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Outline for Today Relationships between two quantitative variables Response and explanatory variables Scatterplot Correlation (r) Properties of correlation Least-squares regression 2
Relationships between Variables So far we have done statistics on one variable at a time (estimation and testing), and we have discussed relationship between two categorical variables (chi- square test of independence) We are now interested in relationship between two quantitative variables and how to use one variable to predict another variable. E.g., the advertising and sale: ( Description ) How sale depends on advertising expenditure? ( Control ) How much to spend on advertising to reach certain goal on sale? ( Prediction ) How much sale do we expect if we spend certain amount of money on advertising? 3

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4 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 independent variable or predictor variable) (frequently - many possible explanatory variables) Example: A study of sale and advertising expenditure sale - response variable Y advertising expenditure - explanatory variable X
Scatterplot 5 Use scatterplot 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 How one variable moves as the other variable moves Explanatory variable (if there is one) --> horizontal axis, Response --> vertical axis Note : To add a categorical variable to the scatterplot, use a different color or symbol for each category.

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Scatterplots Motivating Example : Is household natural gas consumption associated with climate? Annual household natural gas consumption measured in thousands of cubic feet (MCF) Climate as measured by the National Weather Service using heating degree days (HDD) 6
Scatterplots Association between Numerical Variables A graph displaying pairs of values as points on a two-dimensional grid The explanatory variable is placed on the x- axis The response variable is placed on the y-axis 7

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Scatterplots Scatterplot of Natural Gas Consumption (y) versus Heating Degree-Days (x) 8
Scatterplot Scatterplot of Weight versus (against) Height 9

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Scatterplot with Groups Scatterplot of Weight versus (against) Height 10
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