# z-test definition

To compare or evaluate the importance of several statistical studies, most particularly the mean of a sample from a population having a normally distributed distribution or between two independent samples, one can apply a statistical test known as the z-test.

• The similar normal probability distribution that underlies t-tests also underlies z-tests.
• The z-test is the statistical procedure most frequently employed in research methods, which is applied to investigations with high sample sizes (n>30).
• The variance in the z-test scenario is typically known.
• The important factor at every level of significance in the confidence interval for the Z-test is the sample size for all sample sizes, making it more practical than the t-test.
• A z-score is a number that represents the number of standard deviations above or below the population mean.

## z-test formula

For the general populace using a single sample: When n is the number of observations, x, the sample mean, assumable mean, standard deviation, and mean are all present.

## z-test for the difference in mean When n1 and n2 are the numbers of observations for the two samples, x1 and x2 are their respective means, and are their respective standard deviations.

## One sample z-test (one-tailed z-test)

• One sample z-test is performed to evaluate if a certain population parameter, primarily mean, substantially differs from an anticipated value.
• Estimating the correlation between the sample mean and the assumed mean is helpful.
• In this instance, the critical value of the test is determined using the ordinary normal distribution.
• The alternative hypothesis will be adopted in place of the null hypothesis if the sample under test’s z-value meets the requirements for a one-sided test.
• A one-tailed test is used to determine if a population parameter is lower than or higher than a hypothesised value.
• A one-sample z-test assumes that the information is a randomly selected portion of a normally distributed population and has a mean and variance that are the same across all individuals.
• According to this theory, the distribution is symmetric, and the data is continuous.
• Left-sided z-tests and right-sided z-tests are both examples of one-sided z-tests, depending on the alternative hypothesis chosen for the investigation.
• Such a test would be one-sided or, more precisely, left-tailed, and there would only be one rejection area on the left tail of the distribution. For instance, if our H0: 0 = and Ha: 0.
• If H0 = 0 and Ha > 0, the rejection zone is found on the right tail of the curve, but this is also a one-tailed test (right tail).

## Test of two samples (two-tailed z-test)

• For the two-sample z-test, two normally distributed, independent samples are required.
• Using a two-tailed z-test, the relationship between the population parameters of the two samples is investigated.
• In the event that the population parameter does not match the predicted value, the alternative hypothesis is accepted when using the two-tailed z-test.
• When H0 = 0 and Ha = 0—which might indicate that > 0 or 0—are present, the two-tailed test is acceptable.
• As a result, there are two rejection zones in a two-tailed test, one on each tail of the curve.

## z-test examples

• If the sample has a mean height of 67.47 inches, is it reasonable to consider it typical of a broader population with a mean height of 67.39 inches and a standard deviation of 1.30 inches at a 5% level of significance?

Assuming the population’s mean height is 67.39 inches as the null hypothesis, the following may be written:

H0 : µ = 67.39″

Ha: µ ≠ 67.39″

x̄ = 67.47″, σ = 1.30″, n = 400

If the population is assumed to be normal, we may calculate the test statistic z as follows: Using a normal curve area table, a two-tailed test will be used to determine the rejection zones at a 5% level of significance, since Ha is two-sided in the provided query. The results are as follows:

R: | z | > 1.96

Since R: | z | > 1.96 and the observed value of t is 1.231, which is in the acceptance range, H0 is accepted.

## z-test applications

• Z-test is used in research when the variance is known and the sample size is larger.
• A substantial difference between the means of two independent samples is likewise assessed using this method.
• The population percentage of two samples may be compared to an assumed proportion, or the difference between them can be calculated using the z-test.

## 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

• 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
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