4. Here I use the test score data again. Let's say I ran two models to try to explain the relationship between test scores and the percent of children in a school that receive free meals (Rfmeals) Model 1 is a simple regression of test scores on pfmeals. Model 2 is a multiple regression of test scores on pfmeals and pool (percent of parents in a particular school with a college degree). Stata results are below. Model 1: reg tscore pfmeals Source SS df MS Number of obs 4, 889 F (1, 4887) 12921 . 42 Model 288035.863 1 288035.863 Prob > F II 0. 0000 Residual 4, 887 22.2913505 R-squared Adj R-squared 0. 7255 Total 396973 . 693 4, 888 81.2139307 Root MSE 4. 7214 tscore Coef. Std. Err. P>Itl [954 Conf. Interval] pfmeals -. 2478825 . 0021807 -113. 67 0. 000 -. 2521576 - . 2436074 cons 89. 02932 . 1373457 648.21 0. 000 88 . 76006 89.29858 Model 2: . reg tscore pfmeals pcol Source SS df MS Number of obs 4, 889 F (2, 4886) 8201 . 44 Model 305864. 491 2 152932 . 246 Prob > F 0. 0000 Residual 109 . 202 4, 886 18. 6469918 R-squared 0 . 7705 Adj R-squared 0. 7704 II Total 396973. 693 4, 888 81.2139307 Root MSE 4. 3182 tscore Coef. Std. Err. t P>Itl [95$ Conf. Interval] pfmeals -. 1517704 0036932 -41. 09 0. 000 -. 1590107 - . 1445301 pcol 1515683 0049018 30.92 0. 000 . 1419586 . 161178 cons 79. 41244 3354241 236.75 0. 000 78 . 75486 80. 07003 4a. Interpret the coefficient on pfmeals in Model 2 (mathematically and statistically). (5 points) 4b. Is the coefficient on pfmeals in Model 1 biased compared to the coefficient on pfmeals in Model 2. If so, by how much? (6 points) 4c. What 2 conditions would need to be true for bias of pfmeals to exist? Use only the information from Models 1 and 2 above to answer this question. (7 points) 4d. If you regressed pool on pfmeals, what would the coefficient on pfmeals be? (Hint: Use the bias formula.) (7 points) 4e. Your colleague tells you that you should generate a new model (Model 3). This model would be a regression of test scores on pfmeals, pool, and "hungry" (a measure that indicates how hungry children are when they leave school each day). Your colleague says that "hungry" meets the bias criteria, and you agree. Should you include "hungry" in your regression? (7 points)