Statistical Methods

6.1.Introduction & The T-Distribution

Unit 6 - Small samples Tests (n <30)

6.1.Introduction

When the size of the sample is small, the distributions of various statistics are far from normality and hence tests of hypothesis based on normal variate cannot be applied. In such cases tests of hypothesis based or exact sampling distribution of ‘t’ and ‘F’ are applied. When applying these tests it is assumed that the population from which the sample is drawn is normal.
The t - distribution which is popularly known as student’s t distribution is a sampling distribution derived from the parent normal distribution. This distribution is symmetrical about the mean but is slightly flatter than then normal distribution. Unlike the normal distribution it will be different for different size of the sample ‘n’ or the degree of freedom (n-1). When the size of the sample is very small < 30), the t - distribution markedly differs from normal distribution, but as n increases t - distribution resembles more and more a normal distribution (fig.1). The t distribution has mean zero and variance n / (n-2) for n>2. The variable t ranges theoretically from - ∞ to + ∞. The values of ‘t’ have been tabulated for different degrees of ‘freedom at different levels of significance (Fisher and Yates, 1963).

Test of hypothesis based on I distribution are discussed below :