Copyright © 2012 by Nelson Education Limited. Chapter 13 Association Between Variables Measured at...

Copyright © 2012 by Nelson Education Limited.

Chapter 13Association Between Variables

Measured at the Interval-Ratio Level

13-1


• Scattergrams

• Regression and Prediction2

• The Correlation Coefficient (Pearson’s r and r2)

• Testing Pearson’s r for Significance

In this presentation you will learn about:

13-2


• 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


• 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


• 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 13.1.

Scattergrams: An Example

13-5


– Figure 13.1 shows a scattergram displaying the relationship between these variables.

Scattergrams: An Example (continued)

13-6


• Regression line is a single straight line that comes as close as possible to all data points.

Scattergrams: Regression Line

13-7


• 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


o This relationship is moderate in strength.

Scattergrams: Strength of Regression Line (continued)

13-9


• Negative: regression line falls left to right.

• Positive: regression line rises left to right.

Scattergrams: Direction of Regression Line

13-10


– This a positive relationship: As number of children increases, husband’s housework increases.

Scattergrams: Direction of Regression Line

13-11


• 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


• 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


•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



13-15


• 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).


13-16


• 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’ = 1.49+ .69(6) = 5.63o 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


• 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


• 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


Pearson’s r: An Example (continued)

13-20


• 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


• 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


• The value of r squared, r2 (also called the coefficient of determination), provides a PRE interpretation:

r2 = .50 x .50 =.25


13-23


–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 ).


13-24


• 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

Date post:	02-Apr-2015
Category:	Documents
Upload:	zaire-goodwin
View:	213 times
Download:	1 times

Copyright © 2012 by Nelson Education Limited. Chapter 13 Association Between Variables Measured at...

Documents