3.  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.  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. Y = a + b X 1 + c X 2 Where, Y = Dependent Variable X 1 & X 2 = Independent Variable a, b & c = Constants

6.  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 “X 1 & X 2 ” is denoted as R x 1.y.x 2.

7.  The Coefficient of Multiple Correlation is computed by the following formula. R x 1. y. x 2 = r² y.x 1 + r² x 1. x 2 – 2 r y. x 1 r y. x 2 r x 1. x 2 1 - r² y.x 2 Where, r = Simple Coefficient of Correlation having the general formula, r = n ∑ XY – ( ∑ X ) ( ∑ Y ) n ∑ X² - ( ∑ X )² n ∑ Y² - ( ∑ Y )²

9. 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 X 1 + c X 2

10. Data of Assessed Value, Heating Area of Dwelling & Age: HOUSE ASSESSED VALUE ($000) HEATING AREA OF DWELLING (THOUSANDS OF SQUARE FEET) AGE (YEARS) 0184.42.003.42 0277.41.7111.50 0375.71.458.33 0485.91.760.00 0579.11.937.42 0670.41.2032.00 0775.81.5516.00 0885.91.932.00 0978.51.591.75 1079.21.502.75 1186.71.390.00 1279.31.900.00 1374.51.5412.58 1483.81.892.75 1576.81.597.17

12.  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.

15. ASSESSED VALYE (Y) HEATING AREA OF DWELLING (X 1 ) AGE (X 2 )

19. YX1X1 X2X2 Y²X² 1 X² 2 X1YX1YX2YX2YX1X2X1X2 84.42.003.427123.36411.6964168.8288.6486.84 77.41.7111.505990.762.9241132.25132.354890.119.665 75.71.458.335730.492.102569.3889109.765630.58112.0785 85.91.760.007378.813.09760.00151.1840.00 79.11.937.426256.813.724955.0564152.663586.92214.3206 70.41.2032.004956.161.44102484.482252.838.4 75.81.5516.005745.642.4025256117.491212.824.8 85.91.932.007378.813.72494.00165.787171.83.86 78.51.591.756162.252.52813.0625124.815137.3752.7825 79.21.502.756272.642.257.5625118.8217.84.125 86.71.390.007516.891.93210.00120.510.00 79.31.900.006288.493.610.00150.670.00 74.51.5412.585550.252.3716158.2564114.73937.2119.3732 83.81.892.757022.443.57217.5625158.382230.455.1975 76.81.597.175898.242.528151.4089122.112550.65611.4003 ∑Y = 1193.4 ∑X 1 = 24.93 ∑X 2 = 107.67 ∑Y² = 95272.04 ∑X² 1 = 42.2085 ∑X² 2 = 1780.245 ∑X 1 Y = 1992.545 ∑X 2 Y = 8107.142 ∑X 1 X 2 = 162.8426

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27.  15a +42.36440588 = 1193.4  15a = 1193.4 – 42.36440588  a = 1151.035594 15 a = 76.73570627

28. We have, a = 76.73570627 b = 3.43065089 c = - 0.400870445 Now eq. A becomes, Y = 76.73570627 + 3.43065089 X 1 – 0.400870445 X 2

29. By using Regression Equation, Y = 76.73570627 + 3.43065089 (1.10) – 0.400870445 (34.00) Y = 66.879 (in thousands of dollars)

30. The Coefficient of Multiple Correlation is computed by the following formula. R x 1. y. x 2 = r² y.x 1 + r² x 1. x 2 – 2 r y. x 1 r y. x 2 r x 1. x 2 1 - r² y.x 2 First we find r y.x 1, r x 1.x 2 and r y.x 2.

34. Now the Coefficient of Multiple Correlation will be, R x 1. y. x 2 = r² y.x 1 + r² x 1. x 2 – 2 r y. x 1 r y. x 2 r x 1. x 2 1 - r² y.x 2 R x 1. y. x 2 = (0.57)² + (- 0.57)² – 2 (0.57)(- 0.80)(- 0.57) 1 – (- 0.80) ² Rx 1.y.x 2 = 0.60 (Moderate Correlation)

35.  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.

36. Thank you so much  Our honorable teacher Sir Zafar Ali.  The students of BS Commerce (3 rd 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.