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Simple Linear Regression Equation Part 1

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Let us consider the example of linear regression problem solving. We want to receive linear regression formula, the values of standard errors for estimate and coefficients, coefficient of determination, confidence interval and other parameters from the statistical data.

The firm has decided to use linear regression by employing the equation y = a + bx for the annual sales. The prior years's data regarding sales and advertising expenditures are as follows:

Sales($000,000s) (yi) Advertising($000s) (xi)
2871
1431
1950
2160
1635

1. Let us find linear regression equation from using least squares computations.
The following equations can be used to determine the equation for the least squares regression line (in the form of y = a + bx):

regression_equation.gif

In our case n=5 (the number of observations) and:

Regression_sums.gif

Substituting into the two equations gives:
98 = 5a + 247b
5192 = 247a + 13327b

Solving for the two unknowns gives a=4.2 and b=0.31, so expected sales equal 4.2 plus 0.31 times the advertising expenditure,

y = 4.2 + 0.31x

Hence, if the advertising expenditures in the next year is $40 000, the expected sales will be $16 600 000.

The observations are graphed as follows:

Resression.gif

a is called as constant coefficient of linear regression equation and b is called as variable coefficient of linear regression equation.

Therefore, in the Part 1 we have derived:
- the formula of linear regression equation,
- constant and variable coefficients of linear regression equation
from the prior years's observations.

In the Part 2, we'll derive:
- standard error for estimate sales,
- standard error of a,
- standard error of b,
- coefficient of correlation (covariance) r,
- coefficient of determination r2,
- confidence intervals.

Next (Part 2).

Posted by mazoo at March 3, 2005 8:02 PM

Related posts:

Simple Linear Regression Equation Part 2 Mar 21, 2005

Comments

Excellent, so helpful! amazing!

Posted by: N N Tarun at February 13, 2008 8:48 PM