Why Do We Assume the Null Hypothesis Is True?

The short answer: only the null gives you a number to compute with

Every hypothesis test ends by computing a probability. To compute any probability you need a fully specified model: a center, a spread, and a shape. Look at what each hypothesis offers.

• $H_0: p = 0.5$ names one value. From it you get a sampling distribution for $\hat{p}$ ("p-hat") centered at 0.5 with standard deviation $\sqrt{\frac{0.5(0.5)}{n}}$. That is a complete model. You can compute with it.
• $H_a: p > 0.5$ names an infinite set of values: 0.501, 0.6, 0.85, and everything between. Each one implies a different center and a different spread. There is no single distribution to compute against.

That asymmetry is the whole reason. You assume the null not because it is likely, and not because you believe it, but because it is the only hypothesis specific enough to produce a number.

This is also why the null always carries an equals sign. A hypothesis like $p \geq 0.5$ would leave the same problem: no single value, no single distribution. How to know what the null hypothesis is gives the pattern for each AP procedure.

Watch the alternative fail to produce a distribution

Take $n = 100$ and a claim about a proportion. Under the null $H_0: p = 0.5$, the sampling distribution of $\hat{p}$ has standard deviation

One value in, one number out. Now try to do the same under $H_a: p > 0.5$. Every candidate value gives a different answer:

Now ask the question a p-value asks: how likely is a sample proportion of at least 0.61? Under $p = 0.55$ the answer is 0.1139. Under $p = 0.60$ it is 0.4191. Under $p = 0.70$ it is 0.9752. Three different answers from the same alternative hypothesis, because the alternative was never one hypothesis in the first place.

The null gives exactly one answer, so it is the one you can test against. This is the null distribution: the sampling distribution you would see if the null were exactly true.

It is a computational assumption, not a belief

"Assume the null is true" sounds like you are taking a side. You are not. The assumption is a temporary premise you adopt in order to see where it leads.

The structure is the same one used in a proof by contradiction. Suppose the claim holds. Work out what the world would look like. Compare that to what you actually saw. If what you saw is wildly improbable under the supposition, the supposition is in trouble.

The courtroom version is the presumption of innocence. A jury starts from "not guilty" not because they believe it, but because that is the position the evidence has to move them away from. The null gets the same treatment, and for the same reason: it gives the argument a fixed starting point that both sides agree on before any data arrive.

One difference from a mathematical proof is worth naming. A proof by contradiction ends in an impossibility, so you can reject the premise outright. A hypothesis test ends in a probability, so all you can say is that the data are improbable under the null, never impossible. That gap is exactly why a Type I error exists: sometimes the improbable happens and you reject a true null. See Type I vs Type II errors.

The same move appears in every AP procedure

Once you see the pattern, every test on the course looks like the same machine with different fuel. In each case the null supplies the missing number, and the standard error is computed from the null value, not from your data, whenever the null gives you one.

• One proportion. $H_0: p = p_0$ gives $SE = \sqrt{\frac{p_0(1 - p_0)}{n}}$. Note that $p_0$ appears here, not $\hat{p}$. This is the one place students most often substitute the wrong quantity. The confidence interval uses $\hat{p}$ because there is no null value to borrow; the test uses $p_0$ because there is.
• Two proportions. $H_0: p_1 = p_2$ says the two populations share one common proportion, so you pool the samples into one estimate $\hat{p}_c$ and use it for both groups. The null is what licenses the pooling.
• One mean or a mean difference. $H_0: \mu = \mu_0$ centers the sampling distribution of $\bar{x}$ at $\mu_0$, so the standardized statistic is compared to a t-distribution centered at 0. Here the spread has to come from the data, since the null says nothing about $\sigma$, and that substitution is why the reference curve is a t-distribution rather than a normal one. See what degrees of freedom are.
• Two means. $H_0: \mu_1 = \mu_2$ says the difference of the two population means is 0, so the sampling distribution of $\bar{x}_1 - \bar{x}_2$ is centered at 0. Both standard deviations still come from the data, because the null says nothing about either $\sigma$.
• Chi-square for homogeneity or independence. The null generates every expected count through $E = \frac{(\text{row total})(\text{column total})}{\text{grand total}}$. Without the null there are no expected counts, and with no expected counts there is no statistic at all. See chi-square tests explained.

In every row, delete the null and the arithmetic stops. That is the cleanest test of whether you have understood this.

Why you never accept the null

The assumption runs one way only, and the wording of your conclusion has to respect that.

A small p-value says the data are hard to explain under the null, which is evidence against it. A large p-value says the data are perfectly compatible with the null. It does not say the null is true, because the data are usually compatible with plenty of other values too. A sample proportion of 0.52 is consistent with $p = 0.50$, and also with $p = 0.51$ and $p = 0.53$.

So the two allowed conclusions are reject $H_0$ and fail to reject $H_0$. Never "accept $H_0$," and never "prove $H_0$." The phrasing feels pedantic until you notice it is the direct consequence of only ever having computed probabilities in one direction. How to write a hypothesis test conclusion has the exact sentences.

If you want to know which parameter values are actually plausible, a hypothesis test is the wrong tool. Build a confidence interval instead; that is the procedure designed to answer it.

Two misreadings this fixes

Most p-value confusion dissolves once the assumption is clear.

"The p-value is the probability the null is true." It cannot be. The null was assumed true for the entire calculation, so within that calculation its probability is 1. The p-value is $P(\text{data} \mid H_0)$, and swapping the two sides of the bar gives a different quantity that no AP procedure computes.

"A small p-value proves the alternative." The calculation never used the alternative to compute anything. The alternative only decided which tail or tails counted as "at least as extreme." A small p-value says the null explains the data badly, which is an argument by elimination, not a direct measurement of the alternative. That is also why a p-value cannot tell you how big the effect is, only how poorly the null fits it. What is a good p-value works through what the number can and cannot support.