What is the difference between the null hypothesis and the alternative hypothesis?
answer
- two claims, unequal standing
- one of them is the default position
- the equality lives in the null
- data can only argue against the default
basics
~20 sThe null hypothesis (H0) is the default claim of no effect or no difference, written as an equality so its sampling distribution can be computed. The alternative (H1) is the competing claim that only evidence in the data can support.
solid answer
~50 sA hypothesis test sets up two mutually exclusive claims about a population parameter and treats them asymmetrically. The null, `H0`, is the default position — usually "no effect", "no difference", or "the process is on target" — and it is written with an equality such as `mu = 500` for a bottling line that claims a mean fill of 500 ml, because a single fixed value is what lets you compute the distribution of the test statistic. The alternative, `H1`, is the competing claim you would need evidence to support: `mu != 500`, or a directional form such as `mu < 500`. Data can only ever count against `H0`; the machinery is built to reject it or not, never to confirm it. Both statements are about the population parameter, not about the particular sample you drew.
go deeper
Be ready to write both hypotheses for a concrete scenario in one line each, put the equality in the null, and say plainly that the alternative is the claim needing evidence.
Explain why the null must name a specific value: it is the only way to obtain a distribution to measure surprise against, which is also why the alternative can never be the thing you compute under.
Show that you read someone else's stated hypotheses critically — catch a null written about the sample, a desired conclusion parked in the null, or hypotheses that were clearly retro-fitted to the result.
Own the framing decision: which claim sits in the null determines who carries the burden of proof and which mistake the organisation will make routinely, so it is a design choice, not a formality.
## The two claims A hypothesis test is a decision procedure about a **population parameter** — a fixed but unknown number describing the whole population, such as the mean fill volume of every bottle the line will ever produce. You never see it. You see a **sample statistic**, such as the mean of 40 bottles you actually measured, which is a known number once the data are in. The test compares two claims about that parameter: - **Null hypothesis, `H0`** — the default, status-quo claim. For the bottling line advertised at 500 ml: `H0: mu = 500`. It is the position that stands unless the data argue convincingly otherwise. - **Alternative hypothesis, `H1`** — the competing claim, the one you must produce evidence for: `H1: mu != 500`, or a directional version such as `H1: mu < 500` if only underfilling would prompt action. The two must be mutually exclusive and must between them cover the possibilities you care about. They are stated **before** the data are examined. ## Why the null carries the equals sign This is the part candidates most often cannot explain. To decide whether an observed result is surprising, you need to know what results are *expected* when the null is true — that is, you need a probability distribution to measure surprise against. An equality supplies one: if `mu = 500` and you know how the sample mean varies from sample to sample, the whole distribution of possible sample means is pinned down, and you can say how far out into its tail an observation of, say, 497.2 ml sits. The alternative supplies no such thing. `mu != 500` covers 499, 480, 512 and infinitely many other values, each with its own distribution. There is nothing single to compute against. This asymmetry — one hypothesis computable, the other not — is the structural reason the whole procedure runs in only one direction. One-sided nulls such as `H0: mu >= 500` are also written, but the test is carried out at the boundary, `mu = 500`, because that is the case least favourable to rejection: if the data reject at the boundary, they reject for every value further inside the null region. ## Test statistic and critical value Once the two hypotheses are fixed, two more objects follow. The **test statistic** rescales the observed data into a standardised distance from the null value — roughly "how many standard errors is the observed result away from what `H0` claims". Because `H0` names a specific value, the distribution of this statistic when `H0` is true is known. The **critical value** is the threshold on that same scale that separates results you will call surprising from results you will not. It is determined by the significance level `alpha`, the tail probability you are willing to live with when the null is actually true, together with whether the alternative is directional. At `alpha = 0.05`, a two-sided alternative rejects when `|z| > 1.96`; a one-sided alternative in the upper direction rejects when `z > 1.645`. Both the level and the direction belong to the design, not to the results. ## The asymmetry of the verdict There are only two possible outcomes, and they are not mirror images: 1. The statistic lands beyond the critical value — **reject `H0`**. This is a positive conclusion: the data would be unusual if the null were true, so the null is set aside in favour of the alternative. 2. The statistic does not — **fail to reject `H0`**. This is not a positive conclusion about the null. It means the data are compatible with `H0`, but they may also be compatible with plenty of values in `H1` that this sample was too small or too noisy to distinguish. That is why results are phrased as "we reject" or "we fail to reject", never "we accept". The burden of proof rests on the alternative throughout. ## Worked framing A plant claims its line fills bottles to a mean of 500 ml, and quality control wants to know whether the line has drifted in either direction. - `H0: mu = 500` — the line is on target. - `H1: mu != 500` — the line has drifted; both overfilling (a cost) and underfilling (a compliance problem) matter, so the alternative is non-directional. - Fix `alpha` and therefore the critical values before sampling. - Sample bottles, compute the test statistic, compare to the critical value, and report the verdict. Notice the direction of the logic: the plant does not set out to prove the line is fine. It sets out to see whether the evidence is strong enough to say it is not. ## Common errors - **Putting the desired claim in the null.** Whatever you want to demonstrate belongs in `H1`, because only `H1` can be supported by rejection. - **Writing hypotheses about the sample.** "The sample mean equals 500" is not a hypothesis; it is either true or false by inspection. - **Overlapping hypotheses.** `H0: mu <= 500` with `H1: mu <= 505` is not a valid split. - **Claiming the null was proven.** No amount of non-significant data establishes an exact equality.
- Why must the hypotheses be stated about a population parameter rather than a sample statistic?Because the test asks whether the sample is surprising given a claim about the population. Once you have the data, the sample mean is simply a known number — there is nothing left to test about it. Writing `H0` as "the sample mean is 500 ml" makes the hypothesis about the very quantity you measured, which destroys the comparison between what you observed and what the population claim predicts.
- Can the null hypothesis be an inequality such as mu <= 500?Yes. One-sided nulls are written that way, and the test is then carried out at the boundary value `mu = 500` because that is the case least favourable to rejection. If the data are extreme enough to reject at the boundary, they reject for every value further inside the null region, so the arithmetic ends up identical to testing the equality.
- What roles do the test statistic and the critical value play once the hypotheses are fixed?The test statistic rescales the observed data into a standardised distance from the null value, so that its distribution when the null holds is known. The critical value is the threshold on that scale marking the rejection region for the chosen significance level. If the statistic falls beyond it you reject; both the threshold and the direction are settled by the design, not by the data.
It works like a criminal trial: the null is 'not guilty', assumed until the evidence is strong enough to reject it, and an acquittal never proves innocence.
saying these in an interview costs you the question
- Says the null is whatever the researcher hopes to prove.
- States the hypotheses about the sample mean instead of the population mean.
- Claims a test can prove the null hypothesis true.
- Writes two hypotheses that overlap or leave gaps.