Data set: WingLength2 If you completed the helicopter research project, this series of questions will help you

Question:

Data set: WingLength2
If you completed the helicopter research project, this series of questions will help you further investigate the use of regression to determine the optimal wing length of paper helicopters to maximize flight time. It is likely that your data from the project did not appear to lie along a straight line, but rather had a curved pattern. In this exercise, we will use a larger data set collected on six different wing lengths.
Figure 2.7 shows an individual value plot of the data, with five observations for each of the six wing length groups. The curved pattern is quite pronounced, and this makes sense. At some point, the wings are so long that the helicopter does not spin stably or is simply too heavy and falls faster. It appears that there is some optimal wing length around 9 or 10 cm.
a. Using the WingLength2 data set, try several transformations of either the response or the explanatory variable to see if you can alleviate the problem of nonlinearity.
b. It is likely that you cannot successfully find a transformation to solve the nonlinearity problem. A closer look at the residuals (Figure 2.8) helps to explain why. The residuals from the regression follow a somewhat sinusoidal pattern: down, then up, then down again. This is a pattern that is seen in a typical third-degree polynomial (y = ax3 + bx2 + cx + d). To create the appropriate regression model, we will introduce a new method called polynomial regression (instead of simple linear regression. In polynomial regression, we can fit the model
yi = Î²0 + Î²1xi + Î²2xi 2 + Î²3xi 3 + É›i for i = 1, 2,€¦, n where É›i ~ N( 0, Ïƒ2) ( 2.10)
Using statistical computing software, fit the model in Equation (2.10) and report estimates for Î²0, Î²1, Î²2, and Î²3.

Figure 2.7 Flight times for paper helicopters when dropped from a height of 8 feet, each with a small paperclip attached at the bottom of the base of the helicopter.

Figure 2.8 Residuals from an ANOVA and linear regression analysis of the flight times for paper helicopters from six groups: wing lengths 5.5 cm, 6.5 cm (standard), 7.5 cm, 8.5 cm, 19.5 cm, and 10.5 cm.
c. Look at a plot of the residuals from the polynomial regression versus the predicted values (i = b0 + b1xi + b2xi2 + b3xi3) and a normal probability plot of the residuals and comment on the validity of the regression model assumptions for these data.
d. Given your answer to Part B and the fact that estimated flight times are considered to be a smooth (polynomial) function of wing length according to the relationship i = b0 + b1xi + b2xi2 + b3xi3, estimate the optimal wing length that will lead to maximum flight time. Use calculus or visual inspection to identify a maximum that occurs within a reasonable range of wing lengths.
Polynomial regression allows us to create a function relating the explanatory and response variables, and this can be useful for predicting responses for levels of the explanatory variable that were not actually measured as a part of the experiment. Of course, we should take care not to predict responses for levels of the explanatory variable that are quite far from those used in the experiment. The regression model may not extend past the domain we were able to analyze.