One sample z test for the mean - overview
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One sample $z$ test for the mean | Chi-squared test for the relationship between two categorical variables | Mann-Whitney-Wilcoxon test |
You cannot compare more than 3 methods |
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Independent variable | Independent /column variable | Independent/grouping variable | |
None | One categorical with $I$ independent groups ($I \geqslant 2$) | One categorical with 2 independent groups | |
Dependent variable | Dependent /row variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | One of ordinal level | |
Null hypothesis | Null hypothesis | Null hypothesis | |
H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | H0: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| 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:
Formulation 1:
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Alternative 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: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| 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:
Formulation 1:
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Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells. | 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 $W$:
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 $z$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $W$ and of $U$ if H0 were true | |
Standard normal distribution | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | |
Significant? | Significant? | Significant? | |
Two sided:
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| For large samples, the table for standard normal probabilities can be used: Two sided:
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$C\%$ confidence interval for $\mu$ | n.a. | n.a. | |
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). The confidence interval for $\mu$ can also be used as significance test. | - | - | |
Effect size | n.a. | n.a. | |
Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$ | - | - | |
Visual representation | n.a. | n.a. | |
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n.a. | n.a. | Equivalent to | |
- | - | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | |
Example context | Example context | Example context | |
Is the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$ | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | Do men tend to score higher on social economic status than women? | |
n.a. | SPSS | SPSS | |
- | Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
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n.a. | Jamovi | Jamovi | |
- | Frequencies > Independent Samples - $\chi^2$ test of association
| T-Tests > Independent Samples T-Test
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Practice questions | Practice questions | Practice questions | |