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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) (y) _{i} | Advertising($000s) (x) _{i} |

28 | 71 |

14 | 31 |

19 | 50 |

21 | 60 |

16 | 35 |

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**):

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

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:

**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 r^{2},

- confidence intervals.

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