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An explanation of how to use linear regression models for prediction and estimating expected values, as well as calculating the coefficient of correlation and determination. It includes formulas for prediction intervals, confidence intervals for expected values, and the pearson correlation coefficient. An example using wheat yield and fertilizer data is provided.
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STA 100 Lecture 23
a. Prediction:
Given a specific value of the independent variable x, say xg , a 100(1-α)% prediction interval for y is: _ _ y^^ ± tα/2 Se √ 1 + 1/n + ( xg – x )^2 / Σ( x – x )^2
where y^^ = bo + b 1 x.
Example: Wheat yield
b. Estimating the Expected Value:
Given a specific value of the independent variable x, say xg , a 100(1-α)% confidence interval for the expected value of y is: _ _ y^^ ± tα/2 Se √ 1/n + ( xg – x )^2 / Σ( x – x )^2
where y^^ = y^^ = bo + b 1 x.
Example: Wheat yield
b. The coefficient of determination is defined as:
r^2 = 1 – SS(resid) / SS(total)
Example: Wheat yield
c. The analysis of variance table:
Regression Analysis
The regression equation is Yield = 36.4 + 0.0589 Fertilizer
Predictor Coef StDev T P Constant 36.429 5.038 7.23 0. Fertiliz 0.05893 0.01127 5.23 0.
S = 5.961 R-Sq = 84.5% R-Sq(adj) = 81.5%
Analysis of Variance
Source DF SS MS F P Regression 1 972.32 972.32 27.36 0. Residual Error 5 177.68 35. Total 6 1150.
