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Regression - Regression Finding the equation of the line of...

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Height (m) 1·4 1·5 1·6 1·7 1·8 Weight (kg) 50 55 60 65 70 75 80 85 Time (hours) 1 2 3 4 5 6 Concentration 2 4 6 8 10 12 Regression: Finding the equation of the line of best fit Objectives: To find the equation of the least squares regression line of y on x. Background and general principle The aim of regression is to calculate the equation of the line of best fit on a scatter graph. Consider the scatter graph on the right. One possible line of best fit has been drawn on the diagram. Some of the points lie above the line and some lie below it. The vertical distance each point is above or below the line has been added to the diagram. These distances are called deviations or errors – they are symbolised as n d d d ,..., , 2 1 . When drawing in a regression line, the aim is to make the line fit the points as closely as possible. We do this by making the total of the squares of the deviations as small as possible , i.e. we minimise 2 i d . If a line of best fit is found using this principle, it is called the least-squares regression line . Example 1: A patient is given a drip feed containing a particular chemical and its concentration in his blood is measured, in suitable units, at one hour intervals. The doctors believe that a linear relationship will exist between the variables. Time, x (hours) 0 1 2 3 4 5 6 Concentration, y 2.4 4.3 5.0 6.9 9.1 11.4 13.5 We can plot these data on a scatter graph – time would be plotted on the horizontal axis (as it is the independent variable). Time is here referred to as a controlled variable , since the experimenter fixed the value of this variable in advance (measurements were taken every hour). Concentration is the dependent variable as the concentration in the blood is likely to vary according to time. The doctor may wish to estimate the concentration of the chemical in the blood after 3.5 hours. She could do this by finding the equation of the line of best fit. There is a formula which gives the equation of the line of best fit. ε 1
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The equation of the line is bx a y + = where xx xy S S b = and x b y a - = .
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