2.3 Least-Squares Regression - Rocky Mountain...
Transcript of 2.3 Least-Squares Regression - Rocky Mountain...
Example: Price of Toyota PriusWe have data giving the asking price of a 2008 model year ToyotaPrius (response variable), together with the mileage of the car(explanatory variable). The following scatterplot results.
Source: autotrader.com search on 9/9/2012, within 50 miles ofZIP 90001.
Slope and Intercept of a Line
The plot also shows a line with equation y = −0.0667x + 20956,which seems to “fit well” into the data and describe the overalllinear dependence of the response and explanatory variables.
Example: Price of Toyota Prius
In our example,
I slope= −0.0667. Interpretation: for each additional mile,the straight line model predicts that the asking price willdecrease by about $0.07.
I intercept=20956. Interpretation: the straight-line modelpredicts that a 2008 Toyota Prius with zero miles will have anasking price of about $21,000.
Equation of Least-Squares Regression Line
In our example, y = −0.0667x + 20956 (we use y rather than y toindicate that the line gives us predicted, not actual values of y).
Interpreting the Regression Line
I The expression b1 = rsysx
for the slope says that, along the
regression line, a change in one standard deviation in xcorresponds to a change in r standard deviations in y . Ifr ≈ 0, the model predicts little change in y .
I The least-squares regression line always passes through thepoint (x , y).
I Both the slope and especially the intercept are sensitive tooutliers.
Example: Diameter and Height of Redwood Trees
We have the following data giving the diameter of a redwood treeat breast height (in meters, response variable), together with theheight of the tree (in meters, explanatory variable).
x : Diameter 7.22 6.25 7.92 7.10 7.22
y : Height 93.57 91.44 97.54 103.94 87.17
x : Diameter 6.16 6.00 6.90 5.79 6.40
y : Height 80.47 95.71 99.06 65.53 77.72
Example: Diameter and Height of Redwood Trees
We demonstrate how we can find the regression line using aTI-83/TI-83 Plus/TI-84 Plus calculator.
First, we enter the data (STAT, 1:Edit...).
Example: Diameter and Height of Redwood Trees
Make sure the calculator is in “DiagnosticOn” mode (seeinstructions in the previous lecture). Then, select 4:LinReg(ax+b) in the STAT, CALC menu.
Example: Diameter and Height of Redwood Trees
Press ENTER twice.
The linear regression model is y = 10.5x + 18.9.
Example: Diameter and Height of Redwood Trees
The scatterplot and the regression line look like this.
5.5 6.0 6.5 7.0 7.5 8.0 8.5Diameter40
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Example: Diameter and Height of Redwood Trees
In the previous example, note that r2 is also calculated:
So r2 ≈ 0.37 = 37%, and 37% of the variation in the height ofredwood trees is explained by the straight-line regression model.
Regression Analysis Using Excel
Change the layout of the chart by selecting a layout that includesthe regression line (Layout 9 in this case).