Two sample z test overview

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Two sample $z$ test	Paired sample $t$ test	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent/grouping variable	Independent variable
One categorical with 2 independent groups	2 paired groups
Dependent variable	Dependent variable
One quantitative of interval or ratio level	One quantitative of interval or ratio level
Null hypothesis	Null hypothesis
H₀: $\mu_1 = \mu_2$ Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.	H₀: $\mu = \mu_0$ Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.
Alternative hypothesis	Alternative hypothesis
H₁ two sided: $\mu_1 \neq \mu_2$ H₁ right sided: $\mu_1 > \mu_2$ H₁ left sided: $\mu_1 < \mu_2$	H₁ two sided: $\mu \neq \mu_0$ H₁ right sided: $\mu > \mu_0$ H₁ left sided: $\mu < \mu_0$
Assumptions	Assumptions
Within each population, the scores on the dependent variable are normally distributed Population standard deviations $\sigma_1$ and $\sigma_2$ are known Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another	Difference scores are normally distributed in the population Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another
Test statistic	Test statistic
$z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$ Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.	$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$ Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.
Sampling distribution of $z$ if H₀ were true	Sampling distribution of $t$ if H₀ were true
Standard normal distribution	$t$ distribution with $N - 1$ degrees of freedom
Significant?	Significant?
Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$	Two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
$C\%$ confidence interval for $\mu_1 - \mu_2$	$C\%$ confidence interval for $\mu$
$(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$ where the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval). The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test.	$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$ where the critical value $t^$ is the value under the $t_{N-1}$ distribution with the area $C / 100$ between $-t^$ and $t^$ (e.g. $t^$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu$ can also be used as significance test.
n.a.	Effect size
-	Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$
Visual representation	Visual representation

n.a.	Equivalent to
-	One sample $t$ test on the difference scores. Repeated measures ANOVA with one dichotomous within subjects factor.
Example context	Example context
Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women.	Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?
n.a.	SPSS
-	Analyze > Compare Means > Paired-Samples T Test... Put the two paired variables in the boxes below Variable 1 and Variable 2
n.a.	Jamovi
-	T-Tests > Paired Samples T-Test Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line Under Hypothesis, select your alternative hypothesis
Practice questions	Practice questions

Two sample z test - overview