10.1 notes - STAT3000 Section 10.1: Inference about the...

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STAT3000 Section 10.1: Inference about the Regression Model Regression Analysis (4 steps) 1. Determine appropriateness of linear model. a. Random sample b. Linear shape c. No pattern in residual plot d. Equal variance in residual plot 2. Use the sample data to estimate unknown parameters in the model if data are straight enough. 3. Use graphs and statistics to evaluate usefulness of model. 4. When satisfied with the model, use the model to estimate the expected value of y and to predict a future value of y. 193
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Step 1: Model Assumptions Determine appropriateness of linear model. a. Random sample b. Linear shape c. No pattern in residual plot d. Equal variance in residual plot For c. and d. above, consider each setting of x, 1. E( ε ) = 0 2. Var( ε ) = σ 2 ( for each value of x I want my variability and residuals to be the same as every other value of x) figure a. for residuals 3. ε ~ N 4. The values of ε associated with any two observed values of y are independent. 194
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Step 2: Calculate the Model (the equation) The least squares estimate, s 2 , of σ 2 , is s 2 = error SSE SSE = df n-2 SSE = n 2 i i=1 ˆ (y -y) = sum of all (residuals) 2 s = 2 SSE s = n-2 s = estimated standard error of the regression model, also called the standard deviation of the residuals. We want the residuals to be small. Since their mean must be zero, look at variation. The standard deviation of residuals, s, gives a measure of how much the points are spread out around the regression line. To interpret, we expect about 95% of the y’s to be within ± 2s of ˆy . 195
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s 2 , an estimator for σ 2 , is the mean square error (locate on computer output), aka variance around the line, or error variance 10.3 The productivity of a process or an industry is defined as output per unit of input. We can measure the productivity of land used to grow corn by the yield of corn in bushels per acre. Improvements in other inputs (seed, fertilizers, pesticides, and so on) have led to great increases in the productivity of land. The table gives the average corn yields in the United States in the middle of four successive decades: Year 1966 1976 1986 1996 Yield 73.1 88.0 119.
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10.1 notes - STAT3000 Section 10.1: Inference about the...

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