How to Find Intervals for α and β in a Linear Model Given y and x?

AI Thread Summary
In a simple linear model defined by the equation α + β × x = y, the discussion revolves around determining the intervals for α and β given the observed range of y, specifically when x = 4 and y ∈ (-8.51, 23.20). The problem is deemed not well-defined since α and β are interdependent, allowing for multiple combinations that satisfy the equation. To establish unique intervals for α and β, an additional relationship or constraint is necessary. Without this, the problem remains ambiguous and lacks a definitive solution. The need for further information to refine the model is emphasized.
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For a simple linear model:
\alpha+\beta\times x=y

If it is observed that y \in (-8.51,23.20) given x=4

The question is to give intervals of \alpha, \beta, which satisfy y \in (-8.51,23.20) given x=4.

Is this problem identifiable? Can it be found the unique intervals for \alpha, \beta?

I am guessing this is not a well defined problem. How to improve on this?
 
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α and β are connected by a linear equation. This means the you can give any value for one variable and get an interval for the other. To improve you need another relation.
 

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