linear relationship

Review

If the plot of n pairs of data (x , y) for an experiment appear to indicate a "linear relationship" between y and x, then the method of least squares may be used to write a linear relationship between x and y.
The least squares regression line is the line that minimizes the sum of the squares (d1 + d2 + d3 + d4) of the vertical deviation from each data point to the line (see figure below as an example of 4 points).

Figure 1. Linear regression where the sum of vertical distances d1 + d2 + d3 + d4 between observed and predicted (line and its equation) values is minimized.

The least square regression line for the set of n data points is given by the equation of a line in slope intercept form:
y = a x + b
where a and b are given by

Figure 2. Formulas for the constants a and b included in the linear regression .

• Problem 1

  Consider the following set of points: {(-2 , -1) , (1 , 1) , (3 , 2)}
  a) Find the least square regression line for the given data points.
  b) Plot the given points and the regression line in the same rectangular system of axes.

• Problem 2

  a) Find the least square regression line for the following set of data
  
  {(-1 , 0),(0 , 2),(1 , 4),(2 , 5)}
  b) Plot the given points and the regression line in the same rectangular system of axes.

• Problem 3

  The values of y and their corresponding values of y are shown in the table below
  

  x	0	1	2	3	4
  y	2	3	5	4	6

  
  a) Find the least square regression line y = a x + b.
  b) Estimate the value of y when x = 10.

• Problem 4

  The sales of a company (in million dollars) for each year are shown in the table below.
  

  x (year)	2005	2006	2007	2008	2009
  y (sales)	12	19	29	37	45

  
  

  
  
  a) Find the least square regression line y = a x + b.
  b) Use the least squares regression line as a model to estimate the sales of the company in 2012. 

Solution