Index: The Book of Statistical ProofsGeneral TheoremsFrequentist statisticsHypothesis testing ▷ Minimum required sample size

Definition: Consider a hypothesis test concerning parameter $\theta$. We have $H_0: \theta \in \Theta_0$ versus $H_1: \theta \in \Theta_1$, where ${\Theta_0, \Theta_1}$ is a partition of $\Theta$. Let the probability of rejecting the null hypothesis for the test with $n$ observations be $\kappa_n(\theta)$.

Given a pre-specified significance level $\alpha \in [0,1]$ and a desired power $1 - \beta \in [0,1]$, the test must satisfy two conditions:

  1. significance constraint: The probability of a type I error must not exceed $\alpha$: \(\label{eq:type1error} \sup_{\theta \in \Theta_0} \kappa_n(\theta) \le \alpha \; .\)
  2. power constraint: The power of the test must be at least $1 - \beta$ for all parameters corresponding to a minimum detectable effect, $\delta$. This is represented by a specified subset of the alternative hypothesis space, $\Theta_{1,\delta} \subset \Theta_1$: \(\label{eq:power} \inf_{\theta \in \Theta_{1,\delta}} \kappa_n(\theta) \ge 1 - \beta \; .\)

The minimum required sample size, $n_{m}$, is the smallest integer $n$ so that the hypothesis test simultaneously satisfies both the significance and power constraint.

 
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Metadata: ID: D225 | shortcut: mrss | author: alexanderdbolton | date: 2025-07-15, 00:00.