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

Theorem: Let

\[\label{eq:ugkv} \begin{split} y_{1i} &\sim \mathcal{N}(\mu_1, \sigma_1^2), \quad i = 1, \ldots, n_1 \\ y_{2i} &\sim \mathcal{N}(\mu_2, \sigma_2^2), \quad i = 1, \ldots, n_2 \end{split}\]

be a univariate Gaussian data set representing two groups of unequal size $n_1$ and $n_2$ with unknown means $\mu_1$ and $\mu_2$ and unknown variances $\sigma_1^2$ and $\sigma_2^2$. Then, the test statistic

\[\label{eq:z} z = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\]

with sample means $\bar{y}_1$ and $\bar{y}_2$ follows a standard normal distribution

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

under the null hypothesis

\[\label{eq:ztest2-h0} H_0: \; \mu_1-\mu_2 = \mu_\Delta \; .\]

Proof: The sample means are given by

\[\label{eq:mean-samp} \begin{split} \bar{y}_1 &= \frac{1}{n_1} \sum_{i=1}^{n_1} y_{1i} \\ \bar{y}_2 &= \frac{1}{n_2} \sum_{i=1}^{n_2} y_{2i} \; . \end{split}\]

Using the linearity of the expected value, the additivity of the variance under independence and scaling of the variance upon multiplication, the sample means follow a normal distribution

\[\label{eq:mean-samp-dist} \begin{split} \bar{y}_1 &= \frac{1}{n_1} \sum_{i=1}^{n_1} y_{1i} \sim \mathcal{N}\left( \frac{1}{n_1} n_1 \mu_1, \left(\frac{1}{n_1}\right)^2 n_1 \sigma^2 \right) = \mathcal{N}\left( \mu_1, \sigma_1^2/n_1 \right) \\ \bar{y}_2 &= \frac{1}{n_2} \sum_{i=1}^{n_2} y_{2i} \sim \mathcal{N}\left( \frac{1}{n_2} n_2 \mu_2, \left(\frac{1}{n_2}\right)^2 n_2 \sigma^2 \right) = \mathcal{N}\left( \mu_2, \sigma_2^2/n_2 \right) \end{split}\]

and additionally using the invariance of the variance under addition, the distribution of $z = [(\bar{y}1-\bar{y}_2)-\mu\Delta]/\sigma_\Delta$ becomes

\[\label{eq:z-dist-s1} z = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{\sigma_\Delta} \sim \mathcal{N}\left( \frac{(\mu_1-\mu_2)-\mu_\Delta}{\sigma_\Delta}, \left(\frac{1}{\sigma_\Delta}\right)^2 \sigma_\Delta^2 \right) = \mathcal{N}\left( \frac{(\mu_1-\mu_2)-\mu_\Delta}{\sigma_\Delta}, 1 \right)\]

where $\sigma_\Delta$ is the pooled standard deviation

\[\label{eq:std-pool} \sigma_\Delta = \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} \; ,\]

such that, under the null hypothesis in \eqref{eq:ztest2-h0}, we have:

\[\label{eq:z-dist-s2} z \sim \mathcal{N}(0, 1), \quad \text{if } \mu_\Delta = \mu_1-\mu_2 \; .\]

This means that the null hypothesis can be rejected when $z$ is as extreme or more extreme than the critical value obtained from the standard normal distribution using a significance level $\alpha$.

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Metadata: ID: P209 | shortcut: ugkv-ztest2 | author: JoramSoch | date: 2021-03-24, 04:38.