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1. Given that the equation for the sum of the squares is

SS = x2 - (x)2/n

You are presented with the situation that you have two samples of a variable, each sample of an arbitrary number of observations (in the first situation, assume that the numbers are equal, in the second, assume that the numbers are not necessarily equal). Derive an equation from first principles, or set of equations, that describes the relationship between the variances of the two samples, and the variance of the overall dataset that would exist if the two samples were combined.

Generalise this equation to any arbitrary collection of k different samples (where k is the number of different samples).

2. One of the major problems of analysis is the difference between two samples of a variable. We wish to know if the means of the two samples are different.

Let us imagine that you want to know the mean difference in height between males and females of the same age, by sampling age-matched pairs of otherwise randomly selected people.

Derive an equation that describes the sum of the squares of the difference between two samples.

How different would these equations have to be if the individuals that were sampled were male-female non-identical twins pairs rather than randomly selected people.