One sample t test for the mean - overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

One sample $t$ test for the mean
Repeated measures ANCOVA
Independent variableIndependent variable
NoneOne or more within subjects factors (related groups), one or more quantitative control variables of interval or ratio level (covariates), and possibly one or more between subjects factors (independent groups)
Dependent variableDependent variables
One quantitative of interval or ratio levelOne quantitative of interval or ratio level
Null hypothesisTHIS TABLE IS YET TO BE COMPLETED
H0: $\mu = \mu_0$

Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.
-
Alternative hypothesisn.a.
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
-
Assumptionsn.a.
  • Scores are normally distributed in the population
  • Sample is a simple random sample from the population. That is, observations are independent of one another
-
Test statisticn.a.
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 $\mu_0$.
-
Sampling distribution of $t$ if H0 were truen.a.
$t$ distribution with $N - 1$ degrees of freedom-
Significant?n.a.
Two sided: Right sided: Left sided: -
$C\%$ 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 sizen.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 representationn.a.
One sample t test
-
Example contextn.a.
Is the average mental health score of office workers different from $\mu_0 = 50$?-
SPSSn.a.
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
-
Jamovin.a.
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
-
Practice questionsPractice questions