leastsq_f07

leastsq_f07 - Least Squares Regression Fitting a Line to...

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Least Squares Regression Fitting a Line to Bivariate Data
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Linear Relationships Avg. occupants per car 1980: 6/car 1990: 3/car 2000: 1.5/car By the year 2010 every fourth car will have nobody in it! Food for Thought Kind of mathematical relationship between year and avg. no. of occupants per car? Why might relation- ship break down by 2010?
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Basic Terminology Scatterplots, correlation: interested in association between 2 variables (assign x and y arbitrarily) Least squares regression: does one quantitative variable explain or cause changes in another variable?
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Basic Terminology (cont.) Explanatory variable : explains or causes changes in the other variable; the x variable. (independent variable) Response variable : the y -variable; it responds to changes in the x - variable. (dependent variable)
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Examples Fertilizer (x ) corn yield (y ) Advertising $ (x ) store income (y ) Drug dose (x ) blood pressure (y ) Daily temperature (x ) natural gas demand (y ) change in min wage(x) unemployment rate (y)
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Simplest Relationship Simplest equation that describes the dependence of variable y on variable x y = a + bx linear equation graph is line with slope b and y- intercept a
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Graph y x 0 a y=a+bx run rise Slope b=rise/run
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Notation (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x n , y n ) draw the line y=a+bx through the scatterplot , the point on the line corresponding to x i is . x when x y of value observed the is ; x bx when x a y line by the predicted y of value the is ˆ ; ˆ i i = = + = + = i i i i y y bx a y
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Observed y, Predicted y predicted y when x=2.7 yhat = a + bx = a + b*2.7 2.7
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Scatterplot: Fuel Consumption vs Car Weight Fuel Consumption vs Car Weight 2 3 4 5 6 7 1 2 3 4 5 Car Weight (1000 lbs) Fuel consumption (gal/100 miles) Fuel consumption “Best” line?
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Scatterplot with least squares prediction line FUEL CONSUMPTION vs CAR WEIGHT y = 1.639x - 0.3631 r 2 = 0.9538 2 3 4 5 6 7 1.5 2.5 3.5 4.5 WEIGHT (1000 lbs) FUEL CONSUMP. (gal/100 miles)
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How do we draw the line? Residuals ) ( ˆ y predicted y observed residual th : line the from point data th the of deviation vertical the is residual th the i i i i bx a y y y i i i + - = - = - =
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Residuals: graphically Graphical Display of Residuals X X i Y i e i =Y i - Y i Y i positive residual negative residual
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leastsq_f07 - Least Squares Regression Fitting a Line to...

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