One sample t test for the mean - overview

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One sample $t$ test for the mean
One way MANOVA
Independent variableIndependent/grouping variable
NoneOne categorical with $I$ independent groups ($I \geqslant 2$)
Dependent variableDependent variables
One quantitative of interval or ratio levelTwo or more 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.
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Alternative hypothesisn.a.
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
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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
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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 $\mu_0$.
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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.
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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 $\mu_0.$
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Visual representationn.a.
One sample t test
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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
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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
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Practice questionsPractice questions