chapter4 - Relationships Regression PSLS chapter 4 2009 W.H...

Info icon This preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
    Relationships Regression PSLS chapter 4 © 2009 W.H. Freeman and Company
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 (PSLS chapter 4) 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
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. And we would like to make predictions based on the observed association. But which line best describes our data?
Image of page 3

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

View Full Document Right Arrow Icon
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 least-squares regression line The least-squares regression line is the unique line such that the sum of the total vertical (y) distances is zero and sum of the squared vertical (y) distances between the data points and the line is the smallest possible.
Image of page 4
Facts about least-squares regression 1. The distinction between explanatory and response variables is essential in regression. 2. There is a close connection between correlation and the slope of the least-squares line. 3. The least-squares regression line always passes through the point 4. The correlation r describes the strength of a linear 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
Image of page 5

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

View Full Document Right Arrow Icon
is the predicted y value ( y hat) b is the slope a is the y -in tercept 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:
Image of page 6
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.
Image of page 7

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

View Full Document Right Arrow Icon
BEWARE !!! Not all calculators and software use the same convention: Some use instead: ˆ y = a + bx ˆ y = ax + b Make sure you know what YOUR calculator gives you for a and b before you answer homework or exam questions.
Image of page 8
Software output Intercept Slope R 2 r R 2 Intercept Slope
Image of page 9

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

View Full Document Right Arrow Icon
The equation completely describes the regression line.
Image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern