1. Statistics 101
Chapter 3
Section 3

2. Least – Squares Regression
Method for finding a line that summarizes the relationship between two variables

3. Regression Line
A straight line that describes how a response variable y changes as an explanatory variable x changes.
Mathematical model

4. Example 3.8

5. Calculating error
Error = observed – predicted
= 5.1 – 4.9
= 0.2

7. 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

10. What we need
y = a + bx
b = r (sy/ sx)
a = y - bx

11. Try Example 3.9

12. Technology toolbox pg. 154

16. Statistics 101
Chapter 3
Section 3 Part 2

17. 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 slope
A change of one standard deviation in x corresponds to a change of r standard deviations in y

18. 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.

19. 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

22. 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.

24. 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.

26. Curved pattern
A curved pattern shows that the relationship is not linear.

27. Increasing or decreasing spread
Indicates that prediction of y will be less accurate for larger x.

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