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Business Math Essay

1773 words - 8 pages

MSOR 221 Statistical Inference

Chapter 18 Multiple Regression

Model and Required Conditions

For k independent variables (predicting variables) x1, x2, … , xk, the multiple linear regression model is represented by the following equation:

[pic]

where (1, (2, … , (k are population regression coefficients of x1, x2, … , xk respectively, (0 is the constant term, and ( (the Greek letter epsilon) represents the random term (also called the error variable) – the difference between the actual value of Y and the estimated value of Y based on the values of the independent variables. The random term thus accounts for all other independent variables that are not included in the model.
...view middle of the document...

55 |
| |3 |100 |100 |300 |10000 |300 |9 |10000 |95.92 |
| |10 |600 |400 |4000 |240000 |6000 |100 |360000 |379.61 |
| |33 |2800 |1500 |10300 |880000 |18800 |235 |1700000 |1500 |

n = 6

[pic]

By solving the above system of normal equations, we should find the following:

b0 = 6.397 b1 = 20.492 b2 = 0.280

( The sample multiple linear regression equation is:
[pic]

Interpretation of the Regression Coefficients

b1: the approximate change in y if x1 is increased by 1 unit and x2 is held constant.
b2: the approximate change in y if x2 is increased by 1 unit and x1 is held constant.

In Example 1, if x1 is increased by 1 unit and x2 is held constant, then the approximate change in y therefore will be 20.492 units.

Point Estimate

In Example 1, suppose x1 = 4 and x2 = 500, then the point estimate of y equals:
[pic]

The Standard Error of Estimate in Multiple Regression Model

[pic]

where [pic] = the observed y value in the sample
[pic] = the estimated y value calculated from the multiple regression equation

|In Example 1, |[pic] | | | |
| |(17.01)2 | |[pic] |
| |(-5.2)2 | | |
| |(5.27)2 | | |
| |(-41.55)2 | | |
| |(4.08)2 | | |
| |(20.39)2 | | |
| |2502.954 | | | |

Note: [pic] is the point estimate of [pic](the standard deviation of the error variable (.)

Testing the Validity of the Model – The Analysis of Variance (ANOVA) Test

Let’s consider a simple linear regression model:

y

* [pic]= (y / n = the mean of y

* *
[pic]
* *

x

[pic]
← [pic]

[pic] = total deviations
[pic] = total deviations of estimated values from the mean
[pic] = error deviations = [pic]
[pic] = the residual of the ith data point

[pic]
← SST = SSR + SSE
SST = total sum of squared deviations = total variation
SSR = sum of squares resulting from regression = explained variation
SSE = sum of squares...

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