One sample z test for the mean - overview
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One sample $z$ test for the mean | Pearson correlation | Sign test |
You cannot compare more than 3 methods |
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Independent variable | Variable 1 | Independent variable | |
None | One quantitative of interval or ratio level | 2 paired groups | |
Dependent variable | Variable 2 | Dependent variable | |
One quantitative of interval or ratio level | One quantitative of interval or ratio level | One of ordinal 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. | 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. |
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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: $\rho \neq \rho_0$ H1 right sided: $\rho > \rho_0$ H1 left sided: $\rho < \rho_0$ |
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Assumptions | Assumptions of test for correlation | Assumptions | |
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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 $\mu_0$. | Test statistic for testing H0: $\rho = 0$:
| $W = $ number of difference scores that is larger than 0 | |
Sampling distribution of $z$ if H0 were true | Sampling distribution of $t$ and of $z$ if H0 were true | Sampling distribution of $W$ if H0 were true | |
Standard normal distribution | Sampling distribution of $t$:
| The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true. | |
Significant? | Significant? | Significant? | |
Two sided:
| $t$ Test two sided:
| If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
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$C\%$ confidence interval for $\mu$ | Approximate $C$% confidence interval for $\rho$ | n.a. | |
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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 | 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 $\mu_0.$ |
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Visual representation | n.a. | n.a. | |
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n.a. | Equivalent to | Equivalent to | |
- | OLS regression with one independent variable:
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Two sided sign test is equivalent to
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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 there a linear relationship between physical health and mental health? | Do people tend to score higher on mental health after a mindfulness course? | |
n.a. | SPSS | SPSS | |
- | Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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n.a. | Jamovi | Jamovi | |
- | Regression > Correlation Matrix
| Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
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Practice questions | Practice questions | Practice questions | |