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What is the type of t-test used in the significance testing of coefficients/weights of linear regression? Is it one tailed or two tailed?

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That does depend on the hypotesis you want to test. I mean you get the t-statistic in all statistical programs and you know your degrees of freedom, so fell free to calculate is the way you want.

If you mean some kind of 'standard ouput' i assume, that the p-values will in most cases be calculated two-tailed.

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