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Presented By: F AHAD A HMED K HAN A NUM F AROOQ Z AHRA G ULZAR M ARIA S AHER K HAN

MULTIPLE REGRESSION. Presented By: F AHAD A HMED K HAN A NUM F AROOQ Z AHRA G ULZAR M ARIA S AHER K HAN. R OYAL P RESENTERS. INTRODUCTION. BY: M ARIA S AHER K HAN. M ULTIPLE R EGRESSION :-. It was introduced by Karl Pearson in 1908.

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Presented By: F AHAD A HMED K HAN A NUM F AROOQ Z AHRA G ULZAR M ARIA S AHER K HAN

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  1. MULTIPLEREGRESSION Presented By: FAHADAHMEDKHAN ANUMFAROOQ ZAHRAGULZAR MARIASAHERKHAN ROYALPRESENTERS

  2. INTRODUCTION BY: MARIA SAHER KHAN

  3. MULTIPLE REGRESSION:- • It was introduced by Karl Pearson in 1908. • In Statistics, Regression analysis is a method for the prediction of future events. • The relationship between several independent variables or predictors and a dependent variable or criterion is known as Multiple Regression.

  4. EXAMPLES:- • The value of a house depends on the location where it is situated and the condition of the house i.e. Rooms, East Open or West Open and Proper Water Supply etc. • The Salary of an Employee depends on many variables like his education, his experience, his hard work and his skills etc. • We depend on our Parents. If one will die than it will definitely affects our life.

  5. MULTIPLE REGRESSION EQUATION:- Y = a + b X1 + c X2 Where, Y = Dependent Variable X1 & X2 = Independent Variable a, b & c = Constants

  6. MULTIPLE CORRELATION:- • The Coefficient of Multiple Correlation measures the relationship between a dependent variable and the whole group of independent variables. • The Coefficient of Multiple Correlation between “Y” and the two independent variables “X1 & X2” is denoted as Rx1.y.x2.

  7. The Coefficient of Multiple Correlation is computed by the following formula. Rx1.y.x2 = r²y.x1 + r²x1.x2 – 2 ry.x1 ry.x2 rx1.x2 1 - r²y.x2 Where, r = Simple Coefficient of Correlation having the general formula, r = n ∑ XY – ( ∑ X ) ( ∑ Y ) n ∑ X² - ( ∑ X )² n ∑ Y² - ( ∑ Y )²

  8. QUESTION’S EXPLANATION BY: ZAHRA GULZAR

  9. QUESTION:- Following is the data of the Assessed Value (in thousands of dollars) of 15 houses in a certain locality that depends on the Heating Area of Dwelling (thousands of square feet) and Age (in years). Fit a Multiple Regression Equation. Y = a + b X1 + c X2

  10. Data of Assessed Value, Heating Area of Dwelling & Age:

  11. ASSESSED VALUE:- The value at which houses or other fixed assets (i.e. Land & Building) are sold is known as Assessed Value. HEATING AREA OF DWELLING:- The particular area of house which is affected by heat is called the Heating Area of Dwelling. AGE:- The time period since the house has been made of.

  12. REQUIRED:- • Plot a Graph among all the three variables. • Find the Multiple Regression Equation. • Predict the Assessed Value if the Heating area of dwelling is 1.10 and Age is 34.00 years. • Calculate Multiple Correlation.

  13. GRAPH BY: ANUM FAROOQ

  14. SOLUTION:-GRAPH ASSESSED VALYE (Y) AGE (X2) HEATING AREA OF DWELLING (X1)

  15. ASSESSED VALYE (Y) AGE (X2) HEATING AREA OF DWELLING (X1)

  16. FINDING a, b & c :- Given, Y = a + bX1 + cX2  eq. A STEP – I : Apply ∑ on both sides in equation  A , ∑Y = ∑a + ∑bX1 + ∑cX2 ∑a = a1 + a2 + …..+ an ∑a = na ∑Y = na +b∑X1 + c∑X2 eq.1

  17. STEP – II : Apply ∑ and multiply by X1 on both sides in equation A , ∑X1Y = a∑X1 + b∑X²1 + c∑X1X2  eq.2 STEP – III : Apply ∑ and multiply by X2 on both sides in equation  A , ∑X2Y = a∑X2 + b∑X1X2 + c∑X²2  eq.3

  18. STEP – IV : Finding n, ∑Y, ∑X1, ∑X2, ∑X1X2, ∑X²1, ∑²2, ∑X1Y, ∑X2Yfor Multiple Regression. For Multiple Correlation we need ∑Y². The computations needed for the regression equations are given as follows:

  19. REGRESSION EQUATION, PREDICTION & MULTIPLE CORRELATION BY: FAHAD AHMED KHAN

  20. Now eq. 1 , 2 & 3 becomes, 1 => 15a + 24.93b + 107.67c = 1193.4 2 => 24.93a + 42.2085b + 162.8426c = 1992.545 3 => 107.67a + 162.8426b + 1780.245c = 8107.142 Solving equations 1 and 2 , Multiplying eq.  1 by 24.93 and eq.  2 by 15, we get

  21. 1 => 373.95a + 621.5049b + 2684.2131c = 29751.462 2 => 373.95a + 633.1275b + 2442.639c = 29888.175 Now subtracting eq. 1 and eq. 2 , + 373.95a + 621.5049b + 2684.2131c = + 29751.462 + 373.95a + 633.1275b + 2442.639c = + 29888.175 - - - - - 11.6226 b + 241.5741 c = - 136.713 eq.  4

  22. Solving equations 2 and 3 , Multiplying eq.  2 by 107.67 and eq.  3 by 24.93, we get 2=> 2684.2131 a + 4544.589195 b + 17533.26274 c = 214537.3202 3=> 2684.2131 a + 4059.666018 b + 44381.50785 c = 202111.0501 Now subtracting eq. 2 and 3 , + 2684.2131 a + 4544.589195 b + 17533.26274 c = + 214537.3202 + 2684.2131 a + 4059.666018 b + 44381.50785 c = + 202111.0501 - - - - + 484.923177 b - 26848.24511 c = 12426.2701 eq.  5

  23. Solving equations 4 and 5 , Multiplying eq. 4 by 484.923177 and eq. 5 by 11.6226, we get 4 => - 5636.068117 b + 117144.8801 c = - 66295.3023 5 => + 5636.068117 b – 312046.4136 c = + 144425.5669 Now Adding eq. 4 and 5 , 4 => - 5636.068117 b + 117144.8801 c = - 66295.3023 5 => +5636.068117 b – 312046.4136 c = + 144425.5669 - 194901.5335 c = + 78130.2646

  24. c = - 78130.2646 194901.5335 c = - 0.400870445 Now Put (c = - 0.400870445) in eq. 4 to get the value of b, • - 11.6226 b + 241.5741 (- 0.400870445) = - 136.713 • - 11.6226 b – 96.83991697 = - 136.713 • - 11.6226 b = -136.713 + 96.83991697 • - 11.6226 b = - 39.87308303

  25. b = 39.87308303 11.6226 b = 3.43065089 Now Put (b = 3.43065089) and (c = - 0.400870445) in eq. 1 to get the value of a, • 15a + 24.93 (3.43065089) + 107.67 (- 0.400870445) = 1193.4 • 15a + 85.52612669 – 43.16172081 = 1193.4

  26. 15a +42.36440588 = 1193.4 • 15a = 1193.4 – 42.36440588 • a = 1151.035594 15 a = 76.73570627

  27. FINDING REGRESSION EQUATION:- We have, a = 76.73570627 b = 3.43065089 c = - 0.400870445 Now eq. A becomes, Y = 76.73570627 + 3.43065089 X1 – 0.400870445 X2

  28. ESTIMATING Y WHENX1 = 1.10 & X2 = 34.00 :- By using Regression Equation, Y = 76.73570627 + 3.43065089 (1.10) – 0.400870445 (34.00) Y = 66.879 (in thousands of dollars)

  29. FINDING COEFFICIENT OF MULTIPLE CORRELATION:- The Coefficient of Multiple Correlation is computed by the following formula. Rx1.y.x2 = r²y.x1 + r²x1.x2 – 2 ry.x1 ry.x2 rx1.x2 1 - r²y.x2 First we find ry.x1 , rx1.x2 and ry.x2.

  30. FINDING ry.x1 :- By using the formula, ry.x1 = n ∑ YX1 – ( ∑ Y ) ( ∑ X1 ) n ∑ Y² - ( ∑ Y )² n ∑ X1² - ( ∑ X1 )² ry.x1 = (15) (1992.545) – (1193.4) (24.93) (15) (95272.04) - (1193.4)² (15) (42.2085) – (24.93) ² ry.x1 = 0.57

  31. FINDING rx1. x2 :- By using the formula, rX1.X2 = n ∑ X1.X2 – ( ∑ X1 ) ( ∑ X2 ) n ∑ X1² - (∑ X1)² n ∑ X2² - (∑ X2)² rX1.X2 = (15) (162.8426) – (24.93) (107.67) (15) (42.2085) - (24.93)² (15) (1780.2445) – (107.67) ² rX1.X2 = - 0.57

  32. FINDING ry.x2 :- By using the formula, ry.x2 = n ∑ YX2 – ( ∑ Y ) ( ∑ X2 ) n ∑ Y² - ( ∑ Y )² n ∑ X2² - ( ∑ X2 )² ry.x2 = (15) (8107.142) – (1193.4) (107.67) (15) (95272.04) - (1193.4)² (15) (1780.2445) – (107.67) ² ry.x2 = - 0.80

  33. Now the Coefficient of Multiple Correlation will be , Rx1.y.x2 = r²y.x1 + r²x1.x2 – 2 ry.x1 ry.x2 rx1.x2 1 - r²y.x2 Rx1.y.x2 = (0.57)² + (- 0.57)² – 2 (0.57)(- 0.80)(- 0.57) 1 – (- 0.80)² Rx1.y.x2 = 0.60 (Moderate Correlation)

  34. CONCLUSION:- • We can now conclude that the value of each house depends on the time period since it was built and the heating area that affects it. • We have many dependent variables in our life and many independent also. Regression is the best method to know or to measure the relationship between those variables.

  35. ENDOFPRESENTATION Thank you so much • Our honorable teacher Sir Zafar Ali. • The students of BS Commerce (3rd Semester) to cooperate with us. We wish you all the very best for your future. • Please pardon us if we hurt you throughout the Presentation.

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