Compute the R 2 and p-values for the linear, simple quadratic, power, and expone

Question

Compute the R2 and p-values for the linear, simple quadratic, power, and exponential models.

X Y X^2 Y^2 ln(X) ln(Y) 11 18 121 324 2.3979 2.8904 13 15 169 225 2.5649 2.7081 14 12 196 144 2.6391 2.4849 14 17 196 289 2.6391 2.8332 14 19 196 361 2.6391 2.9444 14 28 196 784 2.6391 3.3322 15 24 225 576 2.7081 3.1781 16 20 256 400 2.7726 2.9957 17 18 289 324 2.8332 2.8904 17 19 289 361 2.8332 2.9444 17 24 289 576 2.8332 3.1781 17 42 289 1764 2.8332 3.7377 18 6 324 36 2.8904 1.7918 18 17 324 289 2.8904 2.8332 18 19 324 361 2.8904 2.9444 18 21 324 441 2.8904 3.0445 18 39 324 1521 2.8904 3.6636 19 32 361 1024 2.9444 3.4657 20 21 400 441 2.9957 3.0445 22 17 484 289 3.0910 2.8332 22 31 484 961 3.0910 3.4340 22 34 484 1156 3.0910 3.5264 22 35 484 1225 3.0910 3.5553 22 51 484 2601 3.0910 3.9318 23 29 529 841 3.1355 3.3673 24 20 576 400 3.1781 2.9957 24 31 576 961 3.1781 3.4340 25 40 625 1600 3.2189 3.6889 26 22 676 484 3.2581 3.0910 26 22 676 484 3.2581 3.0910 26 33 676 1089 3.2581 3.4965 26 38 676 1444 3.2581 3.6376 26 43 676 1849 3.2581 3.7612 26 83 676 6889 3.2581 4.4188 27 37 729 1369 3.2958 3.6109 27 62 729 3844 3.2958 4.1271 27 79 729 6241 3.2958 4.3694 28 31 784 961 3.3322 3.4340 28 46 784 2116 3.3322 3.8286 28 53 784 2809 3.3322 3.9703 28 55 784 3025 3.3322 4.0073 28 92 784 8464 3.3322 4.5218 30 19 900 361 3.4012 2.9444 30 34 900 1156 3.4012 3.5264 30 41 900 1681 3.4012 3.7136 30 47 900 2209 3.4012 3.8501 30 50 900 2500 3.4012 3.9120 30 73 900 5329 3.4012 4.2905 31 30 961 900 3.4340 3.4012 31 39 961 1521 3.4340 3.6636 32 60 1024 3600 3.4657 4.0943 33 44 1089 1936 3.4965 3.7842 33 74 1089 5476 3.4965 4.3041 34 79 1156 6241 3.5264 4.3694 34 88 1156 7744 3.5264 4.4773 35 41 1225 1681 3.5553 3.7136 35 60 1225 3600 3.5553 4.0943 35 89 1225 7921 3.5553 4.4886 35 77 1225 5929 3.5553 4.3438 36 43 1296 1849 3.5835 3.7612 36 72 1296 5184 3.5835 4.2767 36 94 1296 8836 3.5835 4.5433 36 133 1296 17689 3.5835 4.8903 36 118 1296 13924 3.5835 4.7707 36 140 1296 19600 3.5835 4.9416 38 91 1444 8281 3.6376 4.5109 38 122 1444 14884 3.6376 4.8040 38 141 1444 19881 3.6376 4.9488 38 170 1444 28900 3.6376 5.1358 39 129 1521 16641 3.6636 4.8598 39 118 1521 13924 3.6636 4.7707 39 98 1521 9604 3.6636 4.5850 40 72 1600 5184 3.6889 4.2767 40 101 1600 10201 3.6889 4.6151 40 125 1600 15625 3.6889 4.8283 40 144 1600 20736 3.6889 4.9698 41 161 1681 25921 3.7136 5.0814 42 155 1764 24025 3.7377 5.0434 2167 4567 65677 402017 255.2443 297.6226 78 n 77 df E[X] 27.7821 E[Y] 58.5513 E[X^2] 842.0128 E[Y^2] 5154.0641 V[X] 71.0818 (using n-1) V[Y] 1748.2246 (using n-1) sX 8.4310 sY 41.8118

Explanation / Answer

I am using excel here to produce result of given peroblems.

(1) Linear :

The regression output

Here R2 = 0.8083

p - values = 3.67 x 10-19 the model is overall significant here.

(2) Simple quadratic model.

R2 = 0.7424

P- value = < 0.0001 for each variable.

(3) For power model; we have to take regression between ln x and lny

R2 = 0.6738

p - value = 3.6 x 10-20

(4) Exponentital model: y = aex

ln y = ln a + x so now we do regession between ln y and x

R2 = 0.7194

Here p- value = 1.14 x 10-22

So, quadraic model is best here.  

SUMMARY OUTPUT Regression Statistics Multiple R 0.808306 R Square 0.653358 Adjusted R Square 0.648797 Standard Error 24.77866 Observations 78 ANOVA df SS MS F Regression 1 87950.66 87950.66 143.2463 Residual 76 46662.63 613.982 Total 77 134613.3 Coefficients Standard Error t Stat P-value Intercept -52.8165 9.718802 -5.43446 6.39E-07 X 4.008622 0.334929 11.96856 3.67E-19

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