One sample t test for the mean - overview
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One sample $t$ test for the mean | Logistic regression | One sample $z$ test for the mean |
You cannot compare more than 3 methods |
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Independent variable | Independent variables | Independent variable | |
None | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | None | |
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 | |
H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | Model chi-squared test for the complete regression model:
| H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | Model chi-squared test for the complete regression model:
| H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | |
Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 $\mu_0$. | 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 $\beta_k$:
| $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
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 $\mu_0$. | |
Sampling distribution of $t$ if H0 were true | Sampling distribution of $X^2$ and of the Wald statistic if H0 were true | Sampling distribution of $z$ if H0 were true | |
$t$ distribution with $N - 1$ degrees of freedom | Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
| Standard normal distribution | |
Significant? | Significant? | Significant? | |
Two sided:
| For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
| Two sided:
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$C\%$ confidence interval for $\mu$ | Wald-type approximate $C\%$ confidence interval for $\beta_k$ | $C\%$ confidence interval for $\mu$ | |
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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. | $b_k \pm z^* \times SE_{b_k}$ 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). | $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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 | Goodness of fit measure $R^2_L$ | Effect size | |
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 $\mu_0.$ | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ 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. | 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 $\mu_0 = 50$? | Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | 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 $\sigma = 3.$ | |
SPSS | SPSS | n.a. | |
Analyze > Compare Means > One-Sample T Test...
| Analyze > Regression > Binary Logistic...
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Jamovi | Jamovi | n.a. | |
T-Tests > One Sample T-Test
| Regression > 2 Outcomes - Binomial
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Practice questions | Practice questions | Practice questions | |