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Suppose I set up a multiple regression with one continuous variable $X_1$, one (2-level) categorical variable $X_2$, and their interaction $X_1 X_2$.

This will result in four parameter estimates ($\beta_0, \beta_1, \beta_2, \beta_3$) and two different lines with different slopes ($\beta _0$ versus $\beta_0 + \beta_2$) and intercepts ($\beta_1$ versus $\beta_1 + \beta_3$), one line for each value of the categorical variable.

Now, could I deduce from the p-values in the model output that I. the slope for the line corresponding to the baseline value of the categorical variable is not statistically significant, but II. the slope for the line corresponding to the other value of the categorical variable is statistically significant?

I think not, because the slope estimates for each line are intertwined. But I am not sure.

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You are correct. Assuming that the two categories of your categorical variable X2 are coded with 0 and 1, the regression slope coefficient for X1 will give you the slope of the regression line in the group that is coded as 0 (I assume that's what you mean by "baseline value"). The p value for that X1 slope coefficient tells you whether the slope is statistically significant in Group 0 (the baseline/reference group).

You do not get the p value for the same slope coefficient in Group = 1 in this analysis--only for the difference in slopes between the two groups. To get the slope coefficient and its p value in Group = 1, you could flip the coding of X2 (0 = 1, 1 = 0) and then rerun the analysis with the reverse coded X2 grouping variable.

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