A Z-test is a statistical hypothesis test in which the test statistic can be approximated by a normal distribution under the null hypothesis.

The name “Z-test” comes from the standard normal distribution, which is also known as Z-distribution.

We can use Z-test for proportion or population mean.

Key Assumptions

The sample size also needs to be large enough (e.g. ) (see central limit theorem).

One Sample Proportion Test

A proportion test determine if the proportion of a certain outcome in a sample differs significantly from an expected proportion. We can either use Binomial test, which is more accurate, especially for small sample size, or z-test (which is a normal approximation)

The test statistic is

where

  • is the sample size
  • is the observed proportion
  • is the null hypothesis proportion (i.e. when )
  • The standard error is , as the population standard deviation is computed by

Example: A random sample found 13,173 boys were born among 25,468 newborn children. The sample proportion of boys was 0.5172. Is this sample evidence that the birth of boys is more common than the birth of girls in the entire population?

Here, we want to test The test statistics is

We are performing a one-tailed test and , so .

Based on the very small p-value, we can conclude that this example is an evidence that baby boys are more common than girls. 1

Z-Test for Mean

To perform a Z-test for a population mean, the population standard deviation must be known from a prior large-sample study. Unlike tests for proportions, it is impossible to calculate the population standard deviation directly. In practice, this information on population mean is often unavailable, and Student’s T-test is often a better alternative.

The test statistic is very similar to the test-statistic of the proportion test, though we need to know the population standard deviation:

where

See Also

Footnotes

  1. S.6 Test of Proportion | STAT ONLINE