Z-test Calculator
The Z-test Calculator is a statistical tool used to determine the probability of a sample statistic (such as the sample mean) being drawn from a population with a known standard deviation.
The Z-test calculator takes the following inputs:
- Sample Mean: The mean or average value of the sample data.
- Population Mean: The hypothesized or known mean of the population.
- Population Standard Deviation: The known or assumed standard deviation of the population.
- Sample Size: The number of observations or data points in the sample.
Using these inputs, the Z-test calculator computes the Z-score, which is a standardized measure of the difference between the sample mean and the population mean, divided by the standard error of the sample mean.
Z-test Calculation Chart
| Sample Mean | Population Mean | Population Standard Deviation | Sample Size | Z-score | P-value |
|---|---|---|---|---|---|
| 10.5 | 10 | 2 | 36 | 1.25 | 0.2119 |
| 10.8 | 10 | 2 | 36 | 2.00 | 0.0455 |
| 9.7 | 10 | 2 | 36 | -1.25 | 0.2119 |
| 10.5 | 10 | 2 | 25 | 1.00 | 0.3173 |
| 10.5 | 10 | 2 | 49 | 1.43 | 0.1531 |
| 11.0 | 10 | 1.5 | 36 | 3.33 | 0.0009 |
| 10.2 | 10 | 1.8 | 36 | 0.56 | 0.5758 |
| 10.5 | 10 | 2.2 | 36 | 1.14 | 0.2549 |
Z-test Formula
The formula for the Z-test is:
Z-score = (Sample Mean - Population Mean) / (Population Standard Deviation / √Sample Size)Where:
Sample Mean: The mean or average value of the sample data.
Population Mean: The hypothesized or known mean of the population.
Population Standard Deviation: The known or assumed standard deviation of the population.
Sample Size: The number of observations or data points in the sample.
Let’s go through an example to understand the Z-test formula better.
A manufacturer claims that the average weight of their product is 10 pounds, with a known population standard deviation of 2 pounds. A sample of 36 products is taken, and the sample mean is found to be 10.5 pounds. We want to determine if the sample mean is significantly different from the claimed population mean.
Plugging the values into the Z-test formula:
Z-score = (Sample Mean – Population Mean) / (Population Standard Deviation / √Sample Size)
Z-score = (10.5 – 10) / (2 / √36)
Z-score = 1.25
This Z-score of 1.25 represents the number of standard deviations the sample mean is from the population mean, assuming the null hypothesis (that the sample mean is equal to the population mean) is true.
To interpret the Z-score, we can use a Z-table or a Z-test calculator to determine the corresponding p-value, which represents the probability of observing a sample mean this far from the population mean, assuming the null hypothesis is true.
In this example, the Z-test calculator would give a p-value of approximately 0.2119. This means that there is a 21.19% chance of observing a sample mean of 10.5 pounds if the population mean is truly 10 pounds, assuming the population standard deviation is 2 pounds and the sample size is 36.
How do you calculate the z-test?
Calculating the z-test involves the following steps:
Determine the Null and Alternative Hypotheses:
Null Hypothesis (H0): The sample mean is equal to the population mean.
Alternative Hypothesis (Ha): The sample mean is not equal to the population mean.
Gather the Required Data:
Sample Mean (x̄)
Population Mean (μ)
Population Standard Deviation (σ)
Sample Size (n)
Calculate the Z-score:
Z-score = (Sample Mean – Population Mean) / (Population Standard Deviation / √Sample Size)
Determine the P-value:
Using a Z-table or Z-test calculator, find the p-value corresponding to the calculated Z-score.
Interpret the Results:
Compare the p-value to the chosen significance level (e.g., 0.05 or 0.01).
If the p-value is less than the significance level, the difference between the sample mean and population mean is statistically significant, and the null hypothesis is rejected.
If the p-value is greater than or equal to the significance level, the difference between the sample mean and population mean is not statistically significant, and the null hypothesis is not rejected.