Use quadratic regression to fit a saturation growth model given...

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We need to find the quadratic equation coefficients a, b, c, where:

F(t)=at²+bt+c. 

QUADRATIC REGRESSION
t F(t) t⁴ tF(t) t²F(t)
0.75 0.8 0.5625 0.421875 0.31640625 0.6 0.45
2 1.3 4 8 16 2.6 5.2
2.5 1.2 6.25 15.625 39.0625 3 7.5
4 1.6 16 64 256 6.4 25.6
6 1.7 36 216 1296 10.2 61.2
8 1.8 64 512 4096 14.4 115.2
8.5 1.7 72.25 614.125 5220.0625 14.45 122.825
31.75 10.1 199.0625 1430.171875 10923.44140625 51.65 337.975

 System of equations is:

(1) 10923.44140625a+1430.171875b+199.0625c=337.975, where 

a∑t⁴+b∑t³+c∑t²=∑(t²F(t));

(2) 1430.171875a+199.0625b+31.75c=51.65, where

a∑t³+b∑t²+c∑t=∑(tF(t));

(3) 199.0625a+31.75b+7c=10.1, where

a∑t²+b∑t+nc=∑F(t), and n=7, the data size.

The arithmetic is very tedious to solve for a, b, c, but the technique is to eliminate c between (1) and (2) to create two equations (4) and (5) containing a and b only. From these a and b are found by eliminating one or other and then using that value to calculate the remaining value. Equation (3) can then be used to calculate c. This done we get:

F(t)=-0.0243t²+0.3391t+0.5968.

F(9.25)=1.6516 approx.

by Top Rated User (894k points)

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