One sample t test for the mean - overview

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One sample $t$ test for the mean
Mann-Whitney-Wilcoxon test
Independent variableIndependent variable
NoneOne categorical with 2 independent 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
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
• The median in population 1 is equal to the median in population 2
Else:
Formulation 1:
• The scores in population 1 are not systematically higher or lower than the scores in population 2
Formulation 2:
• P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.
Alternative hypothesisAlternative hypothesis
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
• Two sided: the median in population 1 is not equal to the median in population 2
• Right sided: the median in population 1 is larger than the median in population 2
• Left sided: the median in population 1 is smaller than the median in population 2
Else:
Formulation 1:
• Two sided: The scores in population 1 are systematically higher or lower than the scores in population 2
• Right sided: The scores in population 1 are systematically higher than the scores in population 2
• Left sided: The scores in population 1 are systematically lower than the scores in population 2
Formulation 2:
• Two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1)
• Right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1)
• Left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 1)
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
Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations 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 $\mu_0$
Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
The second type of test statistic is the Mann-Whitney $U$ statistic:
• $U = W - \dfrac{n_1(n_1 + 1)}{2}$
where $n_1$ is the sample size of group 1

Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.
Sampling distribution of $t$ if H0 were trueSampling distribution of $W$ and of $U$ if H0 were true
$t$ distribution with $N - 1$ degrees of freedom

Sampling distribution of $W$:
For large samples, $W$ is approximately normally distributed with mean $\mu_W$ and standard deviation $\sigma_W$ if the null hypothesis were true. Here \begin{aligned} \mu_W &= \dfrac{n_1(n_1 + n_2 + 1)}{2}\\ \sigma_W &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} Hence, for large samples, the standardized test statistic $$z_W = \dfrac{W - \mu_W}{\sigma_W}\\$$ follows approximately a standard normal distribution if the null hypothesis were true. Note that if your $W$ value is based on group 2, $\mu_W$ becomes $\frac{n_2(n_1 + n_2 + 1)}{2}$.

Sampling distribution of $U$:
For large samples, $U$ is approximately normally distributed with mean $\mu_U$ and standard deviation $\sigma_U$ if the null hypothesis were true. Here \begin{aligned} \mu_U &= \dfrac{n_1 n_2}{2}\\ \sigma_U &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} Hence, for large samples, the standardized test statistic $$z_U = \dfrac{U - \mu_U}{\sigma_U}\\$$ follows approximately a standard normal distribution if the null hypothesis were true.

For small samples, the exact distribution of $W$ or $U$ should be used.

Note: the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated if ties are present in the data.
Significant?Significant?
Two sided:
Right sided:
Left sided:
For large samples, 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.
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n.a.Equivalent to
-If no ties in the data: two sided Mann-Whitney-Wilcoxon test is equivalent to Kruskal-Wallis test with an independent variable with 2 levels ($I = 2$)
Example contextExample context
Is the average mental health score of office workers different from $\mu_0$ = 50?Do men tend to score higher on social economic status than women?
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 Independent Samples...
• Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
• Continue and click OK
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
T-Tests > Independent Samples T-Test
• Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Mann-Whitney U
• Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questions