Chapter 13.3, Problem 12E

To determine

To calculate: The sum of squared differences for the data points $\left(-1,-4\right),\left(1,-3\right),\left(2,0\right),\left(4,5\right),\left(6,9\right)$ and the model $y*=0.14x^2+1.3x-3$ .

Answer to Problem 12E

The value of sum of squared differences is $3.144$ .

Explanation of Solution

Given information:

The data points $\left(-1,-4\right),\left(1,-3\right),\left(2,0\right),\left(4,5\right),\left(6,9\right)$ and the model $y*=0.14x^2+1.3x-3$ .

Formula used:

The sum of squared differences is ${\displaystyle \sum _{i=1}^n{\left(y-y_i*\right)}^2}$ where $y_i$ is the y -coordinate of the point and $y_i*$ is the linear model.

Calculation:

Consider the provided data points $\left(-1,-4\right),\left(1,-3\right),\left(2,0\right),\left(4,5\right),\left(6,9\right)$ and the model $y*=0.14x^2+1.3x-3$ .

Construct a table with data points. Substitute the value of each x to compute the value for each $y*$ .

Then compute the difference between each y and corresponding $y*$ . At last compute the value of ${\left(y-y*\right)}^2$ for each point of data set.

Recall that the sum of squared differences is ${\displaystyle \sum _{i=1}^n{\left(y-y_i*\right)}^2}$ where $y_i$ is the y -coordinate of the point and $y_i*$ is the linear model.

Data points are tabulated as,

  $\begin{array}{cccccc}x& -1& 1& 2& 4& 6\\ y& -4& -3& 0& 5& 9\\ y*& -4.16& -1.56& 0.16& 4.44& 9.84\\ y-y*& 0.16& -1.44& 0.16& 0.56& -0.84\\ {\left(y-y*\right)}^2& 0.0256& 2.0736& 0.0356& 0.3136& 0.7056\end{array}$

Sum up all the values of the last row to obtain the value of sum of squared differences.

Thus, the value of sum of squared differences is $3.144$ .