**z-test definition**

**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**

**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**

**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 (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)**

**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**

**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 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)**

**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**

**References and Sources**

- R. Kothari (1990) Research Methodology. Vishwa Prakasan. India.
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