1. What happens to the minimum detectable effect if you increase the variance of the error term? What does that say about the value of including control variables in your regression model?
2. Suppose that you want to be more careful and decide that you will only reject the null hypothesis at the 1% level instead of the 5% level in a two tailed test. How does this change the minimum detectable effect?
3. For a given MDE, how does deciding to reject the null only at the 1% instead of at the 5% level affect your chances of accepting the alternative hypothesis given that it's true?