A model was fit for a random sample of 100 low birth weight infants born in two teaching hospitals in Boston, Massachusetts, regressing birthweight on the predictors gestational age and toxemia status. The condition toxemia, also known as preeclampsia, is characterized by high blood pressure and protein in urine by the \(20^{\text {th }}\) week of pregnancy; left untreated, toxemia can be life-threatening. Birth weight was measured in grams and gestational age measured in weeks.

(a) Write the model equation.

(b) Interpret the coefficients of the model, and comment on whether the intercept has a meaningful interpretation.

(c) Predict the average birth weight for an infant born to a mother diagnosed with toxemia with gestational age 31 weeks.

(d) Evaluate whether the assumptions for linear regression are reasonably satisfied.

(e) A simple regression model with only toxemia status as a predictor had \(R^{2}=0.0001\) and \(R_{a d j}^{2}=0.010\); in this model, the slope estimate for toxemia status is 7.785 , with \(p=0.907\). The simple regression model and multiple regression model disagree regarding the nature of the association between birth weight and toxemia. Briefly explain a potential reason behind the discrepancy. Which model do you prefer for understanding the relationship between birth weight and toxemia, and why?

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(Intercept) Estimate -1286.200 Std. Error t value Pr(>|t|) 234.918 -5.475 0.0000 toxemia Yes -206.591 51.078 -4.045 0.0001 gestage 84.048 8.251 10.188 0.0000