R version 4.2.3 (2023-03-15 ucrt) -- "Shortstop Beagle"
Copyright (C) 2023 The R Foundation for Statistical Computing
Platform: x86_64-w64-mingw32/x64 (64-bit)

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[Previously saved workspace restored]

> ls()
[1] "pontok"
> pontok
            x        y
1  -1.8128998 1.266693
2  -1.7800062 1.715194
3  -1.5376151 1.530396
4  -1.3854859 2.724910
5  -1.3637005 1.542313
6  -1.0637228 2.884884
7  -1.0034652 2.368573
8  -0.5760387 3.842141
9  -0.5117520 3.261627
10 -0.5016896 3.917820
11 -0.4181832 3.398032
12 -0.2633086 4.391169
13 -0.2026714 3.030846
14 -0.1716494 2.885382
15 -0.1361445 4.168821
16 -0.1220230 4.568679
17  0.1114579 4.560052
18  0.1870899 4.385610
19  0.2119255 3.720698
20  0.4965816 2.757011
21  0.6245640 3.280524
22  0.6773494 3.573479
23  0.6988148 4.350537
24  0.7129744 2.681535
25  0.7543323 3.098144
26  0.9234113 3.063429
27  1.0619507 2.809594
28  1.3010199 2.545490
29  1.4286769 2.060928
30  1.7552736 0.941456
> x
Error: object 'x' not found
> y
Error: object 'y' not found
> attach(pontok)
> x
 [1] -1.8128998 -1.7800062 -1.5376151 -1.3854859 -1.3637005 -1.0637228
 [7] -1.0034652 -0.5760387 -0.5117520 -0.5016896 -0.4181832 -0.2633086
[13] -0.2026714 -0.1716494 -0.1361445 -0.1220230  0.1114579  0.1870899
[19]  0.2119255  0.4965816  0.6245640  0.6773494  0.6988148  0.7129744
[25]  0.7543323  0.9234113  1.0619507  1.3010199  1.4286769  1.7552736
> plot(x,y)
> plot(x,y,type="l")
> reg1 <- lm(y ~ x)
> abline(reg1$coef,col="blue")
> reg2 <- lm(y ~ x + I(x^2))
> reg2

Call:
lm(formula = y ~ x + I(x^2))

Coefficients:
(Intercept)            x       I(x^2)  
    3.82095     -0.03571     -0.85048  

> lines(x,reg2$fitted,col="red")
> plot(x,reg2$resid)
> summary(reg2)

Call:
lm(formula = y ~ x + I(x^2))

Residuals:
     Min       1Q   Median       3Q      Max 
-0.91664 -0.32332 -0.00554  0.34234  0.96987 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.82095    0.13260  28.816  < 2e-16 ***
x           -0.03571    0.10547  -0.339    0.738    
I(x^2)      -0.85048    0.09949  -8.549 3.66e-09 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5348 on 27 degrees of freedom
Multiple R-squared:  0.7388,    Adjusted R-squared:  0.7195 
F-statistic: 38.19 on 2 and 27 DF,  p-value: 1.345e-08

> reg3 <- lm(y ~ I(x^2))
> reg3

Call:
lm(formula = y ~ I(x^2))

Coefficients:
(Intercept)       I(x^2)  
     3.8156      -0.8422  

> plot(x,y)
> lines(x,reg3$fitted)
> anova(reg1,reg2)
Analysis of Variance Table

Model 1: y ~ x
Model 2: y ~ x + I(x^2)
  Res.Df     RSS Df Sum of Sq      F    Pr(>F)    
1     28 28.6218                                  
2     27  7.7217  1      20.9 73.081 3.663e-09 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> anova(reg2,reg3)
Analysis of Variance Table

Model 1: y ~ x + I(x^2)
Model 2: y ~ I(x^2)
  Res.Df    RSS Df Sum of Sq      F Pr(>F)
1     27 7.7217                           
2     28 7.7544 -1 -0.032792 0.1147 0.7375
>