Statistics 101

Chapter 3 Section 3

Least – Squares Regression

Method for finding a line that summarizes the relationship between two variables

Regression Line

A straight line that describes how a response variable y changes as an explanatory variable x changes.

Mathematical model

Example 3.8

Calculating error

Error = observed – predicted = 5.1 – 4.9 = 0.2

Least – squares regression line (LSRL) Line that makes the sum of the

squares of the vertical distances of the data points from the line as small as possible

http://hadm.sph.sc.edu/courses/J716/demos/leastsquares/leastsquaresdemo.html

What we need

y = a + bx b = r (sy/ sx) a = y - bx

Try Example 3.9

Technology toolbox pg. 154

Statistics 101

Chapter 3Section 3 Part 2

Facts about least-squares regression Fact 1: the distinction between

explanatory and response variables is essential

Fact 2: There is a close connection between correlation and the slopeA change of one standard deviation in

x corresponds to a change of r standard deviations in y

More facts

Fact 3: The least-squares regression line always passes through the point (x,y)

Fact 4: the square of the correlation, r2, is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x.

Residuals

Is the difference between an observed value of the response variable and the value predicted by the regression line.

• Residual = observed y – predicted y= y - y

Residuals

If the residual is positive it lies above the line

If the residual is negative it lies below the line

The mean of the least-squares residuals is always zero

If not then it is a roundoff error Technology Toolbox on page 174 shows

how to do a residual plot.

Residual plots

A scatterplot of the regression residuals against the explanatory variable.

To help us assess the fit of a regression line.

If the regression line captures the overall relationship between x and y, the residuals should have no systemic pattern.

Curved pattern

A curved pattern shows that the relationship is not linear.

Increasing or decreasing spread Indicates that prediction of y will be less

accurate for larger x.

Influential Observations

An observation is an influential observation for a statistical calculation if removing it would markedly change the result of the calculation.