Null and alternative hypotheses

A trial with a head start

A hypothesis test puts a claim about a parameter on trial. Two statements are written before any data are examined:

• the null hypothesis $H_0$: the status quo, the "nothing has changed" case, always written with an equality — $H_0:\mu =45$;
• the alternative hypothesis $H_a$: the claim the analyst is trying to establish, written with $<$, $>$ or $\ne$.

$H_0$ gets a head start: it is retained unless the data make it implausible. The burden of proof sits on $H_a$, so the conclusion is either reject $H_0$ in favour of $H_a$ or fail to reject $H_0$ — never accept $H_0$, because a dull case that was not disproved has not been proved either.

Both hypotheses are about the parameter $\mu$, never about the statistic. $H_0:\bar{x}=45$ is meaningless — $\bar{x}$ is known the moment the sample is in.

Worked example — three questions, three alternatives

Kitchener Ridge Logistics has historically taken 45 days to be paid. Three questions a controller might ask:

"Has the new collections process cut mean days-to-pay below 45?" The claim is a reduction. $H_0:\mu =45$, $H_a:\mu <45$. A lower-tailed test.

"Has mean days-to-pay changed?" No direction is claimed. $H_0:\mu =45$, $H_a:\mu \ne 45$. A two-sided test.

"The supplier says invoices are paid within 45 days on average; the controller suspects it takes longer." The supplier's claim is the status quo; the controller wants to show it is wrong in a specific direction. $H_0:\mu =45$, $H_a:\mu >45$. An upper-tailed test.

One more shape the paper uses: a claim stated as an inequality, "mean days-to-pay is at least 45", to be refuted. Write $H_0:\mu =45$ (the boundary of the claim; some texts write $\mu \ge 45$, and either is accepted) and $H_a:\mu <45$. The claim under attack is the null; the direction of the attack is the alternative.

The tail comes from the question

The alternative is chosen from the wording — cut, fallen, longer, more than, changed — and it is chosen before looking at $\bar{x}$. Seeing $\bar{x}=43.1$ and then writing $H_a:\mu <45$ because the data lean that way is a serious error: it doubles the chance of a false positive, exactly as looking at the cards before betting would. If the question asks whether the mean has changed, the test is two-sided even if the sample happens to sit far below 45.

Which claim goes where

A rule of thumb the marker rewards: the claim that would be costly to assert wrongly — "the new process works", "the control has failed", "the drug is effective" — sits in $H_a$, where it needs evidence. The safe default sits in $H_0$. Making the claimant prove the case is what the head start is for. Define $\mu$ in words when you write the hypotheses: $\mu$ = mean days-to-pay for all current invoices — it is a mark, and it fixes what the conclusion sentence will be about.