One sample t test for the mean overview

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One sample $t$ test for the mean	Pearson correlation	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent variable	Variable 1
None	One quantitative of interval or ratio level
Dependent variable	Variable 2
One quantitative of interval or ratio level	One quantitative of interval or ratio level
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: $\rho = \rho_0$ Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.
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: $\rho \neq \rho_0$ H₁ right sided: $\rho > \rho_0$ H₁ left sided: $\rho < \rho_0$
Assumptions	Assumptions of test for correlation
Scores are normally distributed in the population Sample is a simple random sample from the population. That is, observations are independent of one another	In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions) Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear relationship between two variables.
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 $\mu_0$.	Test statistic for testing H0: $\rho = 0$: $t = \dfrac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}} $ where $r$ is the sample correlation $r = \frac{1}{N - 1} \sum_{j}\Big(\frac{x_{j} - \bar{x}}{s_x} \Big) \Big(\frac{y_{j} - \bar{y}}{s_y} \Big)$ and $N$ is the sample size Test statistic for testing values for $\rho$ other than $\rho = 0$: $z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$ $r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$, where $r$ is the sample correlation $\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1 - \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0
Sampling distribution of $t$ if H₀ were true	Sampling distribution of $t$ and of $z$ if H₀ were true
$t$ distribution with $N - 1$ degrees of freedom	Sampling distribution of $t$: $t$ distribution with $N - 2$ degrees of freedom Sampling distribution of $z$: Approximately the standard normal distribution
Significant?	Significant?
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 $\alpha$ Right sided: 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 $\alpha$ Left sided: 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 $\alpha$	$t$ Test 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 $\alpha$ $t$ Test right sided: 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 $\alpha$ $t$ Test left sided: 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 $\alpha$ $z$ Test 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 $\alpha$ $z$ Test right sided: 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 $\alpha$ $z$ Test left sided: 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 $\alpha$
$C\%$ confidence interval for $\mu$	Approximate $C$% confidence interval for $\rho$
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$ 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.	First compute the approximate $C$% confidence interval for $\rho_{Fisher}$: $lower_{Fisher} = r_{Fisher} - z^* \times \sqrt{\dfrac{1}{N - 3}}$ $upper_{Fisher} = r_{Fisher} + z^* \times \sqrt{\dfrac{1}{N - 3}}$ where $r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$ and 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). Then transform back to get the approximate $C$% confidence interval for $\rho$: lower bound = $\dfrac{e^{2 \times lower_{Fisher}} - 1}{e^{2 \times lower_{Fisher}} + 1}$ upper bound = $\dfrac{e^{2 \times upper_{Fisher}} - 1}{e^{2 \times upper_{Fisher}} + 1}$
Effect size	Properties of the Pearson correlation coefficient
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 $\mu_0.$	The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables. The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable. The Pearson correlation coefficient can take on values between -1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship. The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example: the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y - 6$. the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $-3x + 5$ and $2y - 6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $-3$. The Pearson correlation coefficient does not say anything about causality. The Pearson correlation coefficient is sensitive to outliers.
Visual representation	n.a.
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n.a.	Equivalent to
-	OLS regression with one independent variable: $b_1 = r \times \frac{s_y}{s_x}$ Results significance test ($t$ and $p$ value) testing $H_0$: $\beta_1 = 0$ are equivalent to results significance test testing $H_0$: $\rho = 0$
Example context	Example context
Is the average mental health score of office workers different from $\mu_0 = 50$?	Is there a linear relationship between physical health and mental health?
SPSS	SPSS
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	Analyze > Correlate > Bivariate... Put your two variables in the box below Variables
Jamovi	Jamovi
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	Regression > Correlation Matrix Put your two variables in the white box at the right Under Correlation Coefficients, select Pearson (selected by default) Under Hypothesis, select your alternative hypothesis
Practice questions	Practice questions

One sample t test for the mean - overview