# One sample z test for the mean - overview

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One sample $z$ test for the mean | Binomial test for a single proportion | Two sample $z$ test |
You cannot compare more than 3 methods |
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Independent variable | Independent variable | Independent/grouping variable | |

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

Dependent variable | Dependent variable | Dependent variable | |

One quantitative of interval or ratio level | One categorical with 2 independent groups | 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. | H_{0}: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | |

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: $\mu < \mu_0$
| H_{1} two sided: $\pi \neq \pi_0$H _{1} right sided: $\pi > \pi_0$H _{1} left sided: $\pi < \pi_0$
| H_{1} two sided: $\mu_1 \neq \mu_2$H _{1} right sided: $\mu_1 > \mu_2$H _{1} left sided: $\mu_1 < \mu_2$
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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
| - Sample is a simple random sample from the population. That is, observations are independent of one another
| - Within each population, the scores on the dependent variable are normally distributed
- Population standard deviations $\sigma_1$ and $\sigma_2$ are known
- 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
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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 $\mu_0$. | $X$ = number of successes in the sample | $z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$. | |

Sampling distribution of $z$ if H_{0} were true | Sampling distribution of $X$ if H0 were true | Sampling distribution of $z$ if H_{0} were true | |

Standard normal distribution | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | Standard normal distribution | |

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 $\alpha$
| Two sided:
- Check if $X$ observed in sample is in the rejection region or
- Find two sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
- Check if $X$ observed in sample is in the rejection region or
- Find right sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
- Check if $X$ observed in sample is in the rejection region or
- Find left sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
| 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 $\alpha$
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$C\%$ confidence interval for $\mu$ | n.a. | $C\%$ confidence interval for $\mu_1 - \mu_2$ | |

$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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. | - | $(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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 $\mu_1 - \mu_2$ 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. | Visual representation | |

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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 the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women. | |

n.a. | SPSS | n.a. | |

- | Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
- Put your dichotomous variable in the box below Test Variable List
- Fill in the value for $\pi_0$ in the box next to Test Proportion
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n.a. | Jamovi | n.a. | |

- | Frequencies > 2 Outcomes - Binomial test
- Put your dichotomous variable in the white box at the right
- Fill in the value for $\pi_0$ in the box next to Test value
- Under Hypothesis, select your alternative hypothesis
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Practice questions | Practice questions | Practice questions | |