Regression Analysis

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

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.

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.

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.

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

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...