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**Unformatted text preview: **2 Use scatterplots to look at the relationship between two quantitative variables (measured on the same individuals) (First step when studying the relationship) The values of one variable --> horizontal axis The values of the other variable --> vertical axis Each individual appears as a point in the plot Explanatory variable (if there is one) --> horizontal axis, Response --> vertical axis 3 Look at the Federal funds interest rate and the national unemployment rate every 6 months from 2000 to 2006 (Source: U.S. Federal Reserve and the U.S. Department of Labor) Interest Unemploy- Date rate ment rate Jan. 2000 5.45% 4.0% July 2000 6.54% 4.0% . . . . . . . . . Jan. 2006 4.29% 4.7% July 2006 5.24% 4.8% 4 5 Two variables measured on the same individuals are called positively associated if increasing values of one variable tend to occur with increasing values of the other They are negatively associated if increasing values of one variable occur with decreasing values of the other Examples: Per capita GDP of countries is positively associated with life expectancy Federal funds interest rate is negatively associated with the unemployment rate 6 Response variable, denoted as Y, measures the outcome of a study. Y is the variable we want to predict/explain (often called the dependent variable) Explanatory variable, denoted as X, is a variable that may predict/explain (but not necessarily cause) the response variable (often called the predictor variable) (frequently - many possible explanatory variables) Example: A study of sale and advertising expenditure sale - response variable Y advertising expenditure - explanatory variable X 7 The relationship between two variables is said to be linear if the points on the scatterplot lie (approx.) on a straight line. A perfect linear relationship between a response variable (Y) and an explanatory variable (X) is Y = a + bX A positive linear relationship means b > 0 A negative linear relationship means b < 0 What if b = 0? No linear relationship! Correlation is a measure of the strength of the linear relationship between two variables It is usually denoted by r with a range of -1 to 1 r = 1 means the relationship between two variables X and Y is exactly positive linear r = -1 indicates the relationship is exactly negative linear r = 0 indicates a very weak (or no) linear relationship 8 9 Definition: Suppose we have n pairs of observations (x 1 ,y 1 ),,(x n ,y n ) on two variables X and Y. The correlation between X and Y is given by the formula where s x and s y are the SDs of X and Y ( 29 ( 29 =--- = n i y i x i s y y s x x n r 1 1 1 10 11 12 13 r always between 1 and +1 r is 1 or 1 only if points lie exactly on a straight line sign of r indicates a positive or negative association r is unaltered by changes in units of X or Y absolute value of r measures the strength of the linear relationship...

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