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

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The discussion focuses on identifying intervals for the parameters α and β in the linear model defined by the equation α + β × x = y, given the constraints y ∈ (-8.51, 23.20) and x = 4. It concludes that without an additional relationship, the problem is not well-defined, as any value assigned to one parameter allows for a corresponding interval for the other. Therefore, to refine the intervals for α and β, an extra constraint or relation is necessary.

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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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