Is there a difference in the t-test used in a simple regression setting versus the t-test used in a multiple regression setting? What about the F-test in those two situations?

A t-test is used in regression to determine if there is a significant difference between either the slope or the y-intercept and 0. If the slope is found to be significantly different than 0, this is a positive result in that there is likely a meaningful relationship between the regression variables (i.e. x and y). If the intercept is found to be significantly different than 0, this is actually not usually of substantiative interest, but the analysis is automatically included in most statistical programs.

A t-test in simple regression only has one slope and one y-intercept to test for a difference between the given values for each of these and 0. In multiple regression there is still only one y-intercept to test, but there are more than one value for the slope to test (i.e. a value for the slope associated with each additional predictor variable). However, the t-test is still the same and what it is testing for is still the same, there are simply additional t-tests run to accommodate the increased number of variables.

An F-test in simple regression, tests to see if the model is a better fit of the data than the overall mean of the outcome variable. This is the same principle applied in multiple regression. The difference between simple and multiple regression is simply that equation representing multiple regression has additional variables included. The principles behind the F-test however, do not change.