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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. - PowerPoint PPT Presentation
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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.
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