One sample t test for the mean - overview

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One sample $t$ test for the mean
Sign test
Independent variableIndependent variable
None2 paired groups
Dependent variableDependent variable
One quantitative of interval or ratio levelOne of ordinal level
Null hypothesisNull hypothesis
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
  • P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
  • The median of the difference scores is zero in the population
Alternative hypothesisAlternative hypothesis
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
  • Two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair)
  • Right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair)
  • Left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
  • Two sided: the median of the difference scores is different from zero in the population
  • Right sided: the median of the difference scores is larger than zero in the population
  • Left sided: the median of the difference scores is smaller than zero in the population
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 $\mu_0$
$W = $ number of difference scores that is larger than 0
Sampling distribution of $t$ if H0 were trueSampling distribution of $W$ if H0 were true
$t$ distribution with $N - 1$ degrees of freedomThe exact distribution of $W$ under the null hypothesis is the Binomial($n$, $p$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $p = 0.5$.

If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $np = n \times 0.5$ and standard deviation $\sqrt{np(1-p)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately a standard normal distribution if the null hypothesis were true.
Significant?Significant?
Two sided: Right sided: Left sided: If $n$ is small, the table for the binomial distribution should be used:
Two sided:
  • Check if $W$ observed in sample is in the rejection region or
  • Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Right sided:
  • Check if $W$ observed in sample is in the rejection region or
  • Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Left sided:
  • Check if $W$ observed in sample is in the rejection region or
  • Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$

If $n$ is large, the table for standard normal probabilities can be used:
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 $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ 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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n.a.Equivalent to
- Two sided sign test is equivalent to
Example contextExample context
Is the average mental health score of office workers different from $\mu_0$ = 50?Do people tend to score higher on mental health after a mindfulness course?
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 Sign test
JamoviJamovi
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
Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:

ANOVA > Repeated Measures ANOVA - Friedman
  • Put the two paired variables in the box below Measures
Practice questionsPractice questions