One sample t test for the mean overview

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One sample $t$ test for the mean	McNemar's 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 variable	Independent variable
None	2 paired groups
Dependent variable	Dependent variable
One quantitative of interval or ratio level	One categorical with 2 independent groups
Null hypothesis	Null hypothesis
H₀: $\mu = \mu_0$ Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.	Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options: First score of pair is 0, second score of pair is 0 First score of pair is 0, second score of pair is 1 (switched) First score of pair is 1, second score of pair is 0 (switched) First score of pair is 1, second score of pair is 1 The null hypothesis H₀ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) = P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the null hypothesis are: H₀: $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones for the first paired group and $\pi_2$ is the population proportion of ones for the second paired group H₀: for each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)
Alternative hypothesis	Alternative hypothesis
H₁ two sided: $\mu \neq \mu_0$ H₁ right sided: $\mu > \mu_0$ H₁ left sided: $\mu < \mu_0$	The alternative hypothesis H₁ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are: H₁: $\pi_1 \neq \pi_2$ H₁: for each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)
Assumptions	Assumptions
Scores are normally distributed in the population Sample is a simple random sample from the population. That is, observations are independent of one another	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic	Test statistic
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$ Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. 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 $\mu_0$.	$X^2 = \dfrac{(b - c)^2}{b + c}$ Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0.
Sampling distribution of $t$ if H₀ were true	Sampling distribution of $X^2$ if H₀ were true
$t$ distribution with $N - 1$ degrees of freedom	If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$.
Significant?	Significant?
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 $\alpha$	For test statistic $X^2$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$ If $b + c$ is small, the table for the binomial distribution should be used, with as test statistic $b$: Check if $b$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $b$ and check if it is equal to or smaller than $\alpha$
$C\%$ confidence interval for $\mu$	n.a.
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$ 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.	-
Effect size	n.a.
Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$	-
Visual representation	n.a.
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n.a.	Equivalent to
-	Stuart-Maxwell test, with a categorical dependent variable consisting of two independent groups Cochran's Q test, with two related groups Two sided sign test: $b = W$, and $X^2 = z^2$
Example context	Example context
Is the average mental health score of office workers different from $\mu_0 = 50$?	Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?
SPSS	SPSS
Analyze > Compare Means > One-Sample T Test... Put your variable in the box below Test Variable(s) Fill in the value for $\mu_0$ in the box next to Test Value	Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the McNemar test
Jamovi	Jamovi
T-Tests > One Sample T-Test Put your variable in the box below Dependent Variables Under Hypothesis, fill in the value for $\mu_0$ in the box next to Test Value, and select your alternative hypothesis	Frequencies > Paired Samples - McNemar test Put one of the two paired variables in the box below Rows and the other paired variable in the box below Columns
Practice questions	Practice questions

One sample t test for the mean - overview