One sample t test for the mean - overview

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One sample $t$ test for the mean
Marginal Homogeneity test / Stuart-Maxwell test
Independent variableIndependent variable
None2 paired groups
Dependent variableDependent variable
One quantitative of interval or ratio levelOne categorical with $J$ independent groups ($J \geqslant 2$)
Null hypothesisNull hypothesis
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
For each category $j$ of the dependent variable:

$\pi_j$ in the first paired group = $\pi_j$ in the second paired group

Here $\pi_j$ is the population proportion for category $j$
Alternative hypothesisAlternative hypothesis
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
For some categories of the dependent variable, $\pi_j$ in the first paired group $\neq$ $\pi_j$ in the second paired group
AssumptionsAssumptions
  • 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 statisticTest statistic
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $s$ is the sample standard deviation, $N$ 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 $\mu_0$
Computing the test statistic is a bit complicated and involves matrix algebra. You probably won't need to calculate it by hand (unless you are following a technical course)
Sampling distribution of $t$ if H0 were trueSampling distribution of the test statistic if H0 were true
$t$ distribution with $N - 1$ degrees of freedomApproximately a chi-squared distribution with $J - 1$ degrees of freedom
Significant?Significant?
Two sided: Right sided: Left sided: If we denote the test statistic as $X^2$:
  • 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 $\alpha$
$C\%$ 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 $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ 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 contextExample context
Is the average mental health score of office workers different from $\mu_0$ = 50?Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?
SPSSSPSS
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 Marginal Homogeneity test
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