Definition of Z-Test

Z-test is a parametric statistical test generally used to test the hypothesis regarding the equality of the mean(s) of Normal Populations provided the population variance to be known. In this test the distribution of the test statistic under the null hypothesis is said to follow a Standard Normal Distribution that is with mean of the distribution as 0 and the variance as 1.


Structure of Z-Test

Let (x1, x2, ………….., xn) be a random sample of size n from a univariate normal population.

To test H0: μ = μ0     vs   H1: μ ≠ μ0.

The population variance is known σ = σ0 (say).


Now the test statistic is image

This follows a N (0,1) distribution under H0.

Now the test criterion is set as if observed |Z|> c, then the hypothesis is rejected at 100α% level of significance. Here c is the ritical value of the distribution at the chosen level of significance say α, i.e. image

Z-test is also used to test the equality of two means of two independent univariate normal variables with the population variance to be known and equal.

Also for testing the location of bivariate normal population. The testing procedure is bit similar to the process stated above. 

The Graph of Z-Test at 5% Level of Significance


Graphs of Z-Test for different alternatives


Example of Z-Test


A cigarette manufacturer sent to a laboratory presumably identical samples of tobacco. They made five determinations of the nicotine content in milligrams as follows: (i) 21, 25, 27, 18, and 29. Test whether the mean is equal to 26 assuming the variance as 2. (Assume normality)


Here n = 5   

Mean = 24

SD = 4.472

The value of the test statistic is Z = -2.24.

Hence |Z| = 2.24.

Assuming is as a two tailed test and the level of the test as 5% we take the critical value c as

P[|Z|>c] = 0.05.

Therefore c = τ0.025 = 1.96.

Then |Z|>c and the hypothesis is rejected at 5% level of significance and it is concluded that the mean nicotine content is not 26.

Z-Test vs t-Test

T-test is used for testing the hypotheses regarding the means of normal populations when the population variance is unknown.

It is also used to test the equality of population proportion.

Z-test in comparison with Student’s t-test is more convenient. It has different values tabulated for different levels of significance not for different sample size whereas in t-test we will have different critical value for different sample size.

Student’s Z-test


  • Used when you know the standard deviation of the population (σ)

Student’s t-Test


  • Used when you only know the standard deviation of a sample(s)
  • Used if small sample size
  • Can also be used for comparing two samples


Properties of Z-test

Here we discuss some important properties of the Z-test:

  • Z-test gets the power analysis of the test if it is available.
  • It determines the Confidence Interval for the unknown parameter and also gets an estimated value for it.
  • It also takes into account the hypothesized value of the parameter in the test and determines the significance of the null or alternative hypothesis.
  • From Z-test we can also calculate the standard error of the estimated value.
  • Another important property or advantage of Z-test is that the distribution of the test statistic is always known and easy to deal with.


Other Uses of Z-test

  • Z-test can also be used to test hypotheses regarding populations other than normal because of the Central Limit Theorem.
  • Central Limit Theorem (CLT) states that any test statistic can be approximately normally distributed for large samples. Using CLT we can approximate any test statistic as a normal variable and use the Z-test to perform the hypothesis test.
  • This test is also used for testing the maximum likelihood estimators for the parameters in statistical models with parametric approach.
  • In nonparametric tests, sometimes the test statistic in Mann -Whitney U test can be approximated using normal distribution and Z-test is used thereby.


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