Something about ANOVA is confusing me. Say you have a regression model:

y = b0 + b1*x1 (where b0 and b1 are coefficients and x1 is an explanatory variable)

and want to do an F-test on the null hypothesis, b2 = 0 (ie "should I include explanatory variable x2 in my model?")

Is this F-test not equivalent to actually calculating the coefficients for the model:

y = b0 + b1*x1 + b2*x2

and performing a t-test based on the null hypothesis b2 = 0 ?

y = b0 + b1*x1 (where b0 and b1 are coefficients and x1 is an explanatory variable)

and want to do an F-test on the null hypothesis, b2 = 0 (ie "should I include explanatory variable x2 in my model?")

Is this F-test not equivalent to actually calculating the coefficients for the model:

y = b0 + b1*x1 + b2*x2

and performing a t-test based on the null hypothesis b2 = 0 ?

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