Word Count : 849

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We are asked to investigate the effect of Age and Smoking Status on Systolic Blood Pressure. Let us denote: * (Y) = Systolic Blood Pressure (mmHg) * (X_1) = Age (years) * (X_2) = Smoking Status (0 for Non-Smokers, 1 for Smokers) We assume the multiple linear regression model: [ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon ] where (\epsilon) is the error term. Step 1: Construct the design matrix (X) and response vector (Y). Assume the dataset has (n) observations (values are given in the table, but we will illustrate the numerical method without the table). The general structure is: [ X = \begin{bmatrix} 1 & X_{11} & X_{12} \ 1 & X_{21} & X_{22} \ \vdots & \vdots & \vdots \ 1 & X_{n1} & X_{n2} \end{bmatrix}, \quad Y = \begin{bmatrix} Y_1 \ Y_2 \ \vdots \ Y_n \end{bmatrix} ] Step 2: Compute the normal equations for estimating (\beta_0, \beta_1, \beta_2): [ X^T X \beta = X^T Y ] Expanding: [ \begin{bmatrix} n ________ _____ _____ _________ ________ ________ _______ ________ ________ _____ ___ ______.
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