Linear Regression
Problems with Solutions
Linear regression and modelling problems are presented along with their solutions at the bottom of the page. Also a linear regression calculator and grapher may be used to check answers and create more opportunities for practice.
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). The least square regression line for the set of n data points is given by the equation of a line in slope intercept form:
where a and b are given by

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.
Solutions to the Above Problems

a) Let us organize the data in a table.
x y x y x^{ 2} 2 1 2 4 1 1 1 1 3 2 6 9 Σx = 2 Σy = 2 Σxy = 9 Σx^{2 } = 14
We now use the above formula to calculate a and b as follows
a = (nΣx y  ΣxΣy) / (nΣx^{2}  (Σx)^{2}) = (3*9  2*2) / (3*14  2^{2}) = 23/38
b = (1/n)(Σy  a Σx) = (1/3)(2  (23/38)*2) = 5/19
b) We now graph the regression line given by y = a x + b and the given points. 
a) We use a table as follows
x y x y x^{ 2} 1 0 0 1 0 2 0 0 1 4 4 1 2 5 10 4 Σx = 2 Σy = 11 Σx y = 14 Σx^{2 } = 6
We now use the above formula to calculate a and b as follows
a = (nΣx y  ΣxΣy) / (nΣx^{2}  (Σx)^{2}) = (4*14  2*11) / (4*6  2^{2}) = 17/10 = 1.7
b = (1/n)(Σy  a Σx) = (1/4)(11  1.7*2) = 1.9
b) We now graph the regression line given by y = ax + b and the given points. 
a) We use a table to calculate a and b.
x y x y x^{ 2} 0 2 0 0 1 3 3 1 2 5 10 4 3 4 12 9 4 6 24 16 Σx = 10 Σy = 20 Σx y = 49 Σx^{2 } = 30
We now calculate a and b using the least square regression formulas for a and b.
a = (nΣx y  ΣxΣy) / (nΣx^{2}  (Σx)^{2}) = (5*49  10*20) / (5*30  10^{2}) = 0.9
b = (1/n)(Σy  a Σx) = (1/5)(20  0.9*10) = 2.2
b) Now that we have the least square regression line y = 0.9 x + 2.2, substitute x by 10 to find the value of the corresponding y.
y = 0.9 * 10 + 2.2 = 11.2

a) We first change the variable x into t such that t = x  2005 and therefore t represents the number of years after 2005. Using t instead of x makes the numbers smaller and therefore manageable. The table of values becomes.
t (years after 2005) 0 1 2 3 4 y (sales) 12 19 29 37 45
We now use the table to calculate a and b included in the least regression line formula.
t y t y t^{ 2} 0 12 0 0 1 19 19 1 2 29 58 4 3 37 111 9 4 45 180 16 Σx = 10 Σy = 142 Σxy = 368 Σx^{2 } = 30
We now calculate a and b using the least square regression formulas for a and b.
a = (nΣt y  ΣtΣy) / (nΣt^{2}  (Σt)^{2}) = (5*368  10*142) / (5*30  10^{2}) = 8.4
b = (1/n)(Σy  a Σx) = (1/5)(142  8.4*10) = 11.6
b) In 2012, t = 2012  2005 = 7
The estimated sales in 2012 are: y = 8.4 * 7 + 11.6 = 70.4 million dollars.