BPS4e_chapter05

BPS4e_chapter05 - Relationships Regression BPS chapter 5...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
    Relationships Regression BPS chapter 5 © 2006 W.H. Freeman and Company
Background image of page 1

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

View Full Document Right Arrow Icon
Objectives (BPS chapter 5) Regression Regression lines The least-squares regression line Using technology Facts about least-squares regression Residuals Influential observations Cautions about correlation and regression Association does not imply causation
Background image of page 2
Correlation tells us about strength (scatter) and direction of the linear relationship between two quantitative variables. In addition, we would like to have a numerical description of how both variables vary together. For instance, is one variable increasing faster than the other one? And we would like to make predictions based on that numerical description. But which line best describes our data?
Background image of page 3

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

View Full Document Right Arrow Icon
A regression line A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x .
Background image of page 4
Distances between the points and line are squared so all are positive values. This is done so that distances can be properly added (Pythagoras). The regression line The least-squares regression line is the unique line such that the sum of the squared vertical ( y ) distances between the data points and the line is the smallest possible.
Background image of page 5

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

View Full Document Right Arrow Icon
Facts about least-squares regression 1. The distinction between explanatory and response variables is essential in regression. 1. There is a close connection between correlation and the slope of the least-squares line. 1. The least-squares regression line always passes through the point 1. The correlation r describes the strength of a straight-line relationship. The square of the correlation, r 2 , is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x . ( 29 , x y
Background image of page 6
is the predicted y value ( y hat) b is the slope a is the y -intercept ˆ y = ( y - r x s y s x ) + r s y s x x , or ˆ y = a + bx Properties ˆ y "a" is in units of y "b" is in units of y/units of x The least-squares regression line can be shown to have this equation:
Background image of page 7

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

View Full Document Right Arrow Icon
b = r s y s x First we calculate the slope of the line, b , from statistics we already know: r is the correlation s y is the standard deviation of the response variable y s x is the the standard deviation of the explanatory variable x Once we know b , the slope, we can calculate a, the y -intercept : a = y - b x where x and y are the sample means of the x and y variables How to: This means that we don’t have to calculate a lot of squared distances to find the least- squares regression line for a data set. We can instead rely on the equation. But typically, we use a 2-var stats calculator or a stats software.
Background image of page 8
BEWARE !!! Not all calculators and software use the same convention:
Background image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 39

BPS4e_chapter05 - Relationships Regression BPS chapter 5...

This preview shows document pages 1 - 10. Sign up to view the full document.

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