What test to be used when the sample size is less than 30 and population variance is not known?

In simple terms, a hypothesis refers to a supposition which is to be accepted or rejected. There are two hypothesis testing procedures, i.e. parametric test and non-parametric test, wherein the parametric test is based on the fact that the variables are measured on an interval scale, whereas in the non-parametric test, the same is assumed to be measured on an ordinal scale. Now, in the parametric test, there can be two types of test, t-test and z-test.

This article will give you an understanding of the difference between T-test and Z-test in detail.

Content: T-test Vs Z-test

1. Comparison Chart
2. Definition
3. Key Differences
4. Conclusion

Comparison Chart

Definition of T-test

A t-test is a hypothesis test used by the researcher to compare population means for a variable, classified into two categories depending on the less-than interval variable. More precisely, a t-test is used to examine how the means taken from two independent samples differ.

T-test follows t-distribution, which is appropriate when the sample size is small, and the population standard deviation is not known. The shape of a t-distribution is highly affected by the degree of freedom. The degree of freedom implies the number of independent observations in a given set of observations.

Assumptions of T-test:

• All data points are independent.
• The sample size is small. Generally, a sample size exceeding 30 sample units is regarded as large, otherwise small but that should not be less than 5, to apply t-test.
• Sample values are to be taken and recorded accurately.

The test statistic is:

Paired t-test: A statistical test applied when the two samples are dependent and paired observations are taken.

Definition of Z-test

Z-test refers to a univariate statistical analysis used to test the hypothesis that proportions from two independent samples differ greatly. It determines to what extent a data point is away from its mean of the data set, in standard deviation.

The researcher adopts z-test, when the population variance is known, in essence, when there is a large sample size, sample variance is deemed to be approximately equal to the population variance. In this way, it is assumed to be known, despite the fact that only sample data is available and so normal test can be applied.

Assumptions of Z-test:

• All sample observations are independent
• Sample size should be more than 30.
• Distribution of Z is normal, with a mean zero and variance 1.

The test statistic is:

The difference between t-test and z-test can be drawn clearly on the following grounds:

1. The t-test can be understood as a statistical test which is used to compare and analyse whether the means of the two population is different from one another or not when the standard deviation is not known. As against, Z-test is a parametric test, which is applied when the standard deviation is known, to determine, if the means of the two datasets differ from each other.
2. The t-test is based on Student’s t-distribution. On the contrary, z-test relies on the assumption that the distribution of sample means is normal. Both student’s t-distribution and normal distribution appear alike, as both are symmetrical and bell-shaped. However, they differ in the sense that in a t-distribution, there is less space in the centre and more in the tails.
3. One of the important conditions for adopting t-test is that population variance is unknown. Conversely, population variance should be known or assumed to be known in case of a z-test.
4. Z-test is used to when the sample size is large, i.e. n > 30, and t-test is appropriate when the size of the sample is small, in the sense that n < 30.

Conclusion

By and large, t-test and z-test are almost similar tests, but the conditions for their application is different, meaning that t-test is appropriate when the size of the sample is not more than 30 units. However, if it is more than 30 units, z-test must be performed. Similarly, there are other conditions, which makes it clear that which test is to be performed in a given situation.