One sample z test for the mean - overview
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One sample $z$ test for the mean | Pearson correlation |
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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 | |
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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 $\mu_0$. | Test statistic for testing H0: $\rho = 0$:
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Sampling distribution of $z$ if H0 were true | Sampling distribution of $t$ and of $z$ if H0 were true | |
Standard normal distribution | Sampling distribution of $t$:
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Significant? | Significant? | |
Two sided:
| $t$ Test two sided:
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$C\%$ confidence interval for $\mu$ | Approximate $C$% confidence interval for $\rho$ | |
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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. | First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
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Effect size | Properties of the Pearson correlation coefficient | |
Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$ |
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Visual representation | n.a. | |
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n.a. | Equivalent to | |
- | OLS regression with one independent variable:
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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 there a linear relationship between physical health and mental health? | |
n.a. | SPSS | |
- | Analyze > Correlate > Bivariate...
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n.a. | Jamovi | |
- | Regression > Correlation Matrix
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Practice questions | Practice questions | |