# Tests of Linear Combinations of Means, Independent Groups (3 of 7)

1. The fourth step is to calculate the probability of a value of L this different or more different from the value specified in the null hypothesis (zero). Applying the general formula, one obtains

where L is the value of the linear combination in the sample and sL is the standard error of L. The formula for the standard error of L is given below.

This formula is the same as the formula for σL except that MSE (an estimate of ) is used in place of . The formula for MSE (and SL) is the same as the formula used in the calculation of a confidence interval on a linear combination of means.

where is the sample variance of the ith group and "a" is the number of groups. This formula assumes homogeneity of variance and that the "a" sample sizes are equal. A related formula is described elsewhere that can be used with unequal sample sizes.

The degrees of freedom are equal to the sum of the degrees of freedom in the a = 3 groups. Therefore, df = a(n-1) = 3(4-1) = 9. A t table shows that the two-tailed probability value for a t of 2.33 with 9 df is 0.045.