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2. Using the linear model of y = Bo + B12 +, a researcher calculated the hat-matrix H and the residual e using 6 observations. [0.2516 0.1101 0.3931 0.0534 0.1101 0.0818 0.1101 0.2044 0.0157 0.2421 0.2044 0.2233 0.3931 0.1572 0.7704 -0.1352 0.0157 -0.0597 0.0535 0.2421 -0.1352 0.3176 0.2421 0.2233 0.1101 0.2044 0.0157 0.2421 0.2044 0.22327 0.2232 -0.0597 0.2799 0.2233 0.2516 0.0818 (a) Estimate o. That is, find Residual Standard Error s. H = X(X'X)-¹X'= " e= -0.02] 0.12 -0.56 0.88 -0.01 -0.07

2. Using the linear model of y = Bo + B12 +, a researcher calculated the hat-matrix H and the residual e using 6 observations. [0.2516 0.1101 0.3931 0.0534 0.1101 0.0818 0.1101 0.2044 0.0157 0.2421 0.2044 0.2233 0.3931 0.1572 0.7704 -0.1352 0.0157 -0.0597 0.0535 0.2421 -0.1352 0.3176 0.2421 0.2233 0.1101 0.2044 0.0157 0.2421 0.2044 0.22327 0.2232 -0.0597 0.2799 0.2233 0.2516 0.0818 (a) Estimate o. That is, find Residual Standard Error s. H = X(X'X)-¹X'= " e= -0.02] 0.12 -0.56 0.88 -0.01 -0.07

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**Transcription for Educational Use:** --- **2. Using the Linear Model** The linear model is given by: \[ y = \beta_0 + \beta_1 x + \epsilon, \] where a researcher calculated the hat-matrix \( H \) and the residual \( e \) using 6 observations: \[ H = X(X'X)^{-1}X' = \begin{bmatrix} 0.2516 & 0.1101 & 0.3931 & 0.0534 & 0.1101 & 0.0818 \\ 0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.2233 \\ 0.3931 & 0.1572 & 0.7704 & -0.1352 & 0.0157 & -0.0597 \\ 0.0535 & 0.2421 & -0.1352 & 0.3176 & 0.2421 & 0.2233 \\ 0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.22327 \\ 0.0818 & 0.2232 & -0.0597 & 0.2799 & 0.2233 & 0.2516 \end{bmatrix} \] \[ e = \begin{bmatrix} -0.02 \\ 0.12 \\ -0.56 \\ 0.88 \\ -0.01 \\ -0.07 \end{bmatrix} \] --- **(a) Tasks:** 1. **Estimate \( \sigma \) (Residual Standard Error \( s \)):** Calculate the residual standard error from the given data. 2. [Text blurred – not transcribed] 3. [Text blurred – not transcribed] 4. [Text blurred – not transcribed] --- **Explanation of Elements:** - **Hat-Matrix (H):** This is a matrix used in linear regression calculations to project observed data into the space spanned by the predictor variables. The entries in \( H \) indicate the influence of each observation on the fitted values. - **Residuals (e):** These are
Transcribed Image Text:**Transcription for Educational Use:** --- **2. Using the Linear Model** The linear model is given by: \[ y = \beta_0 + \beta_1 x + \epsilon, \] where a researcher calculated the hat-matrix \( H \) and the residual \( e \) using 6 observations: \[ H = X(X'X)^{-1}X' = \begin{bmatrix} 0.2516 & 0.1101 & 0.3931 & 0.0534 & 0.1101 & 0.0818 \\ 0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.2233 \\ 0.3931 & 0.1572 & 0.7704 & -0.1352 & 0.0157 & -0.0597 \\ 0.0535 & 0.2421 & -0.1352 & 0.3176 & 0.2421 & 0.2233 \\ 0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.22327 \\ 0.0818 & 0.2232 & -0.0597 & 0.2799 & 0.2233 & 0.2516 \end{bmatrix} \] \[ e = \begin{bmatrix} -0.02 \\ 0.12 \\ -0.56 \\ 0.88 \\ -0.01 \\ -0.07 \end{bmatrix} \] --- **(a) Tasks:** 1. **Estimate \( \sigma \) (Residual Standard Error \( s \)):** Calculate the residual standard error from the given data. 2. [Text blurred – not transcribed] 3. [Text blurred – not transcribed] 4. [Text blurred – not transcribed] --- **Explanation of Elements:** - **Hat-Matrix (H):** This is a matrix used in linear regression calculations to project observed data into the space spanned by the predictor variables. The entries in \( H \) indicate the influence of each observation on the fitted values. - **Residuals (e):** These are
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table row H equals cell X open parentheses X apostrophe X close parentheses to the power of negative 1 end exponent X apostrophe equals open square brackets table row cell 0.2516 end cell cell 0.1101 end cell cell 0.3931 end cell cell 0.0534 end cell cell 0.1101 end cell cell 0.0818 end cell row cell 0.1101 end cell cell 0.2044 end cell cell 0.0157 end cell cell 0.2421 end cell cell 0.2044 end cell cell 0.2233 end cell row cell 0.3931 end cell cell 0.1572 end cell cell 0.7704 end cell cell negative 0.1352 end cell cell 0.0157 end cell cell negative 0.0597 end cell row cell 0.0535 end cell cell 0.2421 end cell cell negative 0.1352 end cell cell 0.3176 end cell cell 0.2421 end cell cell 0.2233 end cell row cell 0.1101 end cell cell 0.2044 end cell cell 0.0157 end cell cell 0.2421 end cell cell 0.2044 end cell cell 0.22327 end cell row cell 0.0818 end cell cell 0.2232 end cell cell negative 0.0597 end cell cell 0.2799 end cell cell 0.2233 end cell cell 0.2516 end cell end table close square brackets end cell row e equals cell open square brackets table row cell negative 0.02 end cell row cell 0.12 end cell row cell negative 0.56 end cell row cell 0.88 end cell row cell negative 0.01 end cell row cell negative 0.07 end cell end table close square brackets end cell end table

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