Linear Regression
Problems with Solutions






Linear regression and modeling problems are presented. The solutions to these problems are at the bottom of the page.

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 of the vertical deviation from each data point to the line (see figure below).

The least square regression line for the set of n data points is given by

y = ax + b


where a and b are given by

linear regression formulas



  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 rectangualr system of axes.

  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 rectangualr system of axes.

  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 = ax + b.

    b) Estimate the value of y when x = 10.

  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 = ax + 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
  1. 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 Σx2 = 14


    We now use the above formula to calculate a and b as follows

    a = (nΣx y - ΣxΣy) / (nΣx2 - (Σx)2) = (3*9 - 2*2) / (3*14 - 22) = 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 = ax + b and the given points.


    regression line graph problem 1




  2. 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 Σx2 = 6


    We now use the above formula to calculate a and b as follows

    a = (nΣx y - ΣxΣy) / (nΣx2 - (Σx)2) = (4*14 - 2*11) / (4*6 - 22) = 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.


    regression line graph problem 2




  3. 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 Σx2 = 30


    We now calculate a and b using the least square regression formulas for a and b.

    a = (nΣx y - ΣxΣy) / (nΣx2 - (Σx)2) = (5*59 - 10*20) / (5*30 - 102) = 1.9

    b = (1/n)(Σy - a Σx) = (1/5)(20 - 1.9*10) = 0.2

    b) Now that we have the least square regression line y = 1.9 x + 0.2, substitute x by 10 to find the value of the corresponding y.

    y = 1.9 * 10 + 0.2 = 19.2


  4. 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 managable. 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 Σx2 = 30


    We now calculate a and b using the least square regression formulas for a and b.

    a = (nΣt y - ΣtΣy) / (nΣt2 - (Σt)2) = (5*368 - 10*142) / (5*30 - 102) = 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.




More references on
elementary statistics and probabilities.



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Updated: 2 April 2013

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