Question

From the following data obtain the two regression equations.

Sales : 91 97 108 121 67 124 51 73 111 57

Purchase : 71 75 69 97 70 91 39 61 80 47​

Answer 1

The two regression equations are $X = 1.36 Y -5.2$ and  $Y= 0.61X-15.1$ .

Step-by-step explanation:

Let the Sales be represented by variable X and Purchase be represented by variable Y.

The following data is represented below for computing regression equations;

Sales (X)     $X-\bar X$      $(d_x)^{2}$    Purchases (Y)     $Y-\bar Y$      $(d_y)^{2}$       $d_x d_y$

                     ($d_x$)                                                  ($d_y$)

  91                 1             1                71                   1               1             1

  97                7           49              75                  5             25          35

 108               18         324             69                 -1               1           -18

  121               31         961              97                 27           729        837

  67              -23        529             70                  0              0            0

 124               34        1156             91                  21            441        714

  51               -39        1521             39                -31            961       1209

  73               -17         289             61                 -9              81         153

  111                21         441              80                 10            100        210

  57               -33       1089            47                -23           529      759      

$\sum X$ = 900               $\sum (d_x)^{2}$       $\sum Y$ = 700                      $\sum (d_y)^{2}$    3900    

                               = 6360                                              = 2868

Firstly, the mean of the Sales data is given by;

           Mean, $\bar X$  =  $\frac{\sum X}{n}$

                            =  $\frac{900}{10}$  = 90

And, the mean of the Purchase data is given by;

           Mean, $\bar Y$  =  $\frac{\sum Y}{n}$

                            =  $\frac{700}{10}$  = 70

Now, we have to find the regression coefficients;

• X on Y regression coefficient is given by;

                     $bxy=\frac{\sum d_x d_y}{\sum (d_y)^{2} }$

                            =  $\frac{3900}{2868}$  = 1.36

• Y on X regression coefficient is given by;

                     $byx=\frac{\sum d_x d_y}{\sum (d_x)^{2} }$

                            =  $\frac{3900}{6360}$  = 0.61

Now, the regression equation of Sales on Purchases (i.e. X on Y) is given by;

  $(X-\bar X) = bxy(Y- \bar Y)$

   $(X-90) = 1.36 \times (Y- 70)$  

   $(X-90) = 1.36Y- 95.2$

   $X = 1.36 Y -95.2+90$

   $X = 1.36 Y -5.2$

Similarly, the regression equation of Purchase on sale (i.e. Y on X) is given by;

  $(Y-\bar Y) = byx(X- \bar X)$

   $(Y-70) = 0.61 \times (X-90)$  

   $Y-70 = 0.61X-54.9$

   $Y= 0.61X-54.9+70$

   $Y= 0.61X-15.1$          

Answer 2

Step-by-step explanation:

The two regression equations are X = 1.36 Y -5.2X=1.36Y−5.2 and Y= 0.61X-15.1Y=0.61X−15.1 .

Step-by-step explanation:

Let the Sales be represented by variable X and Purchase be represented by variable Y.

The following data is represented below for computing regression equations;

Sales (X) X-\bar XX−

X

ˉ

(d_x)^{2}(d

x

)

2

Purchases (Y) Y-\bar YY−

Y

ˉ

(d_y)^{2}(d

y

)

2

d_x d_yd

x

d

y

(d_xd

x

) (d_yd

y

)

91 1 1 71 1 1 1

97 7 49 75 5 25 35

108 18 324 69 -1 1 -18

121 31 961 97 27 729 837

67 -23 529 70 0 0 0

124 34 1156 91 21 441 714

51 -39 1521 39 -31 961 1209

73 -17 289 61 -9 81 153

111 21 441 80 10 100 210

57 -33 1089 47 -23 529 759

\sum X∑X = 900 \sum (d_x)^{2}∑(d

x

)

2

\sum Y∑Y = 700 \sum (d_y)^{2}∑(d

y

)

2

3900

= 6360 = 2868

Firstly, the mean of the Sales data is given by;

Mean, \bar X

X

ˉ

= \frac{\sum X}{n}

n

∑X

= \frac{900}{10}

10

900

= 90

And, the mean of the Purchase data is given by;

Mean, \bar Y

Y

ˉ

= \frac{\sum Y}{n}

n

∑Y

= \frac{700}{10}

10

700

= 70

Now, we have to find the regression coefficients;

X on Y regression coefficient is given by;

bxy=\frac{\sum d_x d_y}{\sum (d_y)^{2} }bxy=

∑(d

y

)

2

∑d

x

d

y

= \frac{3900}{2868}

2868

3900

= 1.36

Y on X regression coefficient is given by;

byx=\frac{\sum d_x d_y}{\sum (d_x)^{2} }byx=

∑(d

x

)

2

∑d

x

d

y

= \frac{3900}{6360}

6360

3900

= 0.61

Now, the regression equation of Sales on Purchases (i.e. X on Y) is given by;

(X-\bar X) = bxy(Y- \bar Y)(X−

X

ˉ

)=bxy(Y−

Y

ˉ

)

(X-90) = 1.36 \times (Y- 70)(X−90)=1.36×(Y−70)

(X-90) = 1.36Y- 95.2(X−90)=1.36Y−95.2

X = 1.36 Y -95.2+90X=1.36Y−95.2+90

X = 1.36 Y -5.2X=1.36Y−5.2

Similarly, the regression equation of Purchase on sale (i.e. Y on X) is given by;

(Y-\bar Y) = byx(X- \bar X)(Y−

Y

ˉ

)=byx(X−

X

ˉ

)

(Y-70) = 0.61 \times (X-90)(Y−70)=0.61×(X−90)

Y-70 = 0.61X-54.9Y−70=0.61X−54.9

Y= 0.61X-54.9+70Y=0.61X−54.9+70

Y= 0.61X-15.1Y=0.61X−15.1