Z-Test for Hypothesis Testing
A Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-tests are used when sample sizes are large enough for the central limit theorem to be applied.
Where:
- X̄ = Sample mean
- μ = Population mean (hypothesized)
- σ = Population standard deviation
- n = Sample size
Test Type
Select the type of Z-test you want to perform:
One-Sample Z-Test
Test whether the mean of a single sample is significantly different from a known or hypothesized population mean.
One-Sample Z-Test Results
Z-Score
P-Value
Critical Value
Result
Interpretation
Standard Normal Distribution
Two-Sample Z-Test
Test whether the means of two independent samples are significantly different from each other.
Two-Sample Z-Test Results
Z-Score
P-Value
Critical Value
Result
Interpretation
Z-Test for Proportions
Test whether the proportion of a characteristic in a sample is significantly different from a hypothesized population proportion.
Z-Test for Proportions Results
Z-Score
P-Value
Critical Value
Result
Interpretation
Practical Examples
Example 1: Quality Control
A manufacturer claims their products have a mean weight of 500g. A sample of 50 products has a mean weight of 495g with a population standard deviation of 10g. Test if the sample mean is significantly different.
Z = (495 - 500) / (10 / √50) = -5 / 1.414 = -3.536
With α = 0.05 (two-tailed), the critical value is ±1.96. Since -3.536 < -1.96, we reject the null hypothesis.
Example 2: Marketing Campaign
A company claims their new marketing campaign increased conversion rates from 10% to 15%. In a sample of 200 visitors, 35 converted (p̂ = 0.175). Test if this is significantly different from the claimed 15%.
Z = (0.175 - 0.15) / √(0.15×0.85/200) = 0.025 / 0.025 = 1.0
With α = 0.05 (two-tailed), the critical value is ±1.96. Since 1.0 < 1.96, we fail to reject the null hypothesis.
Example 3: Process Improvement
An engineer wants to test if a new process reduces processing time. The current mean is 120 minutes with σ = 15 minutes. A sample of 40 using the new process has a mean of 115 minutes.
Z = (115 - 120) / (15 / √40) = -5 / 2.372 = -2.108
With α = 0.05 (one-tailed left), the critical value is -1.645. Since -2.108 < -1.645, we reject the null hypothesis.
Critical Values for Common Significance Levels
| Significance Level (α) | Two-Tailed Critical Values | One-Tailed Critical Value |
|---|---|---|
| 0.10 (90% confidence) | ±1.645 | ±1.282 |
| 0.05 (95% confidence) | ±1.960 | ±1.645 |
| 0.01 (99% confidence) | ±2.576 | ±2.326 |