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Equation for Line of Best Fit

Given: At a concert, the number of tattoos, x, and the number of piercings, y, that a person had was recorded:

Find: The equation for the line of best fit

x y2 58 75 63 46 8

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The Line of Best Fit The line of best fit results from using the method of least squares. The equation for the

line is determined by its slope, b1, and y-intercept b0. These components are found using formulas 3.6 and 3.7:

Notes: The formula for b1 calls for SS’s “sum of squares”The formula for b0 calls for ’s “summations”Both were found as a part of the preliminary calculations

b1 =SS(xy)

SS(x)and b0 =

1n

[ y - (b1 • x) ]

If you do not know the values of SS(xy), SS(x), x, y and n, you need to obtain them before continuing

(See Animated Tutorial “Bivariate Data - Preliminary Calculations”for assistance in completing these preliminary calculations)

Once you have the values of SS(xy), SS(x), x, y and n, you are prepared to proceed with the calculations of b1 and b0

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Finding the Slope, b1

The preliminary calculations yielded:

Notes: 1) Do not use rounded values for either SS2) Do not round until calculation is completed

SS(x) = 22.8 SS(xy) = 12

The slope, b1, is 0.53

b1 =SS(xy)

SS(x)=

12

22.8= 0.526316 = 0.53

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Finding the y-Intercept, b0

The preliminary calculations yielded:

Notes: 1) Do not use a rounded value for b1

2) Do not round until calculation is completed

The y-intercept, b0, is 3.47

n = 5 x = 24 y = 30

b0 =1n

[ y - (b1 • x) ] =15

[ 30 - ((0.526316) • (24)) ]

=15

[ 30 - (12.631584) ]

=15

[ 17.368416 ] = 3.47368 = 3.47

Not rounded b1 = 0.526316

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The Equation for the Line of Best Fit To write the equation for the line of best fit, simply replace

b1 and b0 in the linear equation model,

The calculated values are: b1= 0.53 and b0= 3.47

y = b0 + b1xˆ

The equation of the line of best fit is: y = 3.47 + 0.53xˆ

Note: y is read ‘y-hat’ or ‘predicted y’ˆ