t-test

The parametric test called the t-test provides a statistical inference about the population by testing the sample that has been drawn from that population in such a manner that it represents the population as a whole.

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This parametric test, called the t-test, is based on the student’s t statistic. This statistic in the t-test is based upon the assumption that the samples are drawn from a normal population. It is assumed in the t-test that the mean of the normal population exists. The shape of the distribution of the t-test is a bell shaped appearance.

The t-test is applicable in those cases where the size of the sample is less than 30. If the sample size is more than 30 and the t-test is carried out on it, then the inference drawn would not be valid as the distribution of the t-test and the normal distribution would not be noticeable.
The parametric test called the t-test is called parametric because it consists of the parameters called the mean and the variance. There are chiefly three types of t-tests: one sample t-test, two independent sample t-tests, and paired sample t-test.

The first type of t-test is applicable in those cases where the testing of one sample is done. For example, if the researcher wants to test whether or not at least 65 % of the students of a particular school would pass their 10 standard board exam, he could use this test. To conduct this type of t-test, a suitable null and alternative hypothesis is created by the researcher. The next step for the researcher is to construct the test statistic. In this case, the test statistic would be t-test. An appropriate level of significance would be selected by the researcher to conduct the t-test of the null hypothesis. The appropriate level of significance for conducting t-test is generally 0.05(which is the same in other significant tests as well). The level of significance refers to the probability that there would be a false rejection of the null hypothesis on which the t-test would be carried out.

Now, the comparison of the tabulated value of the t-test and the calculated value of the t-test is done by the researcher. If the calculated value of the t-test is more than the tabulated value, then the null hypothesis is rejected at that level of significance. In the opposite case of t-test, the null hypothesis is accepted.

Similarly, in the case of the second type of t-test, two independent samples are tested by comparing their significances with the help of the t-test. So, all the steps carried out in the previous step would remain the same, except that the hypothesis assumed by the researcher in this case would be for two independent samples.

Similarly, in the case of the paired sample t-test, the paired type of categories are tested and all the steps would remain the same, except that the hypothesis on which the t-test would be conducted will now be formulated according to the third type of t-test.

t-test

The t-test involves the single interval dependent variable and a dichotomous independent variable if the researcher wishes to conduct the t-test for the difference of means. The t-test can also be used to compare the means for two dependent samples and two independent samples. Additionally, the t-test can be used to test between a sample mean and a known mean, which is also called the t-test for one sample.

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The t-test is a parametric test that makes a very popular and obvious assumption—that of normal distribution or normal population. The researcher should note that if all the assumptions of the t-test are met, then the t-test becomes the most powerful. It is the most powerful test of any particular two sample non-parametric test.

The t-test is basically employed in those cases where the size of the sample is generally less than 30. If, however, the sample size is larger than 30, then instead of using the t-test, the researcher employs the z test.

The t-test is mainly based upon the student’s t distribution. The calculation of the t-test is different for comparison between the independent and the dependent samples, but the inference drawn from the t-test is the same.

The critical value in the t-test is the value that is found in the table of values of the t distribution for a given level of significance. If the value that has been calculated by using the t-test is more than the critical t value, then the null hypothesis that has been assumed in the t-test is rejected. But, if the value that has been calculated by using the t-test is less than the critical t value, then the null hypothesis that is assumed in the t-test is accepted.

The confidence limits in the t-test basically construct the upper bound and the lower bound on an estimate for a given level of significance. The confidence interval in the t-test is the range within these bounds. Such limits are employed in the t-test because such limits provide additional information on the relative meaningfulness of the estimates.

In SPSS, the t-test is conducted by selecting the “compare means” from the “analyze” menu and then by clicking any option, depending upon the type of t-test to be conducted by the researcher in SPSS. If two samples are involved, then the researcher can either employ an independent sample t-test or a paired sample t-test, depending on the type of data.

The following are some assumptions that have been assumed in the t-test:

The first assumption in the t-test is that the distribution, or the population under consideration, is that of normal distribution or normal population. For satisfying this assumption, there are certain tests for normality. The researcher should note that the t-test can draw invalid conclusions when the two samples come from widely different shaped distributions. Some statisticians suggest that the t-test should be normally distributed for the sample size, which is mainly less than 15.

The second assumption made in the t-test is that of the homogeneity of the variances in the sample. SPSS employs a test for testing the homoscedastic nature of the sample in the t-test. This test is called "Levene's Test for Equality of Variances," with F value and corresponding significance. The researcher should note that the t-test will result in invalid inferences if the two samples are unequal in size and also have unequal variances.

The third assumption is that in the t-test it does not matter whether the sample is a dependent or independent sample. This is because the inference drawn from the t-test will remain the same whether the sample is independent or dependent; only the calculation of the t-test will differ.