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Copyright © 2012 by Nelson Education Limited. Chapter 13 Association Between Variables Measured at the Interval-Ratio Level 13-1

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Copyright © 2012 by Nelson Education Limited. Scattergrams Regression and Prediction2 The Correlation Coefficient (Pearson’s r and r 2 ) Testing Pearson’s r for Significance In this presentation you will learn about: 13-2

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Copyright © 2012 by Nelson Education Limited. Scattergrams display relationships between two interval-ratio variables. Scattergrams have two dimensions: –The X (independent) variable is arrayed along the horizontal axis. –The Y (dependent) variable is arrayed along the vertical axis. Each dot on a scattergram is a case. –The dot is placed at the intersection of the case’s scores on X and Y. Scattergrams 13-3

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Copyright © 2012 by Nelson Education Limited. Inspection of the scattergram should always be the first step in assessing the correlation between two interval-ratio variables. –Producing a scattergram before proceeding with the statistical analysis is particularly important since a key requirement the techniques described in Chapter 13 (regression and correlation) is that the two variables have essentially a linear relationship. Scattergrams (continued) 13-4

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Copyright © 2012 by Nelson Education Limited. Data on “number of children” (X) and “hours per week husband spends on housework” (Y) for a sample of 12 families with children, where both husband and wife have jobs outside the home, are provided in Table Scattergrams: An Example 13-5

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Copyright © 2012 by Nelson Education Limited. –Figure 13.1 shows a scattergram displaying the relationship between these variables. Scattergrams: An Example (continued) 13-6

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Copyright © 2012 by Nelson Education Limited. Regression line is a single straight line that comes as close as possible to all data points. Scattergrams: Regression Line 13-7

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Copyright © 2012 by Nelson Education Limited. Regression line indicates strength and direction of the linear relationship between two variables. o The greater the extent to which dots are clustered around the regression line, the stronger the relationship. Scattergrams: Strength of Regression Line 13-8

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Copyright © 2012 by Nelson Education Limited. o This relationship is moderate in strength. Scattergrams: Strength of Regression Line (continued) 13-9

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Copyright © 2012 by Nelson Education Limited. Negative: regression line falls left to right. Positive: regression line rises left to right. Scattergrams: Direction of Regression Line 13-10

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Copyright © 2012 by Nelson Education Limited. –This a positive relationship: As number of children increases, husband’s housework increases. Scattergrams: Direction of Regression Line 13-11

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Copyright © 2012 by Nelson Education Limited. 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 Regression Line: Formula 13-12

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Copyright © 2012 by Nelson Education Limited. Before using the formula for the regression line, a and b must be calculated. –Compute the slope (b ) first, using Formula 13.3 –The Y intercept (a) is computed from Formula 13.4 Regression Analysis 13-13

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Copyright © 2012 by Nelson Education Limited. Table 13.3 has a column for each of the quantities needed to solve the formulas for a and b. Regression Analysis (continued) 13-14

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Copyright © 2012 by Nelson Education Limited. Regression Analysis (continued) 13-15

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Copyright © 2012 by Nelson Education Limited. For the relationship between number of children and husband’s housework: –b (slope) =.69 –a (Y intercept)= 1.49 A slope of.69 means that the amount of time a husband contributes to housekeeping chores increases by.69 (less than one hour per week) for every unit increase of 1 in number of children (for each additional child in the family). The Y intercept means that the regression line crosses the Y axis at Y = 1.49 (or the value of Y when X is 0). Regression Analysis (continued) 13-16

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Copyright © 2012 by Nelson Education Limited. The regression line, Y = a + bX, can be used to predict a score on Y from score on X. For example, how many hours per week could a husband expect to contribute to housework in a family with 6 children? Y’ = (6) = 5.63 o Y’ is used to indicate a predicted value. We would predict that in a dual wage-earner family with six children a husband would contribute 5.63 hours a week to housework. Predicting Y 13-17

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Copyright © 2012 by Nelson Education Limited. Pearson’s r is a measure of association for two interval-ratio variables. Pearson’s r is computed using Formula 13.6 Pearson’s r 13-18

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Copyright © 2012 by Nelson Education Limited. The quantities displayed in Table 13.3 can be substituted directly into Formula 13.6 to calculate r for our sample problem involving dual wage-earner families: Pearson’s r: An Example 13-19

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Copyright © 2012 by Nelson Education Limited. Pearson’s r: An Example (continued) 13-20

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Copyright © 2012 by Nelson Education Limited. Use the guidelines stated in Table 12.3 as a guide to interpret the strength of Pearson’s r. –As before, the relationship between the values and the descriptive terms is arbitrary, so the scale in Table 12.3 is intended as a general guideline only: Interpreting Pearson’s r 13-21

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Copyright © 2012 by Nelson Education Limited. An r of 0.50 indicates a strong and positive linear relationship between the variables. –As the number of children in a family increases, the hourly contribution of husbands to housekeeping duties also increases. Interpreting Pearson’s r (continued) 13-22

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Copyright © 2012 by Nelson Education Limited. The value of r squared, r 2 (also called the coefficient of determination), provides a PRE interpretation: r 2 =.50 x.50 =.25 Interpreting Pearson’s r (continued) 13-23

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Copyright © 2012 by Nelson Education Limited. –When predicting the number of hours per week that husbands in families would devote to housework, we will make 25% fewer errors by basing the predictions on number of children and predicting from the regression line, as opposed to ignoring this variable and predicting the mean of Y for every case. –Another way to say this is that the number of children (X ) explains 25% of the variation in husband’s housework (Y ). Interpreting Pearson’s r (continued) 13-24

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Copyright © 2012 by Nelson Education Limited. In testing Pearson’s r for statistical significance, the null hypothesis states that there is no linear association between the variables in the population. The familiar five-step model should be used to organize the hypothesis testing procedures. –Section 13.8 provides details on testing Pearson’s r for significance. Testing Statistical Significance of r 13-25

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