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I would like to help my (Chemistry) students understand the math behind linear regression.(1) The generally accepted approach for my discipline is to omit any use of calculus and introduce matrix algebra, despite students in our curriculum rarely (if ever) taking a linear algebra course. (See, for example, the discussion starting on page 62 of this book.) Since our students do typically take calculus, it makes more sense to use an approach similar to this cross-validated question which has been satisfactory for determining the predicted values of the slope and intercept for a linear least-squares problem.

I would now like to go a step further and provide some details about how the parameter errors can be obtained. I do not have a problem conveying how the standard error in y (the observed value) is obtained; however, I am unable to explain how the standard errors in the slope and intercept are obtained without resorting to hand-waving and mysterious (to my students) equations (in particular, equations 4-16 and 4-17 in the book linked to above).

Any guidance or suggested references to how I might be able to teach this material to students would be helpful.

(1) which requires that I do a better job at understanding said math.

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One conceptual approach is to pose a thought experiment to the students: Imagine that I've collected this data, run the regression, and obtained this parameter estimate (let's say the slope). Now, this is just an estimate for the population parameter. So, if each of you (the students) were to collect new data, run the analysis, and obtain a parameter, then we would have a larger list of parameter estimates. This list of numbers will have some measure of spread, just like any distribution would. The estimate for that measure of spread is the SE for the slope.

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