One Sample Mean Z-test
Choose Stats > One Sample Mean Z-test
Sample Values: This column contains the sampled values, which must be numerical and continuous.
Summarized Data: If you know the statistical descriptions of the sampled values, fill in N (the count of the sampled values) and the mean. This information will override the sampled values selected above.
Hypothesis Test: The hypothesized mean value of the population must be inputted.
Known Values: The known population standard deviation must be inputted.
Alpha: The significance level used in the calculation. For example, for confidence intervals, the range is (1-alpha)100%.
Aspect |
Z-test |
T-test |
|---|---|---|
Population standard deviation |
Known |
Unknown, estimated from sample |
Sample size |
Typically larger (>30) |
Often smaller (<30) |
Distribution |
Standard normal (z-distribution) |
t-distribution (varies with degrees of freedom) |
Formula |
\(z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\) |
\(t = \frac{\bar{x} - \mu}{s / \sqrt{n}}\) |
Robustness |
Assumes normal distribution |
More robust to slight deviations from normality |
Degrees of freedom |
Not used |
Used, affects t-distribution shape |
Practical use |
Less common (population σ rarely known) |
More common in real-world applications |
A sample output:
---- One Sample Z ----
mean = 1.178
z = -2.880
df = 19.000
u0 = 1.500
Known SD = 0.500
Two-tailed test H0: μ = μ0, H1: μ ≠ μ0: p = 0.004
The p-value is the probability that the population mean equals the specified value from which the samples came.
95.00% range of population mean from which the samples came: (0.959, 1.397)
H0: μ = μ0, H1: μ > μ0 p-value = 0.998
95.00% Lower bound of population mean: 0.994
H0: μ = μ0, H1: μ < μ0 p-value = 0.002
95.00% Upper bound of population mean: 1.362
Hypotheses: Null hypothesis (H0): The population mean (estimated by the sample mean) is equal to the specified value. Alternative hypothesis (H1): The population mean is different from the specified value.
When the p-value is smaller than the significance level, the null hypothesis should be rejected. Alternatively, the p-value is the probability that the population mean equals the hypothesized population mean.
The confidence intervals of the population mean are determined by the percentage size of the range set by alpha.
In a t-test, the null hypothesis (H0) and alternative hypothesis (H1) can indeed be formulated as described, with H0: μ = μ0 and H1: μ > μ0. This is known as a one-tailed or directional test. If H0 is rejected, we accept that the true mean is greater than μ0, which is precisely what H1 states.