CONFIDENCE Vs PREDICTION INTERVALS 12/2/04 Inference for Coefficients Mean Response at X Vs
STAT 141 REGRESSION: CONFIDENCE vs PREDICTION INTERVALS 12/2/04 Inference for coefficients Mean response at x vs. New observation at x Linear Model (or Simple Linear Regression) for the population. (“Simple” means single explanatory variable, in fact we can easily add more variables ) – explanatory variable (independent var / predictor) – response (dependent var) Probability model for linear regression: 2 i ∼ N(0, σ ) independent deviations Yi = α + βxi + i, α + βxi mean response at x = xi 2 Goals: unbiased estimates of the three parameters (α, β, σ ) tests for null hypotheses: α = α0 or β = β0 C.I.’s for α, β or to predictE(Y |X = x0). (A model is our ‘stereotype’ – a simplification for summarizing the variation in data) For example if we simulate data from a temperature model of the form: 1 Y = 65 + x + , x = 1, 2,..., 30 i 3 i i i Model is exactly true, by construction An equivalent statement of the LM model: Assume xi fixed, Yi independent, and 2 Yi|xi ∼ N(µy|xi , σ ), µy|xi = α + βxi, population regression line Remark: Suppose that (Xi,Yi) are a random sample from a bivariate normal distribution with means 2 2 (µX , µY ), variances σX , σY and correlation ρ. Suppose that we condition on the observed values X = xi. Then the data (xi, yi) satisfy the LM model. Indeed, we saw last time that Y |x ∼ N(µ , σ2 ), with i y|xi y|xi 2 2 2 µy|xi = α + βxi, σY |X = (1 − ρ )σY Example: Galton’s fathers and sons: µy|x = 35 + 0.5x ; σ = 2.34 (in inches).
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