Unit 8 lesson 1
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This document discusses linear regression analysis using a single predictor variable. It explains that regression can be used to predict an outcome variable based on a known score on a predictor variable. As an example, it shows how a student's GPA could predict their SAT score. The key elements of a regression equation are defined, including the slope, intercept, and correlation coefficient. SPSS is used to run a regression analysis on mock student data.
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Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
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Direct variation describes the relationship between two quantities where one quantity varies as the other changes proportionally. It can be represented by the equation y = kx, where k is the constant of variation.
Some key points about direct variation:
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Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
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Unit 8 lesson 1
- 1. Unit 8: Regression Lesson 1: Understanding the Single Predictor Regression Equation EDER 6010: Statistics for Educational Research Dr. J. Kyle Roberts University of North Texas Next Slide
- 2. Kyle’s “Mock” Data Next Slide Data from Unit 2 Lesson 1: Reviewing the Homework r = .78 John Meredith Kyle Addie X 1 2 3 4 Y 1 1 1 2 1 2 3 4 X 1 2 3 4 Y
- 3. Remember the Pearson r? “ How well does a single line represent my data?” Next Slide Regression will answer the question: Where should I draw the line? r = .85 r = -.25 r = 1.0
- 4. Equation for a Line Slope = rise/run y = 1 + .5(x) Next Slide y = a + bx Where: “a” is the point at which the line intercepts the y-axis and “b” is the slope of the line
- 5. What is the “Point” of Regression? Regression is about prediction If we know someone’s score on one variable, can we “predict” how well they will perform on another variable? Using students’ gpa to “predict” how they will do on the SAT SAT = 400 + 100(gpa) Therefore, if someone had a gpa of 4.0, then we would “predict” that they would score an 800 on the SAT. Next Slide
- 6. Running a Regression in SPSS Create a dataset utilizing our “mock” data Analyze Regression Linear Next Slide
- 7. SPSS Results y = a + bx Next Slide y = .50 + .30(x) Beta = Pearson r
- 8. Understanding Beta ( β ) β is the correlation between the dependent variable and the independent variable -or- β is the regression coefficient for the standardized (z-scores) variables Next Slide
- 9. What does β tell us? Remember that β is roughly analogous to the Pearson r. Therefore, if we were to square the β we would have a measure of effect size which we refer to as an R 2 . R 2 = effect size for regression -and- η 2 = effect size for ANOVA The effect size tells us how well our regression coefficients are functioning R 2 for the present dataset = .60. Or “x” explains 60% of the variance of “y”. Next Slide
- 10. Utilizing “Dummy” Coded Variables Next Slide Math = students scores on a math achievement variable Gender = male – “0.00” female – “1.00”
- 11. SPSS Results Predicted Math = 84.20 + 8.0(Gender) This means that the “average” male scores 8.0 points less than the “average” female Math male = 84.20 + 8.0(0.0) Math male = 84.20 Math female = 84.20 + 8.0(1.0) Math female = 92.20 Next Slide
- 12. ANOVA and Regression Next Slide Results from an ANOVA Results from the regression
- 13. Unit 8: Regression Lesson 1: Understanding the Single Predictor Regression Equation EDER 6010: Statistics for Educational Research Dr. J. Kyle Roberts University of North Texas