Chapter 13.3, Problem 8E

To determine

To calculate: The sum of squared differences for the data points $\left(-2,6\right),\left(-1,4\right),\left(0,2\right),\left(1,1\right),\left(2,1\right)$ and the model $y*=-1.7x+2.7$ .

Answer to Problem 8E

The value of sum of squared differences is $3.55$ .

Explanation of Solution

Given information:

The data points $\left(-2,6\right),\left(-1,4\right),\left(0,2\right),\left(1,1\right),\left(2,1\right)$ and the model $y*=-1.7x+2.7$ .

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 $\left(-2,6\right),\left(-1,4\right),\left(0,2\right),\left(1,1\right),\left(2,1\right)$ and the model $y*=-1.7x+2.7$ .

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,

  $\begin{array}{cccccc}x& -2& -1& 0& 1& 2\\ y& 6& 4& 2& 1& 1\\ y*& 6.1& 4.4& 2.7& 1& -0.7\\ y-y*& -0.1& -0.4& -0.7& 0& 1.7\\ {\left(y-y*\right)}^2& 0.01& 0.16& 0.49& 0& 2.89\end{array}$

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 $3.55$ .