One sample z test for the mean - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
One sample $z$ test for the mean | Binomial test for a single proportion | $z$ test for a single proportion | Logistic regression |
|
---|---|---|---|---|
Independent variable | Independent variable | Independent variable | Independent variables | |
None | None | None | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | |
Dependent variable | Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with 2 independent groups | One categorical with 2 independent groups | One categorical with 2 independent groups | |
Null hypothesis | 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: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | H0: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | Model chi-squared test for the complete regression model:
| |
Alternative hypothesis | 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 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | Model chi-squared test for the complete regression model:
| |
Assumptions | Assumptions | Assumptions | Assumptions | |
|
|
|
| |
Test statistic | 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 | $z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis. | Model chi-squared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chi-squared test for individual $\beta_k$:
| |
Sampling distribution of $z$ if H0 were true | Sampling distribution of $X$ if H0 were true | Sampling distribution of $z$ if H0 were true | Sampling distribution of $X^2$ and of the Wald statistic if H0 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). | Approximately the standard normal distribution | Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
| |
Significant? | Significant? | Significant? | Significant? | |
Two sided:
| Two sided:
| Two sided:
| For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
| |
$C\%$ confidence interval for $\mu$ | n.a. | Approximate $C\%$ confidence interval for $\pi$ | Wald-type approximate $C\%$ confidence interval for $\beta_k$ | |
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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. | - | Regular (large sample):
| $b_k \pm z^* \times SE_{b_k}$ 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). | |
Effect size | n.a. | n.a. | Goodness of fit measure $R^2_L$ | |
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 $\mu_0.$ | - | - | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit. | |
Visual representation | n.a. | n.a. | n.a. | |
- | - | - | ||
n.a. | n.a. | Equivalent to | n.a. | |
- | - |
| - | |
Example context | 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 proportion of smokers amongst office workers different from $\pi_0 = 0.2$? Use the normal approximation for the sampling distribution of the test statistic. | Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | |
n.a. | SPSS | SPSS | SPSS | |
- | Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Regression > Binary Logistic...
| |
n.a. | Jamovi | Jamovi | Jamovi | |
- | Frequencies > 2 Outcomes - Binomial test
| Frequencies > 2 Outcomes - Binomial test
| Regression > 2 Outcomes - Binomial
| |
Practice questions | Practice questions | Practice questions | Practice questions | |