One sample t test for the mean - overview
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One sample $t$ test for the mean | MANCOVA |
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Independent variable | Independent variables | |
None | One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates) | |
Dependent variable | Dependent variables | |
One quantitative of interval or ratio level | Two or more quantitative of interval or ratio level | |
Null hypothesis | THIS 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 hypothesis | n.a. | |
H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | - | |
Assumptions | n.a. | |
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Test statistic | n.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 true | n.a. | |
$t$ distribution with $N - 1$ degrees of freedom | - | |
Significant? | n.a. | |
Two sided:
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$C\%$ 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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Example context | n.a. | |
Is the average mental health score of office workers different from $\mu_0 = 50$? | - | |
SPSS | n.a. | |
Analyze > Compare Means > One-Sample T Test...
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Jamovi | n.a. | |
T-Tests > One Sample T-Test
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Practice questions | Practice questions | |