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Unformatted text preview: Chapter 7. Correlation and Simple Linear Regression Section 7.1. Correlation Scatterplots • Plot of Y vs X, two quantitative variables, measured on the same individual • X=explanatory variable, Y=response variable • Interpreting scatterplots • Direction (positive, negative) 1 • Strength (strong or moderate or weak ) • Look for outliers. 2 Example: Interpret the following scatterplot. X=percent taking SAT Y=median SAT math score for each state Correlation • A numerical measure of the strength and direction of linear association between two quantitative variables. • Why? Because it is very hard to determine the strength of the linear association by eye. 3 Facts about Correlation coefficient(r) • Correlation does not distinguish between explanatory and response variables. • r is always between 1 and +1. • Interpretation: positive/negative, strong /weak. • r measures only the strength of the linear relationship between x and y. • Outliers can have a strong effect on r. • Correlation is a “pure number,” in other words, it is unitless. • r=+1 means perfect positive linear relationship. • r=1 means perfect negative linear relationship. • It measures the linear relationship between two quantitative variables. The correlation coefficient r=0 doesn’t mean there is no relationship between two variables. It means there is no linear relationship between the variables. 4 CAUTION: Correlation measures the strength of the linear association between two variables. Therefore we can have two variables with a “low” correlation but with a strong association, which is not linear. Example: The variables X and Y are highly correlated and we can see a clear linear pattern. On the other hand, the correlation between the variables X and Z is small, but they also have a strong association. X Y 20 15 10 5 35 30 25 20 15 10 5 Scatterplot of Y vs X Correlations: X, Y Pearson correlation of X and Y = 1.000 5 X Z 20 15 10 5 100 80 60 40 20 Scatterplot of Z vs X Correlations: X, Z Pearson correlation of X and Z = 0.191 Examples: 6 Formula: Correlation Coefficient • Suppose you have n pairs of data points as: (x 1, y 1 ),…, (x n, y n ), • Compute the sample means y x , and sample s.d.s s x and s y , where ∑ ∑ = = = = n i i y n i i x y y n s x x n s 1 2 1 2 ) ( 1 1 , ) ( 1 1 • Compute zscores: x i x s x x z = , y i y s y y z = • Correlation coefficient:   = ∑ = y i x i n i s y y s x x n r 1 1 1 Note: You don’t need to remember this formula. You can use your calculator (should have 2variable functions) or Minitab ( for homework problems). • Why does the formula of r give the same sign (negative/positive) as the graph?...
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 Summer '08
 Kyung
 Statistics

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