IMPORTANCE OF REQUIREMENTS THREE AND FOUR

Requirement Three states that the two populations should be normally distributed. A severe departure from normality seems to have little effect on the conclusions when sample sizes are 30 or more. Of course, the results would be more accurate if the distribution were normal. In reality you can violate this assumption and not worry about it as long as your samples are not extremely small (Hays, 1973).

Requirement Four (homogeneity of variance) states that the populations should have the same variances. When the populations do not have the same variances and the sample sizes are equal, there is little effect on the conclusions reached by the t test. However, when the sample sizes are extremely small and of unequal sizes, the t ratio is affected. One way to handle this problem is to avoid using samples of unequal sizes. However, if this is not possible, and if the situation warrants it, use a nonparametric test instead of a t. If these two solutions are not feasible , the only other possibility is to use a computational formula that computes the standard error of each sample separately and use a corrected number for your degrees of freedom.

The best formula to use for the t test (II) is the following:

The numerator is the actual difference between the means, whereas the denominator is an estimate of the standard error of the difference between the means. The denominator is an estimate of the variability of the difference between the means of the two samples. When you use this formula, you are dividing the observed differences (numerator) by the variation of differences (denominator) that can be expected due to chance. If no significant difference exists between the groups, the ratio will be equal to zero. The further the ratio deviates from zero, the more likely it is that a real difference exists between the groups. The formula for t (II) looks very complicated, but these symbols are in fact old friends . Their meanings are: = mean of group I, = square the individual scores for group I and then find the sum, ( & pound ; X ₁ ) ² = sum the individual scores for group I and then square the sum, N _l = count the number of subjects in group I, and df = N ₁ + N ₂ - 2 degrees of freedom, the number of subjects in group I plus the number of subjects in group II minus 2.