A light wooden uniform metre rule was clamped to the surface of a bench so that

Question

A light wooden uniform metre rule was clamped to the surface of a bench so
that a length L of the metre rule was hanging over the edge. The overhanging
end of the metre rule was loaded with a mass and the depression D produced
at the loaded end was measured. Values of D were obtained for different
lengths L.

Assume that D = KL ^ n, where K and n are constants, and draw a graph that
would enable you to determine the values of n and K. Use a linear regression
analysis of the data to fit the best straight line through the data and determine
the value of D for L = 100 mm from the graph.

L (mm) D (mm) 200 2 300 6 400 13 500 28 600 49 700 75 800 109

Explanation / Answer

PART A.
Mean of X = X / n =    500
Mean of Y = Y / n =   40.2857
(Xi - Mean)^2 =   280000
(Yi - Mean)^2 =   9539.43
(Xi-Mean)*(Yi-Mean) =   49500
b1 = (Xi-Mean)*(Yi-Mean) / (Xi - Mean)^2    
b1 = 49500 / 280000 = 0.177  
bo = Y / n - b1 * X / n  
bo = 40.2857 - 0.177*500 = -48.107  
  
Y = bo + b1 X  
  
Y'= -48.107+0.177*X  

PART B.
Y = -48.107+0.177*100
Y = 128.893

Line of Regression Y on X i.e Y = bo + b1 X X Y (Xi - Mean)^2 (Yi - Mean)^2 (Xi-Mean)*(Yi-Mean) 200 2 90000 1465.795 11485.71 300 6 40000 1175.509 6857.14 400 13 10000 744.509 2728.57 500 28 0 150.938 0 600 49 10000 75.939 871.43 700 75 40000 1205.083 6942.86 800 109 90000 4721.655 20614.29

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