1. A researcher estimates two models:
$$(1)\ \ Pr(Passing\ Test)=\beta_{0}+\beta_{1}Homework\ Grade \\ (2) \ \ Pr(Passing\ Test)=\gamma_{0}+\gamma_{1}Study\ Time+\gamma_{2}Homework\ Grade$$
1. An RMDA professor can only estimate the first equation given the data she has at the end of the course. She is wondering if the relationship of homework grade on passing the test would be smaller or larger when one controls for study time. Unfortunately, she cannot observe study time. She assumes that study time is positively correlated with homework grade, however she is unsure if study time is positively or negatively correlated with passing the test. Give an answer to this professor under those two scenarios.
2. Regression interpretation

1. What is the marginal effect of being male?
2. What is the marginal effect of one more year of age?
3. The average men weights 150 pounds and 62 inches for tall. What is the marginal effect of one more year of age, at age 40?
4. The average men weights 150 pounds and 62 inches for tall. What is the predicted hourly wage for the average men who is a 40 years old?
3. Imagine we have the results from the following regression
$$wage=-11+-0.11weight+0.02weight^{2}+0.28height+3.49male$$
1. What is the marginal effect of an extra pound at 150 pounds?
2. What is the marginal effect of an extra pound at 140 pounds?
4. Imagine you have the following results from a regression:

1. How to the predicted hourly wages changes with height?
5. Let’s say we have this set up, where both female and black are binary variables, where the reference group is not-female and not black respectively:
$$Hourly\ Wage=\beta_{0}+\beta_{1}female+\beta_{2}black$$
1. What is the interpretation of $\beta_1$?
2. What is the interpretation of $\beta_0+\beta_1$?
3. What represents the average hourly wage for black females?
4. What represents the additional effect of being Black on hourly wages relative to not-black?
$$Hourly\ Wage=\beta_{0}+\beta_{1}female+\beta_{2}black+\beta_{3}female*black.$$
5. What is the interpretation of $\beta_1$?
6. What is the interpretation of $\beta_0+\beta_1$?
6. Imagine we have the set up:

$$(1) \ \ Hourly\ Wage=\alpha_{0}+\alpha_{1}Education\ if\ female=1 \\ (2) \ \ Hourly\ Wage=\gamma_{0}+\gamma_{1}Education\ if\ female=0 \\ (3) \ \ Hourly\ Wage=\beta_{0}+\beta_{1}Female+\beta_{2}Education+\beta_{3}Female\times Education$$

1. What does $\alpha_0$ represent? What is the equivalence of $\alpha_0$ in equation (3)?
2. What does $\alpha_1$ represent? What is the equivalence of $\alpha_1$ in equation (3)?
3. What does $\gamma_0$ represent? What is the equivalence of $\gamma_0$ in equation (3)?
4. What does $\gamma_1$ represent? What is the equivalence of $\gamma_1$ in equation (3)?