Chapter 4 notes

# Chapter 4 notes - Chapter 4 Describing Relationships...

This preview shows pages 1–4. Sign up to view the full content.

Chapter 4: Describing Relationships Between Variables Section 4.1: Fitting a Line by Least Squares Suppose that we have two variables x and y and that we wish to describe the relationship between them. In this section, we will find the best linear fit of y versus x . Linear equation: y = β 0 + 1 x , where 0 and 1 are unknown population parameters The goal is to find estimates b 0 and b 1 for the parameters 0 and 1 . Example (to be used throughout this section) : Eight batches of plastic are made and from each batch one test item is molded and its hardness y is measured at time x . The following are the 8 measurements: Time, x 32 72 64 48 16 40 80 56 Hardness, y 230 323 298 255 199 248 359 305 When looking for a relationship between two variables, a first step is to look at a scatterplot. This will allow us to determine if a linear relationship seems to be appropriate for the variables. By looking at this scatterplot we see that there appears to be a strong positive linear relationship between x and y . How do we find an equation for the line that best fits to this data? 1

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

View Full Document
Least squares principle We will fit a line given by y = b 0 + b 1 x , where b 0 and b 1 are estimates for the parameters β 0 and 1 . Note that a straight line will not pass perfectly through every one of our data points. Thus, if we plug a data value x i into the equation y = b 0 + b 1 x , the value we get for y i will not be exactly our data value y i . The idea of fitting a line using least squares is to minimize the squared distances from the actual data value, y i , and the value given by our equation, y i . Thus, we wish to minimize ( ) y y i i i n - = 2 1 The value y i is called the observed value, while the value y i is called the fitted or predicted value. To minimize ( ) y y i i i n - = 2 1 : Plugging y i = b 0 + b 1 x i into the equation yields ( ( ) ) y b b x i i i n - + = 0 1 2 1 . Taking partial derivatives of this with respect to both b 0 and b 1 gives us what are called the Normal Equations: n b b x y i i n i i n 0 1 1 1 + = = = ∑ ∑ b x b x x y i i n i i n i i i n 0 1 1 2 1 1 + = = = = ∑ ∑ ∑ Solving these equations (details omitted) for b 0 and b 1 yields the following: b x x y y x x i i i n i i n 1 1 2 1 = - - - = = ( ) ( ) ( ) and b y b x 0 1 = - Finding the estimates for our example and interpreting them in terms of the problem : 2
Once we have an equation for the fitted line, we can use it to find a predicted/fitted value, y , for a value of x . We must, however, be careful to not extrapolate beyond our data set. Extrapolation is when a value of x beyond the range of our actual x observations is used to find a predicted y . This is a problem because the fitted line was only based on our

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/26/2008 for the course STAT 105 taught by Professor Bingham during the Spring '08 term at Iowa State.

### Page1 / 9

Chapter 4 notes - Chapter 4 Describing Relationships...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online