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).
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.
