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Let us solve this step by step manually. We are given data for Age (X) and Systolic Blood Pressure (Y) for Non-Smokers and Smokers. We are asked to fit a multiple regression model with Smoking Status as a categorical variable and Age as a continuous variable. --- ### Step 1: Encode Smoking Status Let us define a dummy variable: [ S = \begin{cases} 0 & \text{Non-Smoker} \ 1 & \text{Smoker} \end{cases} ] We have 8 Non-Smokers and 7 Smokers. --- ### Step 2: Set up the multiple regression model The model is: [ Y = \beta_0 + \beta_1 X + \beta_2 S + \varepsilon ] Where: * (Y) = Systolic BP * (X) = Age * (S) = Smoking Status (0 = Non-Smoker, 1 = Smoker) We need to calculate (\beta_0, \beta_1, \beta_2) using the least squares method. --- ### Step 3: Calculate necessary sums Combine all data: | Patient | X | Y | S | | ------- | -- | --- | - | | 1 | 40 | 120 | 0 | | 2 | 45 | 124 | 0 | | 3 | 50 | 130 | 0 | | 4 | 55 | 132 | 0 | | 5 | 60 | 138 | 0 | | 6 | 65 | 142 | 0 | | 7 | 70 | 145 | 0 | | 8 | 75 | 150 | 0 | | 1 | 35 | 128 | 1 | | 2 | 42 | 135 | 1 | | 3 | 48 | 142 | 1 | | 4 | 53 | 148 | 1 | | 5 | 58 | 155 | 1 | | 6 | 62 | 158 | 1 | | 7 | 68 | 165 | 1 | We need: (n, \sum X, \sum Y, \sum S, \sum XY, \sum XS, \sum YS, \sum X^2, \sum S^2) #### Non-Smokers (S=0) * n₀ = 8 * ΣX₀ = 40+45+50+55+60+65+70+75 = 460 * ΣY₀ = 120+124+130+132+138+142+145+150 = 1081 ΣX₀² = 40²+45²+50²+55²+60²+65²+70²+75² 40²=1600 45²=2025 50²=2500 55²=3025 60²=3600 65²=4225 70²=4900 75²=5625 ΣX₀² = 1600+2025+2500+3025+3600+4225+4900+5625 = 27,500 ΣXY₀ = 40*120 + 45*124 + ___ ____ ____ ______ ________ ______ ______.
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