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Null Hypothesis

Formula

\[ H_0: \theta = \theta_0 \quad \text{(example form)} \]

Parameters

  • \(H_0\): baseline/default hypothesis
  • \(\theta\): parameter of interest
  • \(\theta_0\): hypothesized value

What it means

The null hypothesis is the reference claim tested against the observed data.

What it's used for

  • Formal hypothesis tests.
  • Defining p-values and rejection thresholds.

Key properties

  • Chosen before seeing results in principled analyses.
  • Often represents "no effect" or a baseline model.

Common gotchas

  • Failing to reject \(H_0\) is not proof that \(H_0\) is true.
  • Poorly chosen nulls can make tests uninformative.

Example

In an A/B test, \(H_0\) may state that the two conversion rates are equal.

How to Compute (Pseudocode)

Input: research question and statistical test setup
Output: null-hypothesis specification H0

define the parameter/contrast of interest
state H0 as a baseline/no-effect/no-difference model for that quantity
pair H0 with an alternative hypothesis H1 and a planned test statistic
return the hypothesis specification

Complexity

  • Time: Not an algorithmic computation; this is a study-design/specification step
  • Space: \(O(1)\) for the formal hypothesis statement
  • Assumptions: Hypothesis specification precedes data analysis and determines downstream test/p-value computations

See also