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Statistics Assignment Help Statistics Homework Help

Alex Gerg

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Statistics Assignment Help | Statistics Homework Help

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Methods of Simple Regression Analysis:

Q 1. From the following data from the regression equations,

Y𝑒 = a + bX and 𝑋𝑒 = a + bY

Use the normal equation method:

X:

Y:

1

15

3

18

5

21

7

23

9

22

Also, estimate the value of Y when X=4, and the value of X when Y=24.

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Solution:

Formulation of the regression equations

X Y 𝑋2 𝑌2 XY

1

3

5

7

9

15

18

21

23

22

1

9

25

49

81

225

324

441

529

484

15

54

105

161

198

Total ∑X=25 ∑Y=99 ∑𝑋2=165 ∑𝑌2=2003

∑XY=533

N=5

(i) Regression equation of X on Y. This is given by

𝑋𝑒 = a + bY

To find the value of the constants a and b in the above formula, the following two normal

equation are to the simultaneously solved.

∑X=Na+b∑Y

∑XY=a∑Y+b∑𝑌2

Substituting the respective values in the above formula we get,

25=5a+99b

533=99a+2003b

Multiplying the equation (i) by 99 and eqn. (ii) by 5 and presenting them in the form of a

subtraction we get,

2475=495a+9801b

(-) 2665=495a+10015b

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_______________________________________

Thus -190= -214b

Or 214=190

b=190/214= .888 approx.

Putting the above values of b in the equ.(i) we get,

25=5a+99(.888)

Or 5a=25-87.912=-62.912

a= -62.912/5=-12.5824=-12.5824

Thus,

Substituting the above values of the constants a and b, we get the regression

equation of X on Y as,

𝐗𝐞 = -12.5824 + 0.888Y

Thus, when Y=24, 𝑋𝑒 = -12.5824 + 0.888(24)

=-12.5824+21.312

=8.7296

(ii) Regression equation of Y on X. this is given by

𝐘𝑒 = a + bX

as under:

To find the values of the constants a and b in the above formula, the following two normal

equations are to be simultaneously solved as under:

∑Y=Na+b∑X

∑XY=a∑X+b∑X2

Substituting the respective values in the above formula we get,

99=5a+25b

533=25a+165b

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Multiplying the equation (1) by 5 and getting the same subtracted from the equation

(ii) we get,

533=25a+165b

(-) 495=25a+125b

_______________________________________

Thus 38= 40b

Or b=38/40=15.05

Putting the above values of b in the equ.(i) we get,

99=5a+25(0.95)

Or 5a=99-23.75=75.25

a= 75.25/5=15.05

thus, a=15.05 and b=0.95

Substituting the above values of the constants a and b, we get the

regression equation of Y on x as,

𝑌𝑒 = 15.05 + 0.95X

Thus, when X=4, 𝑌𝑒 = 15.05 + 0.95(24)

=15.05+3.80

=18.85.

Note:

It may be noted that the above normal equation method of formulating the two regression

equation is verity lengthy and tedious. In order to do away with such difficulties, any of the

following two method of deviation may be used advantageously.

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