Part 1 - Multiple Choice / True and False (8 points) In a multiple-choice question, please select the correct answer. (1 point) In a True and False statement, please correct if it is incorrect. (2 points) 1. In a multiple linear regression with ???? regressors, there are exactly ???? = ???? normal equations, and the solution of these equations results in the best estimations of ???? and ????. 2. ???? is the proportion of variability in the response ???? explained by the model. 3. Is the analysis on individual regression coefficients a marginal test because the regression coefficients depend on all the regressor variables in the model? 4. We assume that the error term in the model has: a. ????() = ???? and ????????????() = ???? b. ????() = 0 and ????????????() = 0 c. ????() = ???? and ????????????() = 0 d. ????() = 0 and ????????????() = ???? 5. The Gauss-Markov theorem states that: a. Least squares estimators are biased but efficient b. Maximum likelihood estimators are the best linear estimators c. Least squares estimators are the best linear unbiased estimators d. Regression coefficients must follow a normal distribution October 9th, 2025 Part 2 - Short Answer 2 points each 1. What is a limitation of ????? 2. Please explain the significance of the regression in multiple linear regression. 3. Why can't scatter plots fully describe multiple regression relationships? October 9th, 2025 Part 3 - Applied Problem Please show your procedure to receive partial credit. Data from ???? = 113 hospitals in the United States are used to assess factors related to the likelihood that a hospital patient will acquire an infection. The variables of interest in this problem are: ???? : Average length of a patient's stay ???? : Average patient's age ???? : Number of X-rays given in the hospital ????: infection risk After running this model to estimate the regression coefficients, the summary output shows the following results: Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.00 1.31 0.76 0.448003 Stay Age 0.31-0.02 0.06-1.60 5.19 0.000001 0.19 3.61 0.43 0.02-0.98 0.330098 0.02 Xray 0.02 0.01 3.41 0.000899 0.01 0.03-0.07 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.602 0.363 0.345 1.085 113.000 Please answer the following questions: 1. What is the final multiple linear regression model of this problem? (2 points) 2. What are the degrees of freedom for the regression? What are the degrees of freedom for the residuals? (2 points) 3. Please comment on the regression coefficient results of the obtained model. (4 points) 4. Considering ???? = 0.05, are all the regressors significant for this model? (Hint: You can use the P-values.) (2 points) October 9th, 2025 5. Test the significance of the entire regression model? Use: ???????? = 73.10, ???????? = 128.28 and ???? = 0.05, compare to ????. , , = 2.69 a. Set the hypotheses (2 points) b. Calculate the test statistic (2 points) c. Make a conclusion (3 points) 6. What can you tell of the ???? and ???? of this model? (4 points) 7. What would be the prediction value (????), under the following scenario: (3 points) Average length of a patient's stay is 5.45 Average patient's age 15 Number of X-rays given in the hospital 42 8. As a researcher, you decided to add two more factors to this model: the number of beds in the hospital (????) and the nurses available on the shift (????). The regression statistics of this new model are presented as follows: Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.650 0.422 0.395 1.043 113.000 Does this model perform significantly better than the initial model? Please explain your answer. What can you conclude from this result? (7 points) October 9th, 2025 Formulas: ???? = ???? = (????????) =???????? =???????? ????????=???????? = ????????/???? ???????? /(????????1)=???????? ???????? ???? =???????? +???????? ???? , ???????? ????????+???? , ???????? ???? , ???? (1+????????(????????) ????????)???? ????+???? , ???? (1+????????(????????) ????????) =1???????? ???????? =1???????? /(????????) ????????/(????1) ???????? =????????=????????1