Statistics · Inference
Null vs Alternative Hypothesis: How to Set Them Up Correctly
Getting the hypotheses wrong dooms the whole test before you’ve calculated anything. Here are the rules, the direction question, and the mistakes that cost the most marks.
Every hypothesis test starts with two statements: the null and the alternative. Set them up wrong and everything downstream — the test statistic, the p-value, the conclusion — is answering the wrong question. It’s the foundation, and it’s where a surprising number of marks quietly leak away.
The good news: there are only a few rules, they’re consistent, and once you internalise them you’ll set up any test correctly in seconds.
The two hypotheses
🔑 Key terms
Null hypothesis (H₀) — the “nothing interesting” statement: no effect, no difference, no relationship. It’s the default the world is assumed to be in until evidence says otherwise. It always contains an equals sign.
Alternative hypothesis (H₁) — the claim you’re actually interested in: there is an effect, a difference, a relationship. It’s what you’re gathering evidence to support.
The whole logic runs like a trial. You assume the null (innocence) and ask whether the evidence is strong enough to reject it. You never “prove” the alternative — you either reject the null or fail to. This is why you can’t “accept the null”: a jury returns “not guilty,” never “innocent.”
💡 Insight — three rules that never change
1. Hypotheses are always about the population, never the sample. You write H₀: μ = 50, using the population parameter μ — never x̄ (the sample mean). You already know the sample mean; there’s nothing to test about it. The test is about the population it came from.
2. The null always gets the equals sign. H₀: μ = 50, or μ ≥ 50, or μ ≤ 50. The alternative gets the strict inequality or ≠. The equals sign lives with the null because you need a specific value to calculate probabilities from.
3. Decide the direction before you see the data. One-sided or two-sided is a choice you make from the research question, in advance — never after peeking at which way the results went.
One-sided or two-sided?
This is the choice that confuses people. It comes down to what your research question is really asking.
| Your question | Alternative | Type |
|---|---|---|
| Is the mean different from 50? | H₁: μ ≠ 50 | Two-sided |
| Is the mean greater than 50? | H₁: μ > 50 | One-sided |
| Is the mean less than 50? | H₁: μ < 50 | One-sided |
Use two-sided when you care about a difference in either direction — which is most of the time, and the safer default. Use one-sided only when the research question is genuinely directional (“does this drug lower blood pressure?”) and a difference the other way would be irrelevant or impossible to act on.
⚠ Common error — choosing one-sided after seeing the data
A one-sided test is easier to pass in its chosen direction — it concentrates all your significance budget on one tail. So switching to one-sided after noticing your result trends that way is a form of cheating: you’re rigging the test to clear the bar. Markers and reviewers treat this as a serious error. The direction must come from the question, decided before the data. When in doubt, go two-sided — it’s the honest default.
⚠ Common error — hypotheses about the sample
Writing H₀: x̄ = 50 instead of H₀: μ = 50 is one of the most common setup mistakes. The sample mean is a number you calculated — there’s nothing probabilistic to test about it. Hypotheses are always claims about the unknown population parameter. Use Greek letters (μ, σ, p, β), not sample statistics.
💡 Insight — the null is a strategic choice
Why is the “boring” statement the one we test? Because you can only calculate probabilities from a specific value. “μ = 50” is precise enough to build a distribution around; “μ is something interesting” isn’t. So we assume the precise, boring thing, work out how surprising the data would be under it, and reject it if the data are surprising enough. The whole machinery needs a fixed number to start from, and the null provides it.
Practice questions
Q1. A company claims its lightbulbs last 1,000 hours on average. You suspect they last less. Write H₀ and H₁.
Q2. A researcher wants to know if a new fertiliser changes crop yield (up or down). Write the hypotheses and state the type.
Q3. A student writes H₀: x̄ = 75. What’s wrong, and how should it read?
Q4. A study is planned as two-sided. After collecting data, the effect looks positive and p = 0.06 two-sided (which would be 0.03 one-sided). The author switches to one-sided to claim significance. What’s the problem?
Worked answers
A1. H₀: μ = 1000 (or μ ≥ 1000); H₁: μ < 1000. One-sided, because your suspicion is directional — you specifically think they last less. The null carries the equals sign; the population mean μ, not the sample.
A2. H₀: μ = μ₀ (yield equals the no-fertiliser value); H₁: μ ≠ μ₀. Two-sided, because “changes… up or down” means you care about a difference in either direction.
A3. The hypothesis is written about the sample mean x̄, but hypotheses must be about the population parameter. It should read H₀: μ = 75. You already know x̄ from your data — the test is about the population μ you can’t observe directly.
A4. The direction was not chosen in advance — it was chosen after seeing which way the data pointed, specifically to cross the significance line. This is p-hacking. A one-sided test is only legitimate if the directional question was set before data collection. Switching post hoc doubles the real false-positive rate for that direction and misrepresents the strength of evidence. The honest report is “p = 0.06, two-sided, not significant at 5%.”
The short version
• H₀ = no effect, contains the equals sign, is assumed true until disproven.
• H₁ = the interesting claim, gets the inequality.
• Always about the population (μ, p), never the sample (x̄).
• Direction (one vs two-sided) is chosen before seeing data. Two-sided is the safe default.
• You reject or fail to reject H₀ — you never “accept” it or “prove” H₁.
• H₁ = the interesting claim, gets the inequality.
• Always about the population (μ, p), never the sample (x̄).
• Direction (one vs two-sided) is chosen before seeing data. Two-sided is the safe default.
• You reject or fail to reject H₀ — you never “accept” it or “prove” H₁.
Once your hypotheses are set, the machinery follows: what the p-value means and the two ways to be wrong.
References
1. Moore, D.S., McCabe, G.P. & Craig, B.A. (2021) Introduction to the Practice of Statistics. 10th edn. New York: W.H. Freeman.
2. Fisher, R.A. (1935) The Design of Experiments. Edinburgh: Oliver & Boyd.
Hypothesis testing gets a deliberately slow chapter in Statistics Made Simple.
Because the logic feels slippery — and there’s a good reason for that, which most courses never tell you.