The maple is a common tree found across the northern hemisphere known for their vibrant autumn colors. Maples are also known for their interesting seed dispersal methods. These seeds, or samaras, are often called helicopters or whirlybirds due to their distinct two-winged appearance. By having two papery wings, the samaras are able to spin to the ground while being picked up by the wind, allowing them to travel farther from the parent tree. As a group of botanists, we are interested in studying the design of maple seeds and the way in which they fly. Samaras typically grow in sets of two wings per seed, with a number of natural growth factors affecting the rate in which they spin, including the weight of the seeds and wings. By studying this design, we are hoping to find which variables allow the seeds to travel farthest, making them more optimal for populating an area.
Experiment Explanation/ How Data Was Collected
For our experiment we chose to test these three factors: number of paperclips, number of staples, and paper type. Our factors varied between notebook or construction paper; 1, 2, or 3 paper clips on the body of the helicopter; and 0, 1, or 2 staples on the model helicopter wings. The number of paperclips were chosen to represent differing body weights. The number of staples were chosen to see if differing wing weights would affect the flight duration. The differing types of paper that the helicopter was constructed out of was used to represent if an overall different weight would affect the flight time. When we created our helicopters, the wings were half of the piece of paper, and the body was cut halfway down into the paper. The wings were not folded. This was kept consistent with all of the helicopters. Our group numbered the helicopters in order to keep them organized while collecting data, and had the same person drop the helicopter from the top of a staircase in Violette Hall for each trial. This helped to keep data consistent. We randomized the order of which the helicopters were flown. We began keeping time from when the helicopter was dropped to the moment it hit the floor for each trial. The factor variations were run five times in a random flight order.
This was a 2 by 3 Factor Analysis experiment. A confounding variable could have been the trial number. As each airplane was flown, wear and tear could have occurred each time it hit the floor, affecting later trials. Our team tried to control for as many third variables as possible. For instance, we had one person drop the plane each time, one person made the measurements for all planes, one person folded the plane into formation, and we had one person placing the paperclips and staples in the same place.
I n t e r a c t i o n s Error
yijkl = M + ?j + ?k + ?l + ??jk + ??jl + ??kl + ???jkl + eijkl
Y= Flight time, ? = Paperclips, ? = Staples, ? = Paper type
i = 1 to 5, j = 1 to 3, k = 0 to 2, l = 1 to 2
S? = S? = S? = S?? = S?? = S?C = S??? = 0 e~Nind(0, s2)
M = Center
After entering our data, we ran the tests for equal variance (see Figure 1 in Appendix) and normality. For the equal variance test, the null hypothesis was that all variances were equal among groups, and the alternate hypothesis was there there was not equal variance. We used the rejection region of a p-value less than 0.05, and since our p-value was 0.206, we could not reject the null hypothesis, and it was safe to assume equal variance. For our normality test, the null hypothesis was that the errors were normal (the alternate hypothesis being that they were not). Since our p-value was greater than the rejection region of 0.05 at a value of 0.422, and AD value of 0.368, it was safe to assume normality as well.
Following the initial tests, we ran the ANOVA (see Figure 2 in Appendix). We chose the General Linear Model because there were multiple factors and interactions to take into account. Upon receiving our output, we first looked to see if there was a three-way interaction effect. Our null hypothesis was that there was no effect, and our alternate hypothesis was that there was. With the rejection region of a p-value less than 0.05, and with our p-value being 0.019, we rejected the null hypothesis and concluded that there was a three-way interaction effect.
Since there was a three-way interaction, we could then stop with the testing. However, if one were to continue, they would see there was a significant paper clip*staple interaction (p-value of 0.0001) and a paper clip*paper type interaction (p-value of 0.015). These are considered two-way interactions and could be used if there had not been a three-way interaction effect.
Our results also indicated that there was a significant paperclip effect (p-value of 0.0001) and staple effect (p-value of 0.0001). These are considered main effects and would only be used if there were no found interaction effects.
We only included the three-way interaction plot (see Figure 3 in Appendix) since that held preference over the other interaction effects and main effects. As confirmed in the plot by the nonparallel lines, there is a three-way interaction occurring in our data. One factor (number of paperclips, number of staples, paper type) does not necessarily produce the highest or lowest outcome over any and all situations. However, according to the data, it appears that the longest flight time, an average mean of 4.51 seconds, was achieved by the helicopter with 1 paper clip, 0 staples, and made of notebook paper. The shortest flight time, an average mean of 2.19 seconds, was achieved by the helicopter with 3 paper clips, 1 staple in each wing, and made of notebook paper. Generally, according to the interaction plot, it appears that 1 paperclip, or a lesser body weight, resulted in the fastest flight time, while 3 paperclips resulted in the slowest. Zero staples resulted in the fastest flight for 1 and 2 paperclips, or a lesser wing weight, but did not change much for 3 paperclips. The notebook paper was fastest for 1 and 2 paperclips, or a lesser overall weight, but slowest for 3 paperclips. The paper type did not have much effect on the staples.
We also used Tukey’s pairwise method as a post-hoc test to compare the statistical significance of the means (see Figure 4 in Appendix). The helicopter with the longest flight time (a mean of 4.51 seconds) had statistical significance with three other helicopters (with means of 4.44, 3.88, and 3.48, in order of longer to shorter flight times). The helicopter with the shortest flight time (a mean of 2.19) had statistical significance with 12 other helicopters (with means of 2.30, 2.398, 2.42, 2.66, 2.70, 2.76, 2.79, 2.83, 2.96, 2.99, 3.00, and 3.31, in order of shorter to longer flight times).
We were able to conclude many things from our experiment. We assumed there was equal variance for all factors when we tested it because we got a larger p-value than .05. Once looking at the ANOVA table, we learned that there was a three-way interaction effect, because the p-value was smaller than .05. At this point, testing could be finished. Looking at figure 3 in the Appendix, it is apparent that the lines are intersecting and nonparallel, further proof that we could conclude a three-way interaction. However, we went on to see if there were any significant effects of the factors. We found that there were two two-way interactions: there was a significant paperclip by staple interaction as well as a paperclip by paper type interaction. As stated before, we would need this information if we weren’t able to find a three-way interaction. There was no evidence that one factor alone affected the helicopter flight time. The longest flight time average was a result of a helicopter with 1 paper clip, 0 staples, and made of notebook paper. The shortest flight time average resulted from a helicopter made with 3 paper clips, 1 staple in each wing, and notebook paper. However, looking at the interaction plot, a general conclusion can be drawn that lesser weight on various parts of the helicopter results in a longer flight time. We recommend for future replications of this experiment to be consistent with all factors of the helicopter: the wing length, the staples and paperclips being in the same positions, and to make sure that the folds are all folded at the same places. If one were to change our experiment, they could include construction paper as well, as a heavier paper option. Because the trials can result to damage to the helicopters and thus could possibly affect the results, the next group might want to make the same helicopter for the five different trials so that the effect of the damage can be kept minimal.
Figure 1: Bartlett’s Test for equal variance:
Figure 2: Normality test using residuals.
General Linear Model: flight time versus paperclips, staples, paper type
Factor Type Levels Values
paperclips Fixed 3 -1, 0, 1
staples Fixed 3 -1, 0, 1
paper type Fixed 2 -1, 1
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
paperclips 2 20.5909 10.2954 42.66 0.000
staples 2 6.4886 3.2443 13.44 0.000
paper type 1 0.0102 0.0102 0.04 0.837
paperclips*staples 4 5.8316 1.4579 6.04 0.000
paperclips*paper type 2 2.1644 1.0822 4.48 0.015
staples*paper type 2 0.5064 0.2532 1.05 0.356
paperclips*staples*paper type 4 3.0563 0.7641 3.17 0.019
Error 72 17.3780 0.2414
Total 89 56.0265
S R-sq R-sq(adj) R-sq(pred)
0.491285 68.98% 61.66% 51.54%
Figure 3: ANOVA Output
Figure 4: Interaction Plot for Paperclip*Staple*Paper Type for flight time.
Comparisons for flight time
Tukey Pairwise Comparisons: Response = flight time, Term = paperclips*staples*paper type
Grouping Information Using the Tukey Method and 95% Confidence
type N Mean Grouping
-1 -1 -1 5 4.512 A
-1 -1 1 5 4.442 A B
-1 0 -1 5 3.882 A B C
0 -1 -1 5 3.480 A B C D
0 0 1 5 3.376 B C D E
-1 0 1 5 3.312 C D E F
-1 1 -1 5 3.004 C D E F
-1 1 1 5 2.990 C D E F
0 1 -1 5 2.962 C D E F
1 1 1 5 2.828 C D E F
0 -1 1 5 2.792 C D E F
1 -1 1 5 2.762 C D E F
1 0 1 5 2.704 D E F
0 0 -1 5 2.656 D E F
1 -1 -1 5 2.424 D E F
0 1 1 5 2.398 D E F
1 1 -1 5 2.302 E F
1 0 -1 5 2.190 F
Means that do not share a letter are significantly different.
* NOTE * Cannot draw the interval plot for the Tukey procedure. Interval plots for
comparisons are illegible with more than 45 intervals.
Figure 5: Tukey’s Pairwise Comparisons (Post Hoc Test)