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Flying plastic helicopters

A company describes an experiment flying plastic helicopters where the objective is to maximize flight time. They used the central composite design shown in Table below. Each run involved a single helicopter made to the following specifications: x1 = wing area (in2), -1 = 11.80 and +1 = 13.00; x2 = wing-length to width ratio, -1 = 2.25 and +1 = 2.78; x3 = base width (in), -1 = 1.00 and +1 = 1.50; and x4 = base length (in), -1 = 1.50 and +1 = 2.50. Each helicopter was flown four times and the average flight time, and the standard deviation of flight time was recorded.
(a) Fit a second-order model to the average flight time response.
(b) Fit a second-order model to the standard deviation of flight time response.
(c) Analyze the residuals for both models from parts (a) and (b). Are transformations on the response(s) necessary? If so, fit the appropriate models.
(d) What design would you recommend maximizing the flight time?
(e) What design would you recommend maximizing the flight time while simultaneously minimizing the standard deviation of flight time?
Std order Run order Wing area Wing ratio Base with Base length Avg. flight time Std. Dev flight time
1 9 -1 -1 -1 -1 3.67 0.052
2 21 1 -1 -1 -1 3.69 0.052
3 14 -1 1 -1 -1 3.74 0.055
4 4 1 1 -1 -1 3.7 0.062
5 2 -1 -1 1 -1 3.72 0.052
6 19 1 -1 1 -1 3.55 0.065
7 22 -1 1 1 -1 3.97 0.052
8 25 1 1 1 -1 3.77 0.098
9 27 -1 -1 -1 -1 3.5 0.079
10 13 1 -1 -1 1 3.73 0.072
11 20 -1 1 -1 1 3.58 0.083
12 6 1 1 -1 1 3.63 0.132
13 12 -1 -1 -1 1 3.44 0.058
14 17 1 -1 1 1 3.55 0.049
15 26 -1 1 1 1 3.7 0.081
16 1 1 1 1 1 3.62 0.051
17 8 2 0 0 0 3.61 0.129
18 15 2 0 0 0 3.64 0.085
19 7 0 -2 0 0 3.55 0.1
20 5 0 2 0 0 3.73 0.063
21 29 0 0 -2 0 3.61 0.051
22 28 0 0 2 0 3.6 0.095
23 16 0 0 0 -2 3.8 0.049
24 18 0 0 0 2 3.6 0.055
25 24 0 0 0 0 3.77 0.032
26 10 0 0 0 0 3.75 0.055
27 23 0 0 0 0 3.7 0.072
28 11 0 0 0 0 3.68 0.055
29 3 0 0 0 0 3.69 0.078
30 30 0 0 0 0 3.66 0.058
Reconsider the plastic helicopter experiment in Problem above. This experiment was actually run in two blocks. Block 1 consisted of the first 16 runs in Table given above1 (standard order runs 1-16) and two center points (standard order runs 25 and 26).
(a) Fit the main-effects plus two-factor interaction models to the block 1 data, using both responses.
(b) For the models in part (a) use the two center points to test for lack of fit. Is there an indication that second-order terms are needed?
(c) Now use the data from block 2 (standard order runs 17-24 and the remaining center points, standard order runs 27-30) to augment block 1 and fit second-order models to both responses. Check the adequacy of the fit for both models. Does blocking seem to have been important in this experiment?
(d) What design would you recommend maximizing the flight time while simultaneously minimizing the standard deviation of flight time?

Sample Solution

Defined as “the sub-set of the selectorate whose support is necessary for the leader to remain in power”[20], the winning coalition, as shown above in Figure 3, is very important in determining whether a non-democratic regime can survive; the larger it becomes as a proportion of the selectorate, the greater the likelihood of the next most popular regime being able to take power. The size itself is mainly influenced by the type of authoritarian regime, and is particularly small in the case of monarchies, which, in the case of hereditary monarchies, only require the approval of a branch of the ruling family in order to survive. As explained by Bueno de Mesquita et al., “in autocratic systems, the winning coalition is often a small group of powerful individuals. [Thus] when a challenger emerges to the sitting leader and proposes an alternative allocation of resources, [the leader thwarts the challenge since he or she] retains a winning coalition”[21]; the size of which is in an inverse relationship with the likelihood of successful challenge, since fewer people must be ‘bought-off’. In fact, “the Selectorate Theory (Bueno de Mesquita et al., 2005) theorises that it is the size difference between the selectorate and the winning coalition […] that is most important”[22] in influencing the survival of non-democratic regimes. This theory has, however, received much criticism. Largely, the extent to which it is true, that having a small winning coalition is the most significant factor affecting the survival of non-democratic regimes, is dependent on how stable the regime appears to be, since “high political instability should reduce the effect of corruption, because actors have less incentive to bribe a government when it is unlikely to survive”[23], meaning the loyalty of the ruler’s winning coalition may become less effective. Thus, in reality, if a challenge to power did arise, the ruler may not be able to rely on his winning coalition if they were, in fact, more confident in the challenger overthrowing the incumbent, as in this circumstance it is highly likely that they would switch allegiances. Furthermore, Clark and Stone argue that Bueno de Mesquita et al.’s analysis “suffers from omitted variable analysis [which] can make the results appear stronger than they are. Once this error is corrected, the results are no longer interesting.”[24] This empirically undermines the foundations of the theory which Bueno de Mesquita et al. try to argue.>

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