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

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

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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.

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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.

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

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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.

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

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

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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.

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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.

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

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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

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Regression Analysis Before using the formula for the regression line, a and b must be calculated. Compute b first, using Formula 15.3 (we won’t do any calculation for this chapter)

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

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Regression Analysis For the relationship between % college educated and turnout: b (slope) =.42 a (Y intercept)= 50.03 Regression formula: Y = 50.03 +.42 X 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.

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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 = 50.3 +.42(10) = 54.5 For X =70, Y = 50.3 +.42(70) = 79.7

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Pearson correlation coefficient But of course, this is just an estimate of turnout based on % college educated, and many other factors also affect voter turnout. How much of the variation in voter turnout depends on % college educated? The relevant statististic is the coefficient of determination (r squared), but first we need to learn about Pearson’s correlation coefficient (r).

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Pearson’s r Pearson’s r is a measure of association for I-R variables. It varies from -1.0 to +1.0 Relationship may be positive (as X increases, Y increases) or negative (as X increases, Y decreases) For the relationship between % college educated and turnout, r =.32. The relationship is positive: as level of education increases, turnout increases. How strong is the relationship? For that we use R squared, but first, let’s look at the calculation process

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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.

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Example of Computation The scores on each variable are displayed in table format: Y = Turnout X = Years of Education CityXY A11.955 B12.160 C12.765 D12.868 E13.070

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Example of Computation Sums are needed to compute b, a, and Pearson’s r. XYX2X2 Y2Y2 XY 11.955141.613025654.5 12.160146.413600726 12.765161.294225825.5 12.868163.844624870.4 13.0701694900910 62.5318782.15203743986.4

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Interpreting Pearson’s r An r of 0.98 indicates an extremely strong relationship between average years of education and voter turnout for these five cities. The coefficient of determination is r 2 =.96. Knowing education level improves our prediction of voter turnout by 96%. This is a PRE measure (like lambda and gamma) We could also say that education explains 96% of the variation in voter turnout.

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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 r 2 is.10. Percent college educated explains 10% of the variation in turnout.

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