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Unformatted text preview: Inference for Regression Inference about the Regression Model and Using the Regression Line Section 10.1 and 10.2 © 2009 W.H. Freeman and Company Simple Linear Regression ! We will deal with analyzing the relationship between two quantitative variables. ! We have graphed this relationship with a scatter plot and calculated the correlation, which are measures of linear relationship. ! If our scatter plot showed a linear relationship, we would calculate the correlation and see if we had a strong linear relationship. ! If we show that our variables are associated linearly, then we can use regression to fit a straight line to the data. ! We can use this line to predict the value of one variable on the basis of the other variable(s). Types of Variables Response (dependent) variable The variable that will be predicted. It measures the outcome of the study. It is the variable of primary interest to the experimenter. Labeled y. Explanatory (independent) variable Variables that are related to the dependent variable. Variables that influence changes in the dependent variable. Label x (or x 1 , x 2 , x 3 , … x k ) where k is the number of independent variables. Illustration A real estate agent wants to more accurately predict the selling price of houses. She believes that the following variables affect the price of a house. Price is the response variable some of the explanatory variables Size (sqft) # of bedrooms Condition (age) Location The data in a scatterplot is a random sample from a population that may exhibit a linear relationship between x and y . (Different sample " different plot.) Now we want to describe the population mean response μ y as a function of the explanatory variable x: μ y = ! + ! 1 x. Also, we want to assess whether the observed relationship is statistically significant . Statistical model for simple linear regression In the population , the linear regression equation is μ y = ! + ! 1 x . Sample data then fits the model: Data = fit + residual y i = ( ! + ! 1 x i ) + ( " i ) where the " i are independent and normally distributed N (0, # ). Linear regression assumes equal variance of y ( # is the same for all values of x ). μ y = ! + ! 1 x The intercept ! , the slope ! 1 , and the standard deviation # of y are the unknown parameters of the regression model . We rely on the random sample data to provide unbiased estimates of these parameters. ! The value of ! from the least-squares regression line is really a prediction of the mean value of y ( μ y ) for a given value of x . ! The least-squares regression line ( ! = b + b 1 x ) obtained from sample data is the best estimate of the true population regression line ( μ y = ! + ! 1 x )....
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This note was uploaded on 02/16/2010 for the course STAT 212 taught by Professor Holt during the Fall '08 term at UVA.
- Fall '08
- Linear Regression