Level of Significance of a Test: the Two-sample t Test Statistic


INSTRUCTIONS

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Introduction

The two-sample t test is used for inferential procedures that compare the value of two population means. The assumptions that allow us to use a t(n1+n2-2) distribution as the sampling distribution of this test statistic are as follows:

  1. Both parent populaions are independent of each other and have distributions belonging to the normal family.
  2. The variance of the two populations is the same; that is,  sigma 1 square = sigma 2 square. Note that this does not imply that we know the value of the populations' variances, only that we know that those value are equal.
  3. The samples are randomly and independently selected.
If these assumptions are satisfied, then for the null hypothesis Ho: mu1-mu2=Do, the two-sample t test statistic is

                        t = ((X1bar - X2bar) - (mu1 - mu2))/square root(Sp square/n1 + Sp square/n2))
  where Sp square is the pooled variance and is the estimator of the common value of the population variances.The pooled variance is calcaulated using

                          Sp square = ((n1-1)s1 square + (n2-1) s2 square)/(n1 + n2 - 2)

If the assumptions are satisfied, we can expect close agreement between the desired value of alpha(level of significance of a test) and the actual proportion of tests that result in rejecting the null hypothesis.