Section 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

Residual – the difference in the actual value and the predicted value. Also known as error in the predicted value. Definitions: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists Residual  y – Where y  the actual value occurring in the population  the predicted value occurring in the sample

HAWKES LEARNING SYSTEMS math courseware specialists TI-84 Plus Instructions: 1.Press STAT, then EDIT 2.Type the x-variable values into L1 3.Type the y-variable values into L2 4.Then highlight L3 and enter the formula with actual values for b 0 +b 1 x. For example, L1. 5.Then highlight L4 and enter the formula L2-L3 Regression, Inference, and Model Building 12.3 Regression Analysis

HAWKES LEARNING SYSTEMS math courseware specialists Determine the residual: The table below gives data from a local school district on a child’s age and their reading level. For this data, a reading level of 4.3 would indicate 3/10 of the year through the fourth grade. Children’s ages are given in years. We know the regression line is. Use this equation to calculate an estimate,, for each value of the independent variable, x, and then use the estimate to calculate the residual for each value of y. Solution: All calculations can be performed at once on a calculator. Age Reading Level Regression, Inference, and Model Building 12.3 Regression Analysis

HAWKES LEARNING SYSTEMS math courseware specialists The results will be as follows: Solution (continued): AgeReading LevelPredicted ValueResidual – – – – – Regression, Inference, and Model Building 12.3 Regression Analysis

Residual: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists The residual of each value reflects how far the original data point is from the point on the regression line. Graphically, the residual is the vertical distance from the original data point to the point on the regression line.

Errors Shown Graphically: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists

Sum of Squared Errors: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists The value calculated by summing the square of the errors is the sum of squared errors, SSE. If the data points are very far from the regression line, then the sum of squared errors will be large. Therefore, the worse the linear model will be at predicting the value of y. If the data points are very close to the regression line, then the sum of squared errors will be small. Therefore, the better the linear model will be at predicting the value of y. The line that fits the data “best” would be the one with the smallest value of SSE.

HAWKES LEARNING SYSTEMS math courseware specialists From the previous example, calculate the sum of squared errors: Calculate the sum of squared errors: Age Reading Level Predicted Value Residual Squared Error – – – – – Regression, Inference, and Model Building 12.3 Regression Analysis Adding the values in the last column we get SSE 

Standard Error of Estimate: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists The standard error of estimate, S e, is a measure of how much the data points deviate from the regression line. This is analogous to how the standard deviation measures how much the data deviates from the sample mean. The smaller the value of the standard error of estimate is, the closer the data points are to the regression line.

HAWKES LEARNING SYSTEMS math courseware specialists Determine the standard error of estimate: Calculate the standard error of estimate for the data given previously for age and reading level. Solution: n  10, and  Age Reading Level Regression, Inference, and Model Building 12.3 Regression Analysis 0.417

Prediction Interval – the confidence interval for the predicted dependent variable. Bivariate Normal Distribution – a distribution where any given fixed value of the independent variable, x 0, and the possible sample values of the dependent variable, y, are normally distributed about the regression line with the mean of the normal distribution equal to and the standard deviation of the normal distribution the same for each value of x 0. Definitions: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists

Bivariate Normal Distribution: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists

Margin of Error: Regression, Inference, and Model Building 12.3 Regression Analysis HAWKES LEARNING SYSTEMS math courseware specialists where d.f.  n – 2 = the sample mean x 0 = the fixed value of x n = the sample size

HAWKES LEARNING SYSTEMS math courseware specialists Prediction Interval for an Individual y: Regression, Inference, and Model Building 12.3 Regression Analysis

HAWKES LEARNING SYSTEMS math courseware specialists Steps to Determine the Prediction Interval for an Individual y: Regression, Inference, and Model Building 12.3 Regression Analysis 1.Find the regression equation for the sample data. 2.Use the regression equation to calculate the point estimate for the given value of x. 3.Calculate the sample statistics necessary to calculate the margin of error. 4.Calculate the margin of error. 5.Construct the prediction interval.

HAWKES LEARNING SYSTEMS math courseware specialists Construct the prediction interval: Construct a 95% prediction interval for the reading level of a child who is 8 years old. Solution: We know from a previous example that the regression equation is. Now we will calculate the point estimate for the given value of x. Age Reading Level Regression, Inference, and Model Building 12.3 Regression Analysis 3.109

HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Calculate the sample statistics necessary to calculate the margin of error: n  10,  10.5, ∑x  105, ∑x 2  1185, t  /2  2.306, S e  Calculate the margin of error: Regression, Inference, and Model Building 12.3 Regression Analysis  1.044

HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Construct the prediction interval: Regression, Inference, and Model Building 12.3 Regression Analysis – <  y < <  y < (2.065, 4.153)

HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals for the Slope and the y-Intercept of the Regression Equation: Regression, Inference, and Model Building 12.3 Regression Analysis Using Microsoft Excel, we can construct confidence intervals for the population slope and y-intercept parameters  1 and  0, respectively.

HAWKES LEARNING SYSTEMS math courseware specialists Construct the prediction interval: Construct a 95% confidence interval for the slope,  1, and y-intercept,  0,of the regression equation for age and reading level. Solution: Begin by entering the data into Microsoft Excel. Age Reading Level Regression, Inference, and Model Building 12.3 Regression Analysis

HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Next, choose DATA ANALYSIS from the TOOLS menu. Choose REGRESSION from the options listed. Enter the necessary information as shown below. Regression, Inference, and Model Building 12.3 Regression Analysis

HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): The results are as follows: Regression, Inference, and Model Building 12.3 Regression Analysis “Multiple R” is the correlation coefficient. “R Square” is just that, r 2. “Standard Error” is the standard error estimate, S e. The intersection of the Residual row and SS column is the SSE.

HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Regression, Inference, and Model Building 12.3 Regression Analysis The blue box contains the values for the coefficients in the regression line. b 0  – and b 1  The red box is upper and lower endpoints of the confidence intervals for the y-intercept and slope. – <  0 < – and <  1 <