Paired sample t test - overview
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Paired sample $t$ test | Binomial test for a single proportion |
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Independent variable | Independent variable | |
2 paired groups | None | |
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 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. | H0: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | |
Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | |
Assumptions | Assumptions | |
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Test statistic | Test statistic | |
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 $\mu_0$. | $X$ = number of successes in the sample | |
Sampling distribution of $t$ if H0 were true | Sampling distribution of $X$ if H0 were true | |
$t$ distribution with $N - 1$ degrees of freedom | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | |
Significant? | Significant? | |
Two sided:
| 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 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 | n.a. | |
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Equivalent to | n.a. | |
| - | |
Example context | Example context | |
Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$? | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | |
SPSS | SPSS | |
Analyze > Compare Means > Paired-Samples T Test...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
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Jamovi | Jamovi | |
T-Tests > Paired Samples T-Test
| Frequencies > 2 Outcomes - Binomial test
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Practice questions | Practice questions | |