Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Univariate Gaussian ▷ Paired t-test

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

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

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

\[\label{eq:t} t = \frac{\bar{d}-\mu_0}{s_d / \sqrt{n}} \quad \text{where} \quad d_i = y_{i1} - y_{i2}\]

with sample mean $\bar{d}$ and sample variance $s^2_d$ follows a Student’s t-distribution with $n-1$ degrees of freedom

\[\label{eq:t-dist} t \sim \mathrm{t}(n-1)\]

under the null hypothesis

\[\label{eq:ttestp-h0} H_0: \; \mu = \mu_0 \; .\]

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 t-test which results in the test statistic given by \eqref{eq:t}.

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Metadata: ID: P206 | shortcut: ug-ttestp | author: JoramSoch | date: 2021-03-12, 09:34.