# One sample z test for the mean - overview

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One sample $z$ test for the mean | Mann-Whitney-Wilcoxon test | One sample $t$ test for the mean |
You cannot compare more than 3 methods |
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Independent variable | Independent/grouping variable | Independent variable | |

None | One categorical with 2 independent groups | None | |

Dependent variable | Dependent variable | Dependent variable | |

One quantitative of interval or ratio level | One of ordinal level | One quantitative of interval or ratio level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ 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:
- H
_{0}: the population median for group 1 is equal to the population median for group 2
Formulation 1: - H
_{0}: the population scores in group 1 are not systematically higher or lower than the population scores in group 2
- H
_{0}: P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
| H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | |

Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

H_{1} two sided: $\mu \neq \mu_0$H _{1} right sided: $\mu > \mu_0$H _{1} 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:
- H
_{1}two sided: the population median for group 1 is not equal to the population median for group 2 - H
_{1}right sided: the population median for group 1 is larger than the population median for group 2 - H
_{1}left sided: the population median for group 1 is smaller than the population median for group 2
Formulation 1: - H
_{1}two sided: the population scores in group 1 are systematically higher or lower than the population scores in group 2 - H
_{1}right sided: the population scores in group 1 are systematically higher than the population scores in group 2 - H
_{1}left sided: the population scores in group 1 are systematically lower than the population scores in group 2
- H
_{1}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) - H
_{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) - H
_{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)
| H_{1} two sided: $\mu \neq \mu_0$H _{1} right sided: $\mu > \mu_0$H _{1} left sided: $\mu < \mu_0$
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Assumptions | Assumptions | Assumptions | |

- Scores are normally distributed in the population
- Population standard deviation $\sigma$ is known
- 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
| - Scores are normally distributed in the population
- Sample is a simple random sample from the population. That is, observations are independent of one another
| |

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 $\mu_0$. | 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 $W$:
- $W$ = sum of ranks in group 1
- $U = W - \dfrac{n_1(n_1 + 1)}{2}$
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. | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 $\mu_0$. | |

Sampling distribution of $z$ if H_{0} were true | Sampling distribution of $W$ and of $U$ if H_{0} were true | Sampling distribution of $t$ if H_{0} were true | |

Standard normal distribution | 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. | $t$ distribution with $N - 1$ degrees of freedom | |

Significant? | Significant? | Significant? | |

Two sided:
- Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
- Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
- Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
- Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
| For large samples, the table for standard normal probabilities can be used: Two sided: - Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
- Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
- Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
- Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
| Two sided:
- Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
- Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
- Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
- Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
| |

$C\%$ confidence interval for $\mu$ | n.a. | $C\%$ confidence interval for $\mu$ | |

$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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. | - | $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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. | |

Effect size | n.a. | Effect size | |

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 $\mu_0.$ | - | 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 $\mu_0.$ | |

Visual representation | n.a. | Visual representation | |

- | |||

n.a. | Equivalent to | n.a. | |

- | 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 $\sigma = 3.$ | Do men tend to score higher on social economic status than women? | Is the average mental health score of office workers different from $\mu_0 = 50$? | |

n.a. | SPSS | SPSS | |

- | 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
| 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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n.a. | Jamovi | Jamovi | |

- | 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
| 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 questions | Practice questions | Practice questions | |