Distance a Paper Airplane Flies? Solution

Introduction




For our experiment, we wanted to test whether or not weight location and amount of weight a ected the distance a paper airplane could y. Our experi-ment is a two factor factorial design where we had three levels for each of our two factors, which were the amount of paper clips on the plane during ight, and the location of those paper clips on the plane. Our levels for the amount of paper clips were one, three, and ve paper clips and were put on the very front of the paper airplane, the back of the wings, and the body in the back of the airplane.




We collected our data on the lawn of the Communications Facility on Wednes-day, May 29th, where we had a tape measure on the ground and threw the paper airplanes at each level three times. We tried to do our experiment relatively quickly so the wind wouldn’t change direction suddenly and kept the same per-son throwing the airplane to keep the experiment consistent. However, there is always some human error involved no matter what. The data we collected is in Figure 1 given below.






Number of Paper Clips








Location of Paper Clips
1
3
5








Front
126, 132, 147
128, 129, 153
114, 130, 129








Back Body
147, 121, 121
122, 175, 122
228, 156, 215








Back Wings
193, 153, 147
210, 126, 131
102, 89, 140








Figure 1: Table of Data Collected.








Analysis of Data







Figure 2: ANOVA Table for Data Collected.




From the ANOVA table seen in Figure 2, we can see that the only signi cant factor from our data is the interaction term for the two factors. We can also

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tell that there is signi cant interaction from the plot given in Figure 3 since the lines for the di erent factor levels are not parallel. Therefore, we will keep all of the factors in the model when doing further analysis of the data.




Figure 3: Interaction Plot.




Since our experiment has a two-factorial design, we will do a Tukey and a Fisher’s LSD pairwise comparison test to see if there is a di erence in mean ight distance at the di erent levels for our location factor.




Now, for the Tukey pairwise comparison with F W ER = 0:05, we will use the following test statistic,




qF W ER;a;edf r
SE


r
731:1
M
= q0:05;3;18




bn
9



= (3:609304)(9:012954)




= 32:53049




Then, nding the treatment means for all of the di erent levels of the location












factor, we get,




y1 = 143




y2 = 144




y3 = 144:778:




Then taking all of the possible combination di erences, we nd




y1
y2 =
1
y1
y3 =
1:778
y2
y3 =
0:7778:



Since the absolute values of all of these di erences are less than the critical value from above, we can conclude that there are no signi cant di erences in the mean ight distance for all of the di erent levels of the location factor.




To check and see if this is actually the case, we will also perform Fisher’s LSD pairwise comparisons, at = 0:05, on the locations factor to con rm the test above. For this comparison, we will have

t =2;edf r








= t0:025;18
r






2
bn
9








M SE


2(731:1

























= (2:100922)(12:74624)




= 26:778:




Once again, none of the di erence between the factor levels are greater than this critical value, so there is no signi cant di erence in mean ight distance between the three di erent levels of location. A graphical representation of this is given in Figure 4 below.




Based on the data we collected and the interaction plot given above, the best combination of our two factors would be the third level of the weight, which was having ve paper clips on the plane, and the second level of location, which was having the paper clips on the back of the body of the plane.This combination had a mean distance traveled of 199:667, which ew 35:33 inches further than the next highest treatment-factor combination.








Figure 4: Graphical Representation of Signi cant Di erences of Location




Factor.







Residuals Analysis













Figure 5: Residual Plots.




Looking at the histogram of the residuals and the normal probability plot from Figure 5, nothing seems to be violating the assumption of normality. We also ran an Anderson-Darling Test on the residuals, which returned a p-value of 0:1462, which is large enough for us to no reject the null hypothesis of normality. Also, the plot of studentized residuals vs. treatment means appear to be randomly


scattered throughout, implying constant variance throughout. Therefore, we can conclude that none of the assumptions for the model are violated.







Conclusion




The distance traveled by the planes may have been in uenced by a few outside factors including the human error of consistently throwing the planes (i.e. initial angle, speed, height of plane) and any wind from collecting the data outdoors. If we were to redo the experiment, we would prefer an indoor location, as well as some type of automated plane launcher in order to stabilize the trajectory and force at which the plane is thrown. Nevertheless it is clear from the interaction plot that with more weight on the body of the plane, the average distance the plane travels will increase. It was interesting to learn that varying the location of the weight as well as the weight itself impacted the travel distance.









































































































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