We are given the data set (0,0),(1,2),(2,16),(3,52),(4,118),(5,223) and we are asked to fit these in a power model `y=ax^b ` :
(1) The easiest way is to input the data in Excel or a graphing utility and perform a power regression yielding `y=2.04x^2.93 ` with a,b rounded to two decimal places.
(2) To determine if the data set will fit a power model well we can plot (lnx,lny) and determine if the plot is approximately a straight line.
Taking the natural logarithm of the coordinates yields the points (0,.6931),(.6931,2.773),(1.0986,3.9512),(1.3862,4.7707),(1.6094,5.4072). These points appear to lie on a line. Performing linear regression on these points yields the line y=.7146+2.929x
So we have `lny=lna+blnx ` Exponentiating both sides with base e gives us:
`y=e^(lna)*e^(blnx)=a*(e^(lnx))^b=ax^b ` where `a=e^(.7146) ` and b=2.929
So the power model is `y~~2.04x^(2.93) `