This is a problem on interpretation of regression equations which have quadratic terms in explanatory variables.

Context of data set: A paper helicopter experiment (http://www.paperhelicopterexperiment.com/) was run to find some optimal dimensions. Explanatory variables are body length and body width (both in cm) of a piece of paper before the folding/cutting is done to produce the helicopter. The response variable is the flight time (in seconds) to land on floor after release from a height of 2.5 m. You can compare multiple regressions from fitting (i) a plane, (ii) a quadratic with the original variables and (iii) a quadratic with centred variables (len.centered=len-6, wid.centered=wid-1.5).
For (ii) and (iii), compare the numerical results with what you would expect based on transformed equations.

For your data set, the response variable is: flight time, in seconds, of the paper helicopter.

flighttime=c(1.3, 1.43, 0.93, 2, 1.21, 1.73, 1.1, 1.41, 1.89, 1.98, 1.98, 1.25, 1, 1.85, 2.08, 1.94, 1.83, 1.69, 1.87, 1.82, 1.61, 1.69, 1.86, 0.93, 1.87)

The explanatory variables are:
(i) length of paper before folding
len=c(6.1, 6.1, 6.6, 6.6, 5.6, 6.1, 5.6, 6.1, 6.1, 6.6, 5.6, 5.6, 6.6, 6.6, 6.6, 6.6, 5.6, 5.6, 5.6, 6.1, 5.6, 5.6, 6.6, 6.6, 6.1)

(ii) width of paper before folding
wid=c(0.7, 0.7, 0.7, 2.1, 0.7, 1.4, 0.7, 0.7, 1.4, 1.4, 1.4, 0.7, 0.7, 1.4, 2.1, 1.4, 1.4, 2.1, 1.4, 2.1, 2.1, 2.1, 2.1, 0.7, 1.4)

For the $i$th case, let $x_{i1}$ be the value of length and $x^*_{i1}=x_{i1}-6$ be value of centered length; let $x_{i2}$ be the value of width and $x^*_{i2}=x_{i2}-1.5$ be value of centered width; and let $y_{i}$ be the value of flight time.
Consider two regression models that are quadratic in the original variables and quadratic in the centered variables.
and
To answer the parts below, several separate regressions could be done or you can answer based on theory.

Part a)
Compare the least squares coefficients for the two quadratic models. Maybe some coefficients are invariant to the centering.
Which of the following are correct? Possibly more than one item is correct.

A. ${\hat\beta}^*_5={\hat\beta}_5$
B. ${\hat\beta}^*_3={\hat\beta}_3$
C. ${\hat\beta}^*_1={\hat\beta}_1$
D. ${\hat\beta}^*_2\ne{\hat\beta}_2$
E. ${\hat\beta}^*_4\ne{\hat\beta}_4$
F. ${\hat\beta}^*_4={\hat\beta}_4$
G. ${\hat\beta}^*_2={\hat\beta}_2$
H. ${\hat\beta}^*_1\ne{\hat\beta}_1$
I. ${\hat\beta}^*_0={\hat\beta}_0$
J. ${\hat\beta}^*_3\ne{\hat\beta}_3$
K. ${\hat\beta}^*_0\ne{\hat\beta}_0$
L. ${\hat\beta}^*_5\ne{\hat\beta}_5$


Part b)
What is the relationship of the least squares coefficient ${\hat\beta}_1^*$ with centered length and width variables, and the least squares coefficient ${\hat\beta}_0,{\hat\beta}_1,\ldots, {\hat\beta}_5$ for the original length and width variables.
Which of the following are correct? There might be more than one correct answer.

A. ${\hat\beta}^*_1= {\hat\beta}_1-12{\hat\beta}_3-1.5{\hat\beta}_5$
B. ${\hat\beta}^*_1= {\hat\beta}_1$
C. ${\hat\beta}^*_1= {\hat\beta}_1+12{\hat\beta}_3+1.5{\hat\beta}_5$
D. ${\hat\beta}^*_1= {\hat\beta}_1+12{\hat\beta}_3$
E. ${\hat\beta}^*_1= {\hat\beta}_1-6{\hat\beta}_3$
F. ${\hat\beta}^*_1= {\hat\beta}_1-12{\hat\beta}_3$
G. ${\hat\beta}^*_1= {\hat\beta}_1+6{\hat\beta}_3$
H. None of the above



Part c)
Which of following would suggest that quadratic model is a better fit to the data?
There might be more than one correct answer.

A. At least one of $\beta_1,\beta_2$ is significant
B. At least one of $\beta_1^*,\beta_2^*$ is significant
C. Adjusted R2 is more than marginally larger for the quadratic
D. From the physics of the experiment, flight time cannot be linear in length and width of the paper over a wide range of inputs
E. At least one of $\beta_3,\beta_4,\beta_5$ is significant
F. None of the above



Part d)
If $\beta_2$ is not significant, then choose appropriate answers below.
There might be more than one correct answer.

A. width can be dropped from the model for an improved fit with regression on $x_1,x_1^2,x_2^2,x_1x_2$
B. one cannot make a conclusion because the coefficient of width in the quadratic model is not invariant to shifting / centering
C. width should not be dropped from the model for an improved fit
D. None of the above

Hint: