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Let’s solve this step by step, manually, using the Newton-Raphson method for logistic regression and then apply the Hosmer-Lemeshow test. We are given: * Initial parameter estimates: (\beta_0 = -2.148), (\beta_1 = 0.36) * Observations: (x_i) (concentration), (y_i) (recovered), (n_i) (total subjects) We want one iteration of Newton-Raphson. --- Step 1: Logistic regression basics For a logistic regression: [ \pi_i = \frac{e^{\beta_0 + \beta_1 x_i}}{1 + e^{\beta_0 + \beta_1 x_i}} ] The Newton-Raphson update is: [ \boldsymbol{\beta}^{(new)} = \boldsymbol{\beta}^{(old)} + (\mathbf{X}^\top \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^\top (\mathbf{y} - \boldsymbol{\mu}) ] Where: * (\mathbf{X} = \begin{bmatrix} 1 & x_1 \ \vdots & \vdots \ 1 & x_n \end{bmatrix}) * (\mathbf{W} = \text{diag}(n_i \pi_i (1-\pi_i))) * (\boldsymbol{\mu} = n_i \pi_i) --- Step 2: Compute (\pi_i) with initial estimates [ \pi_i = \frac{e^{\beta_0 + \beta_1 x_i}}{1 + e^{\beta_0 + \beta_1 x_i}} ] * For (x_1 = 2): [ \eta_1 = -2.148 + 0.36 \cdot 2 = -2.148 + 0.72 = -1.428 ] [ \pi_1 = \frac{e^{-1.428}}{1 + e^{-1.428}} \approx \frac{0.240}{1 + ____ ________ _______ _________ _________ _____ ___.
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