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Chapter 15

Chapter 15. Association Between Variables Measured at the Interval-Ratio Level. Chapter Outline. Introduction Scattergrams Regression and Prediction The Computation of a and b The Correlation Coefficient (Pearson’s r). Chapter Outline. Interpreting the Correlation Coefficient: r 2

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Chapter 15

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  1. Chapter 15 Association Between Variables Measured at the Interval-Ratio Level

  2. Chapter Outline • Introduction • Scattergrams • Regression and Prediction • The Computation of a and b • The Correlation Coefficient(Pearson’s r)

  3. Chapter Outline • Interpreting the Correlation Coefficient: r 2 • The Correlation Matrix • Testing Pearson’s r for Significance • Interpreting Statistics: The Correlates of Crime

  4. This Presentation • Scattergrams • Graphs that display relationships between two interval-ratio variables. • The Regression Line • Summarizes the linear relationship between X and Y. Predicts score on Y from score on X. • Pearson’s r • Preferred measure of association for two I-R variables.

  5. Scattergrams • Scattergrams have two dimensions: • The X (independent) variable is arrayed along the horizontal axis. • The Y (dependent) variable is arrayed along the vertical axis.

  6. Scattergrams • Each dot on a scattergram is a case. • The dot is placed at the intersection of the case’s scores on X and Y.

  7. Scattergra ms • Shows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 states.

  8. Scattergrams • Horizontal X axis - % of population of a state with a college education. • Scores range from 15.3% to 34.6% and increase from left to right.

  9. Scattergrams • Vertical (Y) axis is voter turnout. • Scores range from 44.1% to 70.4% and increase from bottom to top

  10. Scattergrams: Regression Line • A single straight line that comes as close as possible to all data points. • Indicates strength and direction of the relationship.

  11. Scattergrams:Strength of Regression Line • The greater the extent to which dots are clustered around the regression line, the stronger the relationship. • This relationship is weak to moderate in strength.

  12. Scattergrams: Direction of Regression Line • Positive: regression line rises left to right. • Negative: regression line falls left to right. • This a positive relationship: As % college educated increases, turnout increases.

  13. Scattergrams • Inspection of the scattergram should always be the first step in assessing the correlation between two I-R variables

  14. The Regression Line: Formula • This formula defines the regression line: • Y = a + bX • Where: • Y = score on the dependent variable • a = the Y intercept or the point where the regression line crosses the Y axis. • b = the slope of the regression line or the amount of change produced in Y by a unit change in X • X = score on the independent variable

  15. Regression Analysis • Before using the formula for the regression line, a and b must be calculated. • Compute b first, using Formula 15.3:

  16. Regression Analysis • The Y intercept (a) is computed from Formula 15.4:

  17. Regression Analysis • For the relationship between % college educated and turnout: • b (slope) = .42 • a (Y intercept)= 50.03 • A slope of .42 means that turnout increases by .42 (less than half a percent) for every unit increase of 1 in % college educated. • The Y intercept means that the regression line crosses the Y axis at Y = 50.03.

  18. Predicting Y • What turnout would be expected in a state where only 10% of the population was college educated? • What turnout would be expected in a state where 70% of the population was college educated? • This is a positive relationship so the value for Y increases as X increases: • For X =10, Y = 54.5 • For X =70, Y = 79.7

  19. Pearson’s r • Pearson’s r is a measure of association for I-R variables. • For the relationship between % college educated and turnout, r =.32. • This relationship is positive and weak to moderate. • As level of education increases, turnout increases.

  20. Example of Computation • The computation and interpretation of a, b, and Pearson’s r will be illustrated using Problem 15.1. • The variables are: • Voter turnout (Y) • Average years of school (X) • The sample is 5 cities. • This is only to simplify computations, 5 is much too small a sample for serious research.

  21. Example of Computation • The scores on each variable are displayed in table format: • Y = Turnout • X = Years of Education

  22. Example of Computation • Sums are needed to compute b, a, and Pearson’s r.

  23. Interpreting Pearson’s r • An r of 0.98 indicates an extremely strong relationship between years of education and voter turnout for these five cities. • The coefficient of determination is r2 = .96. Education, by itself, explains 96% of the variation in voter turnout.

  24. Interpreting Pearson’s r • Our first example provides a more realistic value for r. • The r between turnout and % college educated for the 50 states was: • r = .32 • This is a weak to moderate, positive relationship. • The value of r2 is .10.Percent college educated explains 10% of the variation in turnout.

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