T-test
The idea behind parametric tests is to provide the researcher with a statistical inference about the population by conducting statistically significant tests (like t-test) on the sample drawn from the population. The parametric test called t-test is based on a student’s t statistic. This statistic assumes that variables are drawn from the normal population. The mean of the population in this statistic of t-test has been assumed to be known. The distribution, called t-distribution, has a similar shape to that of a normal distribution, i.e. a bell shaped appearance.
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The parametric test called t-test is useful for testing those samples whose size is less than 30. The reason behind this is that if the size of the sample is more than 30, then the distribution of the t-test and the normal distribution will not be distinguishable.
The parametric test is used for conducting statistically significant tests in the testing of hypotheses. There are basically three types of t-tests: one sample t-test, two independent sample t-test and paired t-test.
In the case of a one sample t-test, if a researcher in the field of psychology is working on a study where he wants to make sure that at least 65% of students will pass the IQ test, he can use the t-test. So, one sample t-test will be used after the hypothesis has been formulated in this particular case. The parametric test is then calculated by selecting an appropriate formula of t-test. In this case, the appropriate formula will be a t-test for a single mean. A selection of the level of significance is conducted to check the t-test of the null hypothesis. Usually, the researcher takes 0.05 as the appropriate level of significance while conducting the t-test. The level of significance refers to the minimum probability that there will be a false rejection of the null hypothesis. Now, if the value calculated from the t-test is more than the tabulated value, then the null hypothesis gets rejected at a particular level of significance. Similarly, if the value calculated from the t-test is less than the tabulated value, then the null hypothesis gets accepted at a particular level of significance.
In two independent sample t-tests, two samples that are not at all related to each other are tested. The main idea behind two independent sample t-tests is to draw out a statistical inference about the comparison of two independent samples of data. For example, in the field of psychology, if the researcher wants to compare the IQ level of students living in region A and region B, then a two independent sample t-test is useful. The region A and the region B are not at all related to each other, i.e. they are independent of each other. The procedure for conducting this t-test is the same, except that now the sample number is double instead of single. Also, in the case of t-test for single mean and two independent sample t-tests, there are different formulas for the degree of freedom. The degree of freedom is referred to as the restriction that a researcher puts forward while conducting parametric tests, like t-test in this case.
The paired sample t-test refers to that type of sample in which the variables form paired categories. For example, if a researcher wants to compare male and female smokers, paired sample t-test comes into play if the variable is in following form: male chain smoker and female chain smoker, male occasional smoker and female occasional smoker, etc.
Reference:
Introduction to the theory of statistics: Mood A.M., Graybill F.A., Boes D.C