CAS Exam MAS-I Spring 2019, Question 32
Study guide: Exam MAS-I
You are fitting the following linear regression model with an intercept:
y = X\beta + e; \quad e_i \sim N(0, \sigma^2)and are given the following values used in the model:
X = \begin{bmatrix}
1 & 0 & 1 & 9 \\
1 & 1 & 1 & 15 \\
1 & 1 & 1 & 8 \\
0 & 1 & 1 & 7 \\
0 & 1 & 1 & 6 \\
0 & 0 & 1 & 6
\end{bmatrix}; \quad
y = \begin{bmatrix}
21 \\
32 \\
19 \\
17 \\
15 \\
15
\end{bmatrix}; \quad
X^T X = \begin{bmatrix}
3 & 2 & 3 & 32 \\
2 & 4 & 4 & 36 \\
3 & 4 & 6 & 51 \\
32 & 36 & 51 & 491
\end{bmatrix}(X^T X)^{-1} = \begin{bmatrix}
1.38 & 0.25 & 0.54 & -0.16 \\
0.25 & 0.84 & -0.20 & -0.06 \\
0.54 & -0.20 & 1.75 & -0.20 \\
-0.16 & -0.06 & -0.20 & 0.04
\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}(X^T X)^{-1} X^T y = \begin{bmatrix}
0.297 \\
-0.032 \\
3.943 \\
1.854
\end{bmatrix}; \quad
X(X^T X)^{-1} X^T y = \begin{bmatrix}
20.93 \\
32.03 \\
19.04 \\
16.89 \\
15.04 \\
15.07
\end{bmatrix}; \quad
\sigma^2 = 0.012657Calculate the modeled estimate of the intercept parameter.
- A) Less than 0
- B) At least 0, but less than 1
- C) At least 1, but less than 2
- D) At least 2, but less than 3
- E) At least 3
Answer this question and see the worked solution