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I couldn't quite think of how best to title this, so recommendations are welcome. Same goes for the tags (I don't have the reputation to use the tags that I thought were appropriate). The question is this:

"Suppose you have N pairs of observations, (x,y), and you have a model with some unknown parameters, B, that estimates the relationship between x and y, F(x,B) -> y. Now suppose you determine B using the method of least-squares (and, implicitly, that all the assumptions of least-squares are satisfied). The parameters, B, are themselves random variables, each with its own variance. Is there any way to estimate the reduction (or increase) in the variance of B that would result from applying the same method of least-squares to N+1 pairs of observations?"

The question is asked in the context of experimentation. If each data point costs $X, an affirmative answer to the question would go a long way in determining whether or not to continue testing.

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I would think about starting with a power analysis: i.e. how many data points do you need to measure the effect (or parameter) that you are interested to a specified level of confidence, ceteris paribus? Then, you estimate a cost.

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