Null vs Alternative Hypotheses Explained with Examples

The null hypothesis (H₀) is the default claim you start with, and the alternative hypothesis (H₁) is the claim you are looking for evidence of. Think of a courtroom: the accused is presumed innocent (H₀) until the evidence proves guilt (H₁) beyond reasonable doubt. Statistics works the same way.

Getting these two statements right matters because every later step — the test statistic, the p-value, the final conclusion — depends on them. If the hypotheses are written backwards, even perfect arithmetic answers the wrong question.

Key Concepts

• Null hypothesis (H₀): the status-quo statement, always containing an equality: =, ≤, or ≥. Example: μ = 500 hours.
• Alternative hypothesis (H₁): the research claim, never containing equality: ≠, >, or <. Example: μ > 500 hours.
• Two-tailed test: H₁ uses ≠. You reject H₀ for extreme values on either side — used when the claim is just "different".
• Right-tailed test: H₁ uses >. You reject H₀ only for large values — used for claims like "more than".
• Left-tailed test: H₁ uses <. You reject H₀ only for small values — used for claims like "less than".

How It Works

To write hypotheses from a claim, follow these steps:

1. Identify the parameter (mean μ, proportion p, variance σ²) and the claimed value.
2. Translate the wording into H₁ first. "More than 500" becomes μ > 500; "different from 50%" becomes p ≠ 0.5.
3. Write H₀ as the logical opposite, always with equality. If H₁ is μ > 500, then H₀ is μ ≤ 500.
4. Let H₁ decide the tail. The inequality in H₁ tells you which tail holds the rejection region.
5. Never put the sample statistic in a hypothesis. Hypotheses are about parameters (μ, p), never about x̄ or p̂.

Worked Example

A company claims its batteries last more than 500 hours. You test 36 batteries: x̄ = 520 hours, s = 40 hours. Test at α = 0.05.

1. Hypotheses: H₀: μ ≤ 500; H₁: μ > 500 (right-tailed, matching "more than").
2. Standard error: s / √n = 40 / 6 = 6.67 hours.
3. Test statistic: t = (520 − 500) / 6.67 = 20 / 6.67 = 3.00.
4. Critical value: for df = 35 at α = 0.05 (one-sided), t* ≈ 1.69.
5. Decision: 3.00 > 1.69, so the statistic falls in the rejection region — reject H₀.
6. Conclusion: the sample supports the company's claim that average battery life exceeds 500 hours.

Common Misconceptions

• H₀ does not always say "no effect". It is simply the equality version of the claim being examined — in the battery example, H₀: μ ≤ 500 is the statement we try to overturn.
• You can never "accept" H₀. Failing to reject it means the evidence was insufficient, not that the null is proven true.
• Do not swap H₀ and H₁ to get the answer you want. The research claim — the thing you hope to prove — always goes in H₁, because only rejection of H₀ counts as statistical evidence for it.

Real-World Applications

• Product claims: regulators test a manufacturer's "lasts longer" advertisement exactly like the battery example.
• Medicine: a trial sets H₀: "new drug is no better than the old one" and looks for evidence of improvement in H₁.
• Environmental checks: H₀: "pollution is within the legal limit" versus H₁: "pollution exceeds the limit", tested on river samples.

H₀ states the default with equality, H₁ states the claim worth proving, and the tail direction follows the wording. Write them correctly, and the rest of the test almost writes itself.