Question

Use a transformation to linearize this equation and then employ linear regression to determine parameters a and

b. based on your analysis, predict y at x=1.6.

Answer 1

Given the equation


`y=\left( (a+√(x))/(b√(x)) \right)^2`

which models the data tabulated below:


`\begin{tabular} c x&y\\[1ex] 0.5&10.4\\ 1&5.8\\ 2&3.3\\ 3&2.4\\ 4&2 \end{tabular}`

The linear regression equation is given by


`y=a+bx`

where:
`b= \frac{\Sigma xy-n\bar{x}\bar{y}}{\Sigma x^2-n\bar{x}^2}` and
`a=\bar{y}-b\bar{x}`

We extend the given table as follows:


`\begin{tabular} c x&y&x^2&xy\\[1ex] 0.5&10.4&0.25&5.2\\ 1&5.8&1&5.8\\ 2&3.3&4&6.6\\ 3&2.4&9&7.2\\ 4&2&16&8\\[1ex] \Sigma x=10.5&\Sigma y=23.9&\Sigma x^2=30.25&\Sigma xy=32.8 \end{tabular} \\ \\ \\ \bar{x}= (\Sigma x)/(n) = (10.5)/(5) =2.1 \\ \\ \bar{y}=(\Sigma y)/(n) = (23.9)/(5) =4.78`

Thus,


`b= (32.8-5(2.1)(4.78))/(30.25-5(2.1)^2) \\ \\ = (32.8-50.19)/(30.25-22.05) = (-17.39)/(8.2) \\ \\ =-2.12`

and


`a=4.78-(-2.12)(2.1)=4.78+4.454=9.234`

Therefore, the linearlized form of the equation is y = 9.234 - 2.12x



Part B:

At x = 1.6,


`y=9.234-2.12(1.6)=9.234-3.392=5.842`