Sampling Distribution of a Linear Combination of Means (1 of 4)
Assume there are k populations each with the same variance (σ²). Further assume that (1) n subjects are sampled randomly from each population with the mean computed for each sample and (2) a linear combination of these means is computed by multiplying each mean by a coefficient and summing the results. Let the linear combination be designated by the letter "L." If this sampling procedure were repeated over and over again, a different value of L would be obtained each time. It is this distribution of the values of L that makes up the sampling distribution of a linear combination of means. The importance of linear combinations of means can be seen in the section "Confidence interval on linear combination of means" where it is shown that many experimental hypotheses can be stated in terms of linear combinations of the mean and that the choice of coefficients determines the hypothesis tested. The formula for L is:
L = a1M1 + a2M2 + ... + akMk
where M1is the mean of the numbers sampled from Population 1, M2 is the mean of the numbers sampled from Population 2, etc. The coefficient a1 is used to multiply the first mean, a2 is used to multiply the second mean, etc.