Question

2) Use the least square regressing to fit a curve on the form: y = a + bx’ suitable for this data X 0 1 2 3 4 5 у 1.0 3.0 15 20 140 250 Compute the standard error of estimate and the correlation coefficient Final answer: y = -5.4577 + 2.0522 x, r² = 0.977

Please can you solve it on a paper

Answer 1

We know that for a polynomial equation of the form

The polynomial regression model Yi Bo + Bixi + B2x} + ... + Bmx + εi (i = 1, 2, ..., n) =

Then the model can be written as a system of linear equations as follows

[1 Yi 21 x" E1 Y2 1 22 c! m 2 . Bo Bi B2 E2 חוז. 1 13 Y3 + = 3 E3 ... Lyn ] [i 2 27 Lßm Len.

So we have

Bo 0 y ya YS & Ez Ez ey Yo E6 15 20 घ Ez 3 Eu Eig B3 140 250 127 64 125 رمی کے 1= Bot & 32 Bot Bz+E2 15: BoT883 + Ez 20= $+27B3 +64 140 Bo + 648₂ +85 150 - Bo + 12553 +5 Xra 0 1 & 27 64 125 225 0-001 Xp.x= 225 20 515 (P+X) = (0.2831 & X5 ZX= (0.2831 -0.0031 0.0000829 -0.0031 0.8439 -0105 0.2583 0.1493 0.2799 0.0022 0.0072 TO-0031 -0.003 -0.0024 -0.000+ * 3 15 20 140 250

HT (45.457. 2:05 22 Bo = -5.457. ß3= 2.6522 so regression equation is gewon by -5.457 +2:0522.23

So the R² can be computed as follows

R² = Sum of (y_predicted - y_mean)² / Sum of (y-y_mean)²

= 50864/69126/52061.5 = .977

х у 0 1 1 3 2 15 (X - Mean(x)) (Y - Mean(Y)) (X - Mean(X)) * (Y - Mean(Y)) -2.5 -70.5 176.25 -1.5 -68.5 102.75 -0.5 -56.5 28.25 0.5 -51.5 -25.75 1.5 68.5 102.75 2.5 178.5 446.25 830.5 (X - Mean(x))? Y predicted (Ypredicted - Ymean? 6.25 -5.457 5922.379849 2.25 -3.4048 5610.729063 0.25 10.9606 3665.018952 0.25 49.9524 464.2990658 2.25 125.8838 2957.597702 6.25 251.068 32244.66662 17.5 50864.69126 (y - Ymean)? 4970.25 4692.25 3192.25 2652.25 4692.25 31862.25 3 20 4 140 250 5 15 52061.5 429 71.5 Mean 2.5 R? 0.97701164 y= a + bx a -5.457 2.0522 b

Correlation coefficient is calculated as follows

the formula known as the Pearson correlation coefficient, shown below: р = (–a)(*_*)()

So for our data we find that corelation coefficient = 1/5 * 0* 0 = 0

X у 0 1 1 3 2 15 (X - Mean(X)) (X - Mean(x))/stdev(x) -2.5 -1.33630621 -1.5 -0.801783726 -0.5 -0.267261242 0.5 0.267261242 1.5 0.801783726 2.5 1.33630621 0 0 (Y - Mean(Y)) (Y - Mean(Y))/(stdev(y)) -70.5 -0.690900935 -68.5 -0.671300909 -56.5 -0.55370075 -51.5 -0.504700683 68.5 0.671300909 178.5 1.749302369 0 0 3 20 4 140 5 250 15 429 2.5 71.5 1.87082869 102.040678 Mean Std dev

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