z-test definition
z-test is a statistical tool used for the comparison or determination of the significance of several statistical measures, particularly the mean in a sample from a normally distributed population or between two independent samples.
- Like t-tests, z tests are also based on normal probability distribution.
- Z-test is the most commonly used statistical tool in research methodology, with it being used for studies where the sample size is large (n>30).
- In the case of the z-test, the variance is usually known.
- Z-test is more convenient than t-test as the critical value at each significance level in the confidence interval is the sample for all sample sizes.
- A z-score is a number indicating how many standard deviations above or below the mean of the population is.
z-test formula
For the normal population with one sample:
where x̄ is the mean of the sample, and µ is the assumed mean, σ is the standard deviation, and n is the number of observations.
z-test for the difference in mean:
where x̄1 and x̄2 are the means of two samples, σ is the standard deviation of the samples, and n1 and n2 are the numbers of observations of two samples.
One sample z-test (one-tailed z-test)
- One sample z-test is used to determine whether a particular population parameter, which is mostly mean, significantly different from an assumed value.
- It helps to estimate the relationship between the mean of the sample and the assumed mean.
- In this case, the standard normal distribution is used to calculate the critical value of the test.
- If the z-value of the sample being tested falls into the criteria for the one-sided tets, the alternative hypothesis will be accepted instead of the null hypothesis.
- A one-tailed test would be used when the study has to test whether the population parameter being tested is either lower than or higher than some hypothesized value.
- A one-sample z-test assumes that data are a random sample collected from a normally distributed population that all have the same mean and same variance.
- This hypothesis implies that the data is continuous, and the distribution is symmetric.
- Based on the alternative hypothesis set for a study, a one-sided z-test can be either a left-sided z-test or a right-sided z-test.
- For instance, if our H_{0}: µ_{0} = µ and H_{a}: µ < µ_{0}, such a test would be a one-sided test or more precisely, a left-tailed test and there is one rejection area only on the left tail of the distribution.
- However, if H_{0}: µ = µ_{0} and H_{a}: µ > µ_{0}, this is also a one-tailed test (right tail), and the rejection region is present on the right tail of the curve.
Two sample z-test (two-tailed z-test)
- In the case of two sample z-test, two normally distributed independent samples are required.
- A two-tailed z-test is performed to determine the relationship between the population parameters of the two samples.
- In the case of the two-tailed z-test, the alternative hypothesis is accepted as long as the population parameter is not equal to the assumed value.
- The two-tailed test is appropriate when we have H_{0}: µ = µ_{0} and H_{a}: µ ≠ µ_{0} which may mean µ > µ_{0} or µ < µ_{0}
- Thus, in a two-tailed test, there are two rejection regions, one on each tail of the curve.
z-test examples
If a sample of 400 male workers has a mean height of 67.47 inches, is it reasonable to regard the sample as a sample from a large population with a mean height of 67.39 inches and a standard deviation of 1.30 inches at a 5% level of significance?
Taking the null hypothesis that the mean height of the population is equal to 67.39 inches, we can write:
H_{0} : µ = 67.39“
H_{a}: µ ≠ 67.39“
x̄ = 67.47“, σ = 1.30“, n = 400
Assuming the population to be normal, we can work out the test statistic z as under:
As H_{a} is two-sided in the given question, we shall be applying a two-tailed test for determining the rejection regions at a 5% level of significance which comes to as under, using normal curve area table:
R : | z | > 1.96
The observed value of t is 1.231 which is in the acceptance region since R: | z | > 1.96, and thus, H_{0} is accepted.
z-test applications
- Z-test is performed in studies where the sample size is larger, and the variance is known.
- It is also used to determine if there is a significant difference between the mean of two independent samples.
- The z-test can also be used to compare the population proportion to an assumed proportion or to determine the difference between the population proportion of two samples.
Z-test vs T-test (8 major differences)
Basis for comparison |
T-test |
Z-test |
Definition | The t-test is a test in statistics that is used for testing hypotheses regarding the mean of a small sample taken population when the standard deviation of the population is not known. | z-test is a statistical tool used for the comparison or determination of the significance of several statistical measures, particularly the mean in a sample from a normally distributed population or between two independent samples. |
Sample size | The t-test is usually performed in samples of a smaller size (n≤30). | z-test is generally performed in samples of a larger size (n>30). |
Type of distribution of population | t-test is performed on samples distributed on the basis of t-distribution. | z-tets is performed on samples that are normally distributed. |
Assumptions | A t-test is not based on the assumption that all key points on the sample are independent. | z-test is based on the assumption that all key points on the sample are independent. |
Variance or standard deviation | Variance or standard deviation is not known in the t-test. | Variance or standard deviation is known in z-test. |
Distribution | The sample values are to be recorded or calculated by the researcher. | In a normal distribution, the average is considered 0 and the variance as 1. |
Population parameters | In addition, to the mean, the t-test can also be used to compare partial or simple correlations among two samples. | In addition, to mean, z-test can also be used to compare the population proportion. |
Convenience | t-tests are less convenient as they have separate critical values for different sample sizes. | z-test is more convenient as it has the same critical value for different sample sizes. |
References and Sources
- C.R. Kothari (1990) Research Methodology. Vishwa Prakasan. India.
- https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/One-Sample_Z-Tests.pdf
- https://www.wallstreetmojo.com/z-test-vs-t-test/
- https://sites.google.com/site/fundamentalstatistics/chapter-13
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- 2% – https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/One-Sample_Z-Tests.pdf
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