arrive. The null is a benchmark to measure evidence against, not a claim the test ever settles.