Why do we need to estimate the variance of $\hat{\beta}_1|x_1, x_2,\ldots, x_n$ when we want to test a hypothesis about $\beta_1$?
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Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – utobi Oct 16 '22 at 21:26
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Possible duplicate of https://stats.stackexchange.com/questions/485676/meaning-of-standard-error-of-the-coefficients-in-a-regression-model/485680#485680 and the comments provided there – Christoph Hanck Oct 17 '22 at 09:21