C8 Regression and Prediction

• prediction error is the difference between the actual and predicted values of Y.
• When the bivariate trend is reasonably linear, a line of“bestfit”easily can be found and used for purposes of predicting values of Y from X. Such a line is called a regression line.
• For r ±1 00, each case would fall exactly on the regression line, and prediction would be errorless. But when the correlation is not perfect, as in the present instance, there necessarily will be prediction error. This is a difference between the actual and predicted values of Y.
• In a sense, each Yˆ is an estimate of the mean of Y values corresponding to a particular value of X.
• The regression line is a“running mean.”
• Coefficient of determination Formula: r2 = 1 - RSS/TSS.
• R-square (r^2) measures how well the model fits the data. It ranges from 0 to 1 with higher values indicating better fit.
• Standard Error of Estimate (SE): Measures the average distance between the observed values and the fitted line. A smaller SE indicates more accuracy in predictions.
• Confidence Interval for Slope: A range of possible slope estimates based on the sampling distribution of the slope estimator.