Z-test

Illustration of the null hypothesis region in a Z-test.

The Z-test is a type of statistical test that determines if there is a significant difference between the means of two groups. It is used when the population variance is known and the sample size is large (typically n > 30). The Z-test is based on the standard normal distribution and is commonly used in hypothesis testing.

Hypothesis Testing

In hypothesis testing, the Z-test is used to test the null hypothesis (H_) against an alternative hypothesis (H_). The null hypothesis typically states that there is no effect or no difference, while the alternative hypothesis suggests that there is an effect or a difference.

The test statistic is calculated using the formula:

Z = \( \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \)

where:

• \( \bar{x} \) is the sample mean,
• \( \mu \) is the population mean,
• \( \sigma \) is the population standard deviation,
• \( n \) is the sample size.

Critical Region

The critical region is determined by the significance level (_), which is the probability of rejecting the null hypothesis when it is true. Common significance levels are 0.05, 0.01, and 0.10. The critical value is found from the standard normal distribution table.

If the calculated Z value falls into the critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

Assumptions

The Z-test assumes that:

• The data follows a normal distribution.
• The sample size is large enough for the Central Limit Theorem to apply.
• The population variance is known.

Applications

Z-tests are used in various fields such as medicine, psychology, and economics to compare sample data against known population parameters. They are particularly useful in quality control and clinical trials.

Related Pages

• T-test
• Chi-squared test
• ANOVA
• P-value
• Confidence interval