Chapter 3: Examining Relationships

1. Chapter 3: Examining Relationships By: Ericka Brundage, Christina Chaudoir, Robin Day, Kathryn Turner

2. The relationship between two variables can be strongly influenced by other variables that are lurking in the background A response variable measures an outcome of a study An explanatory variable helps explain or influences changes in a response variable Calling one variable explanatory and the other response doesn�t necessarily mean that changes in one cause changes in the other

3. 3.1: Scatterplots and Correlation Scatterplot shows the relationship between 2 quantitative variables Interpreting a Scatterplot: Look for the overall pattern and for striking deviations from that pattern You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship An outlier, an individual value that falls outside the overall pattern of the relationship

4. Scatterplots Positive association is when above-average values of one tend to accompany above-average values of the other, and below-average values also tend to occur together Negative association is when above-average values of one tend to accompany below-average values of the other, and vice versa

5. Correlation The correlation measures the direction and strength of the linear relationship Makes no distinction between explanatory and response variables R does not change when we change the unites of measurement of x, y, or both Positive r indicates positive association between the variables, and negative r indicates negative association R is always a number between -1 and 1

6. Correlation Requires that both variables be quantitative, so that it makes sense to do the arithmetic indicated by the formula for r Correlation does not describe curved relationships between variables, no matter how strong they are Not resistant Not a complete summary of two-variable data

7. 3.2: Least-Squares Regression A regression line summarizes the relationship between two variables, but only when one of the variables helps explain or predict the other y = a + bx where b is the slope and a is the y-intercept Extrapolation is the use of a regression line for prediction outside the range of values of the explanatory variable x used to obtain the line

8. 3.2 Cont�d. The least-squares regression line of y on x is the line that makes the sum of the squared vertical distances of the data points from the line as small as possible A residual is the difference between an observed value of the response variable and the value predicted by the regression line Residual = observed y � predicted y A residual plot is a scatterplot of the regression residuals against the explanatory variable

9. 3.2 Cont�d. The coefficient of determination r2 is the fraction of the variation in the values of y that is explained by the LSRL

10. 3.3: Correlation and Regression Wisdom Correlation and regression describe only linear relationships Extrapolation often produces unreliable predictions Correlation is not resistant An outlier is an observation that lies outside the overall pattern of the other observations A point is influential if removing it would markedly change the result of the calculation

11. 3.3 Cont�d. A lurking variable is a variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables Keep in mind that association does not imply causation