Null and alternative hypotheses (H₀ H₁)

How to read it

OpenStax’s Introductory Statistics describes H₀ as “a statement of no difference between the variables” and Hₐ as a claim about the population that contradicts H₀, “what we conclude when we reject H₀”. Penn State’s STAT 500 course notes write the alternative as Hₐ or H₁ and call it the research hypothesis; NIST’s Engineering Statistics Handbook writes H₀ and Hₐ. Both hypotheses are statements about the population, so they are written with the population’s letters: OpenStax’s examples include H₀: μ = 2.0 against Hₐ: μ ≠ 2.0 for a mean (μ), and H₀: ρ = 0 for a correlation (ρ).

Also searched as: alternative hypothesis, alternative hypothesis symbol, h0, H₀ H₁, h1, ha, null and alternative hypotheses symbol, null hypothesis symbol.

Reject or fail to reject

A test ends in one of two decisions, “reject H₀” or “do not reject H₀” (OpenStax). NIST’s handbook explains why the second is not called accepting it: to accept a hypothesis does not mean that it is true, only that there is no evidence to believe otherwise. Fisher put it more strongly in 1935: the null hypothesis “is never proved or established, but is possibly disproved”. How strong the evidence against H₀ is, is given by the P-value (p).

Which sign goes in H₀

OpenStax’s rule is that H₀ always has a symbol with an equal in it (=, ≤ or ≥) and Hₐ never does (≠, < or >). It adds that many researchers write = in the null hypothesis even when the alternative has > or <.

Look-alikes

The small 0 is a zero: Unicode’s SUBSCRIPT ZERO, U+2080, with SUBSCRIPT ONE, U+2081, for H₁. The α that appears beside H₀ in a test is the chance of rejecting H₀ when it is true, the Type I error.