Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Univariate Gaussian with known variance ▷ Paired z-test

Theorem: Let $y_{i1}$ and $y_{i2}$ with $i = 1, \ldots, n$ be paired observations, such that

$\label{eq:ugkv} y_{i1} \sim \mathcal{N}(y_{i2} + \mu, \sigma^2), \quad i = 1, \ldots, n$

is a univariate Gaussian data set with unknown shift $\mu$ and known variance $\sigma^2$. Then, the test statistic

$\label{eq:z} z = \sqrt{n} \, \frac{\bar{d}-\mu_0}{\sigma} \quad \text{where} \quad d_i = y_{i1} - y_{i2}$

with sample mean $\bar{d}$ follows a standard normal distribution

$\label{eq:z-dist} z \sim \mathcal{N}(0, 1)$

under the null hypothesis

$\label{eq:ztestp-h0} H_0: \; \mu = \mu_0 \; .$

Proof: Define the pair-wise difference $d_i = y_{i1} - y_{i2}$ which is, according to the linearity of the expected value and the invariance of the variance under addition, distributed as

$\label{eq:d-dist} d_i = y_{i1} - y_{i2} \sim \mathcal{N}(\mu, \sigma^2), \quad i = 1, \ldots, n \; .$

Therefore, $d_1, \ldots, d_n$ satisfy the conditions of the one-sample z-test which results in the test statistic given by \eqref{eq:z}.

Sources:

Metadata: ID: P210 | shortcut: ugkv-ztestp | author: JoramSoch | date: 2021-03-24, 05:10.