10. Multiple Regression

North South University (NSU)Course title: Applied StatisticsChapter: Multiple Regression

Multiple regression

A statistical process by which several variables are used to predict another variable is called multiple regression.

Sample multiple regression

The equation

is the sample multiple regression of on . Where is the dependent variable

are the independent variable.

is the intercept made by the regression line

are the regression coefficient represent the rate of change of the dependent variable per unit change in each of the independent variables when the other independent variables are held constant.

Estimating of regression coefficients

Let us consider the regression model with only two independent variables. So the regression equation can be written as

The coefficient estimators can be solved for the following forms:

.

Multiple Regression

Where, is the sample correlation between and .

is the sample correlation between and .

is the sample correlation between and .

is the sample standard deviation for .



Example

Given the following set of dataa) Calculate the multiple regression plane.b) Predict when and .

25 3.5 5.0

30 6.7 4.2

11 1.5 8.5

22 0.3 1.4

27 4.6 3.6

19 2.0 1.3

Solution

The multiple regression plane of on and is

(a) We know the least square estimates of the coefficients are

.

of 6 Dr. Mohd. Muzibur RahmanProfessor, Department of Statistics, JU

Multiple Regression

From the data we get,

So,

So, the estimated multiple regression plane of on and is

(b) When and , the predicted value of is

Example: The following information has been gathered from a random sample of apartment rents in a city. We are trying to predict rent (in dollars per month) based on the size of the apartment (number of rooms) and the distance from town (in miles).

Rent ($) Number of rooms Distance from town360 2 11000 6 1450 3 2525 4 3350 2 10300 1 4

(a) Calculate the least squares equation that best relates these three variables.

(b) If some one is looking for a two bedrooms apartment and 2 miles from the town,

what rent should be expect to pay?


Multiple Regression

Solution: Our least square equation is


.


So,

So, the least squares equation of rent on number of rooms and distance from town is

(b) If some one is looking for a two bedrooms apartment and 2 miles from town, the expected rent would be


Multiple Regression

Example: Mr. X, owner and general manager of the campus stationary store, is concerned about the sales behavior of a CD sold at the store. He realizes that there are many factors that might help explain sales, but believes that advertising and price are major determinants. He has collected the following data:

Sales (unit sold) Advertising (no. of Ads) Price (Tk.)

33 3 125

61 6 115

70 10 140

82 13 130

17 9 145

24 6 140

(a) Calculate the least squares equation to predict sales from advertising and price.

(b) If advertising is 7 and price is Tk. 132, what sales would you predict?

Solution: Our least square equation of sales on advertising and price is


.


So,


Multiple Regression

So, the least squares equation of sales on advertising and price is

(b) If the number of advertising is 7 and price is Tk. 132, then the predicted sales would be


10. Multiple Regression

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