What is Z-Test? | Definition, Statistics, Calculation and Examples

Z-test is a form of statistical tool that is used to find out whether the means of two distribution vary even with known variances and large sample size.

It is a form of hypothesis test that is used to decide whether to accept a null hypothesis or not. As a statistical test, it is univariate, and the test statistic result is expected to follow a standard normal distribution.

It is employed only when there are a known standard deviation and a large sample size (n>30).

What are Z-scores?

Z-scores or Z-statistics are numbers that represent how much the test statistic results have deviated above or below the mean distribution.

For example, a Z-score of +1.45 signifies that the test statistic result is 1.45 standard deviations above the mean. In contrast, a Z-score of -1.45 implies that the observation has deviated 1.45 below the mean population.

When should a Z-test be conducted?

The following conditions should prevail to perform a Z-test.

1. The sample size must be more than 30.
2. The sample data should always be random. Otherwise, the test statistic results may turn out to be inaccurate.
3. The data points must not be similar. Besides that, they must not overlap with each other.
4. The data must reflect a standard normal distribution.
5. The standard deviation of the population has to be known.
6. If the population standard deviation is unknown, it should be assumed that the sample variance is equal to the population variance.

However, in case, the distribution variation is not known, and the sample data is below 30, a T-test will prove to be more suitable than a Z-test.

How to conduct a Z-test?

The following steps should be followed to conduct a Z-test:

1. First of all, the null (H0) and the alternative hypothesis needs to be stated.
2. Now, select the alpha level.
3. The Z table has to be employed to define the criticality of Z.
4. Now, the Z state statistic has to be calculated.
5. Once, the test statistic result is obtained, compare it with the critical value z. 
6. The comparison will lead you to the conclusion if the null hypotheses (H0) can be accepted or not.

Z-test Calculation

The following formula can be employed to compute a Z-test:

Z- test = (x̄ – μ) / (σ / √n)

where, 

• x̄ = Sample Mean
• μ = Population Mean
• σ = Standard Deviation of Population
• n = Number of Observation

Example

Suppose the IQ score of a particular class is 113. The mean IQ of India is 100 with a standard deviation of 15. Is the intelligence quotient of this class significantly above than the mean IQ?

Tests that can be employed as Z-tests

The following are some significant tests that one can consider to run as Z-tests:

1. One-sample location test.
2. Two-sample location test.
3. Paired difference test.
4. Maximum likelihood estimate.

Advantages of Z-test

The following are some significant advantages of the Z-test.

1. It is a straightforward and reliable test.
2. A Z-score can be used for a comparison of raw scores obtained from different tests.
3. While comparing a set of raw scores, the Z-score considers both the average value and the variability of those scores.

Disadvantages of Z-test

Despite its various advantages, the Z-test suffers from some significant limitations:

1. Z-test requires a known standard deviation which is not always possible.
2. It cannot be conducted with a smaller sample size (less than 30).

References

1. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1420-9101.2011.02297.x
2. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1420-9101.2010.02226.x