Which of the following test is used when the population SD is known and N 30?

Z-test is the statistical hypothesis used to determine whether the two samples’ means calculated are different if the standard deviation is available and the sample is large. In contrast, the T-test determines how averages of different data sets differ in case the standard deviation or the variance is unknown.

Z-tests and T-tests are the two statistical methods that involve data analysis, which has applications in science, business, and many other disciplines. The T-test is a univariate hypothesis test based on T-statistics, wherein the mean, i.e., the average, is known, and population variance, i.e., the standard deviation, is approximated from the sample. On the other hand, Z-test is also a univariate test based on a standard normal distributionStandard Normal DistributionThe standard normal distribution is a symmetric probability distribution about the average or the mean, depicting that the data near the average or the mean are occurring more frequently than the data far from the average or the norm. Thus, the score is termed “Z-score”.read more.

Table of contents

You are free to use this image on your website, templates, etc., Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked
For eg:
Source: Z-Test vs T-Test (wallstreetmojo.com)

Uses

#1 – Z-Test

Z-test FormulaZ-test FormulaZ-test formula is applied hypothesis testing for data with a large sample size. It denotes the value acquired by dividing the population standard deviation from the difference between the sample mean, and the population mean.read more, as mentioned earlier, are the statistical calculations that one can use to compare population averages to a sample’s. The Z-test will tell you how far, in standard deviationsStandard DeviationsStandard deviation (SD) is a popular statistical tool represented by the Greek letter 'σ' to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data's reliability.read more terms, a data point is from the average of a data set. A Z-test will compare a sample to a defined population one typically uses for dealing with problems relating to large samples (i.e., n > 30). Mostly, they are very useful when the standard deviation is known.

#2 – T-Test

T-testsT-testsA T-test is a method to identify whether the means of two groups differ from one another significantly. It is an inferential statistics approach that facilitates the hypothesis testing.read more are also calculations one can use to test a hypothesis. Still, they are very useful when determining if there is a statistically significant comparison between the two independent sample groups. In other words, a t-test asks whether the comparison between the averages of 2 groups is unlikely to have occurred due to random chance. Usually, T-tests are more appropriate when dealing with problems with a limited sample sizeSample SizeThe sample size formula depicts the relevant population range on which an experiment or survey is conducted. It is measured using the population size, the critical value of normal distribution at the required confidence level, sample proportion and margin of error.read more (i.e., n < 30).

Z-Test vs. T-Test Infographics

Here we provide you with the top 5 differences between the z-test vs. t-test you must know.

You are free to use this image on your website, templates, etc., Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked
For eg:
Source: Z-Test vs T-Test (wallstreetmojo.com)

Key Differences

• One of the essential conditions for conducting a T-test is that the population standard deviation or the variance is unknown. Conversely, the population variance formulaPopulation Variance FormulaPopulation variance can be calculated using this formula: σ2 = ∑ni=1 (xi – μ)2 / N, where, σ2 is population variance, x1, x2, x3,…..xn are the observations, N is the number of observations and µ is the mean of the data set.read more, should be assumed to be known or known in the case of a Z-test.
• The t-test, as mentioned earlier, is based on student’s t-distributionT-distributionThe formula to calculate T distribution is T=x¯−μ/s√N. Where x̄ is the sample mean, μ is the population mean, s is the standard deviation, N is the size of the given sample.read more. On the contrary, the Z-test assumes that the distribution of sample means will be normal. The normal distribution and the student’s T- distribution appear the same, as both are bell-shapedBell-shapedBell Curve graph portrays a normal distribution which is a type of continuous probability. It gets its name from the shape of the graph which resembles to a bell. read more and symmetrical. However, they differ in one of the cases with less space in the center and more in their tails in T-distribution.
• Z-test is used as given in the above table when the sample size is large, which is n > 30, and the t-test is appropriate when the sample size is not big, which is small, i.e., that n < 30.

Z-Test vs. T-Test Comparative Table

• All data points are independent.
• Normal Distribution for Z, with an average zero and variance = 1.

• All data points are not dependent.
• Sample values are to be recorded and taken accurately.

Conclusion

By and to a larger extent, these tests are almost similar. Still, the comparison comes only to their conditions for their application, meaning that the T-test is more appropriate and applicable when the sample size is not more than thirty units. However, if it is greater than thirty units, one should use a Z-test. Similarly, other conditions will clarify which test to perform in a situation.

There are also different tests like the F-test, two-tailed vs. single-tailed, etc., so statisticians must be careful after analyzing the situation and then deciding which one to use. Below is a sample chart for what we discussed above.

Recommended Articles

This article has been a guide to Z-Test vs. T-Test. Here, we discuss the top 5 differences between these hypothesis testing, infographics, and a comparative table. You may also look at the following articles: –

What test will be used when n 30 but the population standard deviation is known?

What statistical test to use if n=30?

When N 30 and the population standard deviation is not known?

What test is used when the sample size is below 30 and the population standard deviation is unknown?