Texts: QUESTION 4
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Consider the linear regression model
Y = β0 + β1Xi + β2X2 + εi i = 1, 2, ..., n
where:
Y, Xi, and X2 are the blood pressure, age in years, and weight in kg of the patient respectively; β0, β1, and β2 are the regression coefficients; and εi are independent and normally distributed errors with mean 0 and variance σ^2.
a) Show that the method of least squares estimators of β1 and β2 are β1 = Σ(xi - x̄)(yi - ȳ) / Σ(xi - x̄)^2 and β2 = Σ(xi - x̄)(zi - z̄) / Σ(xi - x̄)^2, which solve the equations:
Σyi = nβ0 + β1Σxi + β2Σxi^2 and Σxiyi = β0Σxi + β1Σxi^2 + β2Σxi^3 i = 1, 2, ..., n
b) In matrix notation, the equations in part a can be expressed as AX = Z, where A is a 2x2 matrix and Z is a 2x1 vector.
c) Given:
Σyi = 104328 Σxi = 324235 Σxi^2 = 43737 Σxi^3 = 421273 Σzi = 135458 n = 11
Fit the linear regression model Y = β0 + β1Xi + β2X2 to the data. That is, find the least squares estimates of β1 and β2.
d) Predict the blood pressure of a patient aged 60 years and weighing 200 kg.
