# Sampling Distribution of a Linear Combination of Means (2 of 4)

Assuming the means from which L is computed are

independent, the mean and standard deviation of the sampling distribution
of L are:

μ

_{L} = a

_{1} μ

_{1}
+ a

_{2} μ

_{2} + ... + a

_{k}
μ

_{k
}and

where μ

_{i} is the mean for population
i, σ² is the variance of each population,
and n is the number of elements sampled from each population. Consider
an example application using the sampling distribution of L. Assume that
on a test of reading ability, the population means for 10, 12, and 14
year olds are 60, 68, and 80 respectively. Further assume that the variance
within each of these three populations is 100. Then, μ

_{1}
= 60, μ

_{2} = 68, μ

_{3}
= 80, and σ² = 100. If eight 10-year-olds,
eight 12- year-olds and eight 14-year-olds are sampled randomly, what is
the probability that the mean for the 14 year olds will be 15 or more
points higher than the average of the means for the two younger groups?