ch11_slides

ch11_slides - LINEAR REGRESSION AND CORRELATION •...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: LINEAR REGRESSION AND CORRELATION • Consider bivariate data consisting of ordered pairs of numerical values ( x, y ). • Often such data arise by setting an x variable at certain fixed values (which we will call levels ) and taking a random sample from the population of Y that is assumed to exist at each of the levels of x . • Here we are thinking of x as not being a random variable , because we are considering only selected fixed values of x (for sampling purposes). • However, Y is a random variable and we define Y on the population that exists at each level of x . 1 • Graphically, a scatterplot of the data depicting the y-values obtained by sampling the populations at each of the pre- selected x-values might appear as follows:- 6 r r r r r r r r r r r r r r r r r r r r r r r x y 2 Our objectives given such data are usually twofold: • Summarize the characteristics of the Y populations across values of x – Fit the Model • Interpolate between levels of X to estimate parameters of Y populations from which samples were not taken – Prediction • The center of our attention is usually on the means of the Y populations, E ( Y ) , and especially their relationship to one another . • Considering various relationships among these population means is called parametric modeling . 3 • The simple linear regression model says that the populations at each x-value are normally distributed and that the means of these normal distributions all fall on a straight line, called the regression line. • Chapters 11 and 12 are mostly about investigating to what extent the relationship among the population means is linear . 4 • Let us begin by considering a linear relationship among population means. • The equation of a straight line through means E ( Y ) across x-values can be written as E ( Y ) = β + β 1 x • Here β is the intercept and β 1 is the slope of the line.- 6 x E(Y) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! β β 1 1 5 • The y-values observed at each x-value is assumed to be a random sample from a normal distribution with the mean E ( Y ) = β + β 1 x , i.e., the mean is a linear function of x . • The variance of the normal distributions at each x-value is assumed to be the same (or constant). • Thus the y-values can be related to the x-values through the relationship y = β + β 1 x + (1) • Here is a random variable (called random error ) with mean zero i.e, ( E ( ) = 0) , and variance σ 2 . • This model says that sample values are random distances from the line μ = β + β 1 x at each x-value. 6 • The unknown constants in Equation (1), β , β 1 , and σ 2 are called the parameters of the model. • The next question we consider is “How do we proceed to derive a good approximating line through Y population means, given only samples from some of the Y populations?” • In other words, we need to obtain good estimates of the parameters of the model using the observed data....
View Full Document

Page1 / 71

ch11_slides - LINEAR REGRESSION AND CORRELATION •...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online