Line of Best Fit - WheelesClassroom

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Transcript Line of Best Fit - WheelesClassroom

Least Square Regression Line
Line of Best Fit
 Our objective is to fit a line in the scatterplot that fits
the data the best
 As just seen, the best fit would minimize the sum of
squares.
 Line of best fit looks like:
ŷ  b  b x
0
1
That’s a hat on the y, meaning that it is a prediction not
the actual y values. VERY IMPORTANT!!!
Need a slope and y-intercept
Need a point and slope
 An obvious point is the mean of the x and the mean of
the y.
(x, y)
 This point is the middle of both variables.
Slope in the z’s
 If we look at the scatterplot of the z-scores we find that
the line of best fit must go through (0,0)
 The slope of the line that minimizes the sum of
squares in the z-scores will always be r.
 This tells you that for each increase of 1 standard
deviation in x there is a change of r standard
deviations in y.
Example: Square Foot vs. Selling Price
for Houses in Boulder, CO (Table 2.3)
r  0.677
Here is the scatterplot of the z-scores with
the line that minimizes the sum of squares.
Slope in the actual scatterplot
 Since the slope of the line in the z-scores compares the
standard deviations we include these back to get the slope of
the line in the scatterplot of the data.
 Thus the slope of the line in the regular scatterplot becomes
sy
b1  r
sx
 Interpretation of the slope:
 For every increase of 1 unit in x, there is an
increase/decrease of b1 units in y
House sales
s price  45135.6 ssquare _ feet  640.15
r  0.677
sy
45135.6
b1  r 
0.677  47.734
sx
640.15
Interpretation: For every increase of 1
square foot the selling price increases
by $47.73.
Finding intercept, b0
 Now that we have the slope, we only need a
point that the line runs through to get the
intercept.
 We have one: ( x , y )
 So the equation for intercept becomes:
b0  y  b1 x
 Interpretation of the intercept is generally
meaningless. So be careful!
House Sales
x  2627.42 y  177330
b1  47.734
b0  y  b1 x  177330  47.734(2627.42)
b0  51912.73
yˆ  51912.73  47.734 x
price  51912.73  47.734( square feet )