Fit a model of the form logy = b0 + b1x. Report the regression outputs and predi

Question

Fit a model of the form logy = b0 + b1x. Report the regression outputs and predict the value of y at x = 8.8491.

x y 5,7379 41,190185 6,9068 139,672445 7,4784 237,816650 7,0299 163,841582 6,9977 172,035354 7,2119 202,046931 6,3912 65,287461 6,2840 85,216920 7,1649 163,301796 6,7259 125,524380 7,0322 138,365675 6,7037 97,651010 6,1147 51,904215 7,3941 237,127981 7,8491 332,985851 6,9601 164,959502 6,5811 132,608344 6,4262 82,855655 7,5490 179,504450 7,1783 216,112693 6,8017 110,697361 7,1801 162,649894 6,8801 129,541332 6,7551 91,049198 6,3809 76,822687 6,4245 98,760724 7,3916 254,042100 6,9892 182,874477

Explanation / Answer

Solution

The given data along with z = logy to base 10 are given at the bottom.

Treating z as dependent variable and x as independent variable, the requisite Excel Computations are given below:

n

28

xbar

68,757

zbar

4.83631067

Sxx

603186963

Szz

0.02476322

Sxz

3860.62508

b1

6.4004E-06

b0

4.39623823

Thus, the fitted model is: logy = 4.396 + 0.0000064x ANSWER 1

[Formulae used in the above calculations are:

Mean X = Xbar = (1/n)sum of xi over I = 1, 2, …., n; ……………….(1)

Sxx = sum of (xi – Xbar)2 over i = 1, 2, …., n ………………………………..(2)

Mean Z = Zbar =(1/n)sum of zi over i = 1, 2, …., n;…………….(4)

Szz = sum of (zi – Zbar)2 over i = 1, 2, …., n ………………………………………………(5)

Sxz = sum of {(xi – Xbar)(zi – Zbar)} over i = 1, 2, …., n………(7)

Estimated Regression of Z on X is given by: Z = b0 + b1X, where

b1 = Sxz/Sxx and b0 = Zbar – b1Xbar..…………………….(8)]

Predicted value of logy at x = 8.8491 is: logycap = 4.396 + (0.0000064 x 88491)

= 4.962

So, Predicted value of y at x = 8.8491 is: 104.962 = 91622.05 ANSWER 2

x, y and z = logy values

I

x

y

Z = log10(y)

1

57,379

41,190,185

7.61479374

2

69,068

139,672,445

8.14511074

3

74,784

237,816,650

8.37624226

4

70,299

163,841,582

8.21442413

5

69,977

172,035,354

8.23561771

6

72,119

202,046,931

8.30545226

7

63,912

65,287,461

7.81482978

8

62,840

85,216,920

7.93052583

9

71,649

163,301,796

8.21299096

10

67,259

125,524,380

8.09872808

11

70,322

138,365,675

8.14102837

12

67,037

97,651,010

7.98967674

13

61,147

51,904,215

7.71520263

14

73,941

237,127,981

8.3749828

15

78,491

332,985,851

8.52242578

16

69,601

164,959,502

8.21737734

17

65,811

132,608,344

8.12257085

18

64,262

82,855,655

7.91832215

19

75,490

179,504,450

8.25407522

20

71,783

216,112,693

8.33468028

21

68,017

110,697,361

8.04413727

22

71,801

162,649,894

8.21125378

23

68,801

129,541,332

8.11240836

24

67,551

91,049,198

7.95927612

25

63,809

76,822,687

7.88548949

26

64,245

98,760,724

7.99458427

27

73,916

254,042,100

8.40490569

28

69,892

182,874,477

8.2621531

n

28

xbar

68,757

zbar

4.83631067

Sxx

603186963

Szz

0.02476322

Sxz

3860.62508

b1

6.4004E-06

b0

4.39623823

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