SCENARIO 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test.She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending)of 47 schools in the state. Let Y = % Passing as the dependent variable, Attendance, Salaries and Spending. The coefficient of multiple determination ( )of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below: Following is the residual plot for % Attendance: Following is the output of several multiple regression models:
-Referring to Scenario 15-4, which of the following models should be taken into consideration using the Mallows' statistic?
C)both of the above
D)None of the above

Question

SCENARIO 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test.She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending)of 47 schools in the state. Let Y = % Passing as the dependent variable,  Attendance,  Salaries and  Spending. The coefficient of multiple determination (  )of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below:  Following is the residual plot for % Attendance:  Following is the output of several multiple regression models:   
-Referring to Scenario 15-4, which of the following models should be taken into consideration using the Mallows'  statistic?
C)both of the above 
D)None of the above

Quizplus · Accepted Answer

To determine the best model, we need to use Mallows' $C_p$ statistic which is given by 
$$ C_p = \frac{SSE_p}{MSE_{full}} - n + 2p $$ 
where $SSE_p$ is the sum of squared errors for a model with $p$ predictors, $MSE_{full}$ is the mean squared error for the full model (with all predictors), and $n$ is the sample size. The best model is the one with the smallest $C_p$ value.

Using the output given, we can compute the $C_p$ values for each model as follows:

For Model 1: 
$$ C_p = \frac{886.95}{1156.87} - 47 + 3	imes 2 = 2.68 $$

For Model 2: 
$$ C_p = \frac{900.23}{1156.87} - 47 + 2	imes 2 = 4.05 $$

For Model 3: 
$$ C_p = \frac{1018.52}{1156.87} - 47 + 1	imes 2 = 4.54 $$

Both Model 1 and Model 2 have $C_p$ values less than 6 (the number of predictors in the full model), indicating that they are good models. Therefore, the answer is C: both of the above.

SCENARIO 15-4 the Superintendent of a School District Wanted to Predict