One-Sample Z-Test
Introduction
The one-sample z-test can be used to determine if the mean in a population differs from a pre-defined value, based on a sample. This test is often used if there is a large sample size. For smaller sample sizes, a Student t-test is usually used. Strickly the test requires the population standard deviation to be known, but often the sample standard deviation is used as an estimate for this.
Performing the Test
with SPSS
Formulas
The test statistic is given by:
\(Z=\frac{\bar{x}-\mu_{H_0}}{SE}\)
With:
- \(SE = \sqrt{\frac{\sigma^2}{n}}\)
- \(\sigma^2\approx s^2=\frac{\sum_{i=1}^n\left(x_i-\bar{x}\right)^2}{n-1}\)
- \(\bar{x}=\frac{\sum_{i=1}^n x_i}{n}\)
Symbols used:
- \(n\) is the sample size
- \(x_i\) the i-th score
- \(\bar{x}\) the arithmetic mean (average)
- \(\sigma^2\) the population variance
- \(s^2\) the sample variance
The p-value (two-tailed) is then calculated using:
\(sig. = 2\times\left(1-\Phi\left(z\right)\right)\)
where \(\Phi\left(\dots\right)\) is the cumulative density function of the standard normal distribution.
Interpreting the Result
The assumption about the population for this test (the null hypothesis) is that the mean is a specific value.
The test provides a p-value, which is the probability of a test statistic as from the sample, or even more extreme, if the assumption about the population would be true. If this p-value (significance) is below a pre-defined threshold (the significance level \(\alpha\) ), the assumption about the population is rejected. We then speak of a (statistically) significant result. The threshold is usually set at 0.05. Anything below is then considered low.
If the assumption is rejected, we conclude that the mean in the population will be different than the one used in the test.
Note that if we do not reject the assumption, it does not mean we accept it, we simply state that there is insufficient evidence to reject it.
Writing the results
Writing up the results of the test uses the format (APA, 2019 p. 182):
z = <z-value>, p = <p-value>
So for example:
A one-sample Z test, indicated that the mean was significantly different from 53, z = 3.10, p < .001.
The p-value is shown with three decimal places, and no 0 before the decimal sign. If the p-value is below .0005, it can be reported as p < .001.
APA (2019, p. 88) states to also report an effect size measure.
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