' Consider the ElantraSales data for the analysis. The Elantra…

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Step 1: 1. Formulating the Linear Regression Model The linear regression model can be represented as:  Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 + ϵ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_4 X_4 + \epsilon 2. Calculate the Coefficients  To find the coefficients, we use the Normal Equation:  β = ( X T X ) − 1 X T Y \beta = (X^T X)^{-1} X^T Y Where: X X X is the matrix of independent variables with a column of ones for the intercept Y Y Y is the vector of dependent variable ElantraSales Step-by-Step Calculation: Construct the Matrix  X X X and Vector  Y Y Y : X = [ 1 X 11 X 12 X 13 X 14 1 X 21 X 22 X 23 X 24 ⋮ ⋮ ⋮ ⋮ ⋮ 1 X n 1 X n 2 X n 3 X n 4 ] X = \begin{bmatrix} 1 & X_{11} & X_{12} & X_{13} & X_{14} \ 1 & X_{21} & X_{22} & X_{23} & X_{24} \ \vdots & \vdots & \vdots & \vdots & \vdots \ 1 & X_{n1} & X_{n2} & X_{n3} & X_{n4} \end{bmatrix} ​ Y = [ Y 1 Y 2 ⋮ Y n ] Y = \begin{bmatrix} Y_1 \ Y_2 \ \vdots \ Y_n \end{bmatrix} ​ Calculate  X T X X^T X X T X : X T X = [ ∑ i = 1 n 1 ∑ i = 1 n X i 1 ∑ i = 1 n X i 2 ∑ i = 1 n X i 3 ∑ i = 1 n X i 4 ∑ i = 1 n X i 1 ∑ i = 1 n X i 1 2 ∑ i = 1 n X i 1 X i 2 ∑ i = 1 n X i 1 X i 3 ∑ i = 1 n X i 1 X i 4 ∑ i = 1 n X i 2 ∑ i = 1 n X i 2 X i 1 ∑ i = 1 n X i 2 2 ∑ i = 1 n X i 2 X i 3 ∑ i = 1 n X i 2 X i 4 ∑ i = 1 n X i 3 ∑ i = 1 n X i 3 X i 1 ∑ i = 1 n X i 3 X i 2 ∑ i = 1 n X i 3 2 ∑ i = 1 n X i 3 X i 4 ∑ i = 1 n X i 4 ∑ i = 1 n X i 4 X i 1 ∑ i = 1 n X i 4 X i 2 ∑ i = 1 n X i 4 X i 3 ∑ i = 1 n X i 4 2 ] X^T X = \begin{bmatrix} \sum_{i=1}^{n} 1 & \sum_{i=1}^{n} X_{i1} & \sum_{i=1}^{n} X_{i2} & \sum_{i=1}^{n} X_{i3} & \sum_{i=1}^{n} X_{i4} \ \sum_{i=1}^{n} X_{i1} & \sum_{i=1}^{n} X_{i1}^2 & \sum_{i=1}^{n} X_{i1} X_{i2} & \sum_{i=1}^{n} X_{i1} X_{i3} & \sum_{i=1}^{n} X_{i1} X_{i4} \ \sum_{i=1}^{n} X_{i2} & \sum_{i=1}^{n} X_{i2} X_{i1} & \sum_{i=1}^{n} X_{i2}^2 & \sum_{i=1}^{n} X_{i2} X_{i3} & \sum_{i=1}^{n} X_{i2} X_{i4} \ \sum_{i=1}^{n} X_{i3} & \sum_{i=1}^{n} X_{i3} X_{i1} & \sum_{i=1}^{n} X_{i3} X_{i2} & \sum_{i=1}^{n} X_{i3}^2 & \sum_{i=1}^{n} X_{i3} X_{i4} \ \sum_{i=1}^{n} X_{i4} & \sum_{i=1}^{n} X_{i4} X_{i1} & \sum_{i=1}^{n} X_{i4} X_{i2} & \sum_{i=1}^{n} X_{i4} X_{i3} & \sum_{i=1}^{n} X_{i4}^2 \end{bmatrix} ​ Calculate  X T Y X^T Y X T Y = [ ∑ i = 1 n Y i ∑ i = 1 n X i 1 Y i ∑ i = 1 n X i 2 Y i ∑ i = 1 n X i 3 Y i ∑ i = 1 n X i 4 Y i ] X^T Y = \begin{bmatrix} \sum_{i=1}^{n} Y_i \ \sum_{i=1}^{n} X_{i1} Y_i \ \sum_{i=1}^{n} X_{i2} Y_i \ \sum_{i=1}^{n} X_{i3} Y_i \ \sum_{i=1}^{n} X_{i4} Y_i \end{bmatrix} ​ Calculate  ( X T X ) − 1 (X^T X)^{-1} Use matrix inversion techniques to find  ( X T X ) − 1 (X^T X)^{-1} Calculate  β 3. Calculate Predicted Values  Y ^ = X β \hat{Y} = X \beta 4. Calculate R-squared Value R 2 = 1 − ∑ i = 1 n ( Y i − Y ^ i ) 2 ∑ i = 1 n ( Y i − Y ˉ ) 2 R^2 = 1 - \frac{\sum_{i=1}^{n} (Y_i - \hat{Y}_i)^2}{\sum_{i=1}^{n} (Y_i - \bar{Y})^2} ​ Where: Y ^ i \hat{Y}_i Y ^ i ​ are the predicted values Y i Y_i Y i ​ are the observed values Y ˉ \bar{Y} Y ˉ is the mean of the observed values 5. Calculate the Standard Errors and T-values Calculate Residuals: ϵ i = Y i − Y ^ i \epsilon_i = Y_i - \hat{Y}_i ​ Calculate Residual Sum of Squares (RSS): R S S = ∑ i = 1 n ϵ i 2 RSS = \sum_{i=1}^{n} \epsilon_i^2 ​ Calculate the Variance of the Residuals: σ 2 = R S S n − k − 1 \sigma^2 = \frac{RSS}{n - k - 1} ​ Where  k k k is the number of predictors. Calculate the Standard Errors of the Coefficients: S E ( β j ) = σ 2 ( X T X ) j j − 1 SE(\beta_j) = \sqrt{\sigma^2 (X^T X)^{-1}_{jj}} ​ Calculate the T-values: t j = β j S E ( β j ) t_j = \frac{\beta_j}{SE(\beta_j)} ​ ​ 6. Calculate P-values Use the T-distribution to find the p-values corresponding to the T-values. 7. Identify the Most Significant Variable The variable with the lowest p-value is considered the most significant predictor. Final Answer:

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