CAS Exam MAS-I Fall 2018, Question 35
Study guide: Exam MAS-I
You are fitting a linear regression model of the form:
y = X\beta + e;\ e_i \sim N(0, \sigma^2)
and are given the following values used in this model:
X = \begin{bmatrix} 1 & 0 & 1 & 9 \\ 1 & 1 & 1 & 15 \\ 1 & 1 & 1 & 8 \\ 1 & 1 & 0 & 7 \\ 1 & 1 & 0 & 6 \\ 1 & 0 & 0 & 6 \end{bmatrix}; \quad y = \begin{bmatrix} 19 \\ 32 \\ 19 \\ 17 \\ 13 \\ 15 \end{bmatrix}; \quad X^T X = \begin{bmatrix} 6 & 4 & 3 & 51 \\ 4 & 4 & 2 & 36 \\ 3 & 2 & 3 & 32 \\ 51 & 36 & 32 & 491 \end{bmatrix}(X^T X)^{-1} = \begin{bmatrix} 1.75 & -0.20 & 0.54 & -0.20 \\ -0.20 & 0.84 & 0.25 & -0.06 \\ 0.54 & 0.25 & 1.38 & -0.16 \\ -0.20 & -0.06 & -0.16 & 0.04 \end{bmatrix}; \quad (X^T X)^{-1} X^T y = \begin{bmatrix} 2.335 \\ 0.297 \\ -0.196 \\ 1.968 \end{bmatrix}H = X(X^T X)^{-1} X^T = \begin{bmatrix} 0.684 & 0.070 & 0.247 & -0.171 & -0.146 & 0.316 \\ 0.070 & 0.975 & -0.044 & 0.108 & -0.038 & -0.070 \\ 0.247 & -0.044 & 0.797 & 0.063 & 0.184 & -0.247 \\ -0.171 & 0.108 & 0.063 & 0.418 & 0.411 & 0.171 \\ -0.146 & -0.038 & 0.184 & 0.411 & 0.443 & 0.146 \\ 0.316 & -0.070 & -0.247 & 0.171 & 0.146 & 0.684 \end{bmatrix}Calculate the residual for the 5^{\text{th}} observation.
- A) Less than -1
- B) At least -1, but less than 0
- C) At least 0, but less than 1
- D) At least 1, but less than 2
- E) At least 2
Answer this question and see the worked solution