Question

Find the least square line Y=a+bX for the data points (-1,10), (0,9),(1,7),(2,5),(3,4),(4,3),(5,0),
(6,-1)

Answer 1

Answer:

Answer:Step 1: Calculate the mean of the x -values and the mean of the y -values.

X¯¯¯=∑i=1nxinY¯¯¯=∑i=1nyin

Step 2: The following formula gives the slope of the line of best fit:

  m=∑i=1n(xi−X¯¯¯)(yi−Y¯¯¯)∑i=1n(xi−X¯¯¯)2

Step 3: Compute the y -intercept of the line by using the formula:

b=Y¯¯¯−mX¯¯¯

Step 4: Use the slope m and the y -intercept b to form the equation of the line.

Step-by-step explanation:

Answer 2

Answer:

The least square line is   y = 5.23957-0.8103x

Step-by-step explanation:

   Write the given data points as

          $x_{i}$ =   -1     0   1    2    3   4   5   6

          $y_{i}$ =   10    9   7    5    4   3   0  -1

   Mean of $x_{i}$  values = (-1+0+2+3+4+5+6)/8 = 19/8 = 2

   Mean of $y_{i}$  values = (10+9+7+5+4+3+0+-1) = 37/8 = 4.625

  Straight line equation is y = a+bx

 The normal equations are

    ∑y = an +b∑x    ---------(i)

  ∑xy = a∑x + b∑x² --------(ii)

      Here n= 8

    now we find the all the values and substitute in the above equations

                    x        y         x²            xy

                   -1       10         1            -10

                    0       9         0             0

                    1        7          1              7

                    2       5         4             10

                    3       4         9             12

                    4       3         16           12

                    5       0        25           0

                    6       -1        36          -6

             ∑x =19 ∑y=37 ∑x²=92   ∑xy=25

      substitute the above values  

            37 = 10a + 19b ------------(iii)

            25 = 19a + 92b ------------(iv)

         multiply equation (iii) with '19'

            37×19 = 19(10a+19b)

               703 = 190a+361b -------------(v)

         multiply equation(iv) with '10'

            10×25 = 10(19a+92b)

                250 = 190a+920b ------------(vi)

             subtract equation (v) and (vi)

            (190a+361b)-(190a+920b) = 703-250

                                    361b-920b = 453

                                            -559b = 453  

                                                     b = -453/559

                                                      b = -0.8103

         substitute value of b= -0.8103 in equation(iii)

                     37 = 10a+19(-0.8103)

                     37 = 10a-15.3957

                    10a = 37+15.3957

                     10a = 52.3957

                         a = 5.23957

    substitute values of a and b in the straight line equation is

             y = a+bx

             y = 5.23957+(-0.8103)x

             y = 5.23957-0.8103x

  Hence, the least square line is   y = 5.23957-0.8103x

 

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