Question

From the following data obtain the two regression equations.
Sales :
Purchase :
91
71
97
75
108
69
121
97
67
70
124
91
51
39
73
61
111
80
57
47​

Answer 1

Answer:

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