Z-Test- Definition, Formula, Examples, Uses, Z-Test Vs T-Test

Z Test Definition

To compare or evaluate the importance of several statistical studies, particularly the mean of a sample from a population having a normally distributed 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 investigate 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 means, assumable mean, standard deviation, and mean are all present.

Z Test for the Difference in Mean

When n1 and n2 are the observations for the two samples, x1 and x2 are their respective means and 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 determines if a population parameter is lower than or higher than a hypothesized 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 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).

Z 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.

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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 

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 in 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 the 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 is 1.
Population parameters In addition, to the mean, the t-test can also be used to compare partial or simple correlations between two samples. In addition, to the mean, the 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.


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