Helicopter Theme and Variations

Transcription

1 Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the helicopter is dropped The mount of weight dded to the helicopter Whether the helicopter is dropped indoors or outdoors The mteril from which the helicopter is mde Which direction the rotors re bent (clockwise vs. counterclockwise) wilder vritions could include: Vritions on the bsic design of the helicopter Punching holes in the helicopter rotors Some possible response vribles re: How long the helicopter tkes to fll How fr the helicopter lnds from desired (point) trget Whether the helicopter hits desired (lrge) trget wilder vritions could include: Whether someone is ble to ctch the helicopter How ttrctive the flight of the helicopter is, subjectively judged by n observer Useful mterils for helicopter experiments: Helicopters of severl designs Stopwtches Mesuring tpes or meter sticks Pper clips (for vrying weight) or hevy binder clips for outdoor drops Msking tpe (to mrk drop heights on verticl wll nd to mrk trget) Pendulum (to locte point directly beneth drop) Crds, dice, or rndom digit tble A bsket or binder clip with long string ttched will gretly ese the return of helicopter from its lnding site to the pltform from which it ws dropped. NCSSM pge 1

2 Some prticulr experimentl designs re: One smple t: 1. Do we live in vcuum? If ir resistnce hs no effect on the fll of helicopter, then the durtion of its fll should be clculble from the physics lw 1 2 2h m h= 2 gt ; or, T =, where g = 9.8 g 2, h is the height of the fll in meters, nd s T is the time of the fll in seconds. Hve one person drop single helicopter 3 times from fixed height nd hve nother person time the descents. Clculte the theoreticl time to fll bsed on the ssumption of no ir resistnce, nd cll it 25m T. (E.g. from height of 5 meters, T = 9.8m s 2 1sec.) Using one-smple t test of significnce, test the hypothesis µ time = T vs. µ time > T. (Hopefully, the effect of ir resistnce cn be detected with very high level of significnce!) 2. How long does this helicopter tke to fll, on verge? One person drops single helicopter 3 times from fixed height. Estimte the men time for it to fll using one-smple t confidence intervl. In this cse, the popultion of inference is only drops by this prticulr dropper with his or her prticulr helicopter. In order to generlize to helicopter design (e.g., ll long-rotor helicopters) use rndom smple of 3 different long-rotor helicopters. Note: The extrpoltion to the popultion of ll humn droppers my be vlid, but it could only be done bsed on belief, not on sttisticl evidence. We often do this without thinking bout it. E.g., suppose tht we re testing long-rotor helicopters vs. short-rotor helicopters for some property, nd for convenience, we mke ll the long-rotors blue nd ll the short-rotors yellow. How do we know tht ny observed differences re due to rotor length nd not to color? We believe color hs no effect bsed on plusible resons for different helicopter behvior, but there is no sttisticl evidence tht the confounding vrible of color did not hve n effect. 3. How close to trget cn I get this helicopter to lnd, on verge? This vrition is just like experiment 2, bove, except tht the response vrible mesured is the distnce tht the helicopter lnds from desired point trget, rther thn the time it took to fll. NCSSM pge 2

3 Pired differences t: 4. Does this long-rotor helicopter tke different length of time to fll, on verge, thn this short-rotor helicopter? Mke one long-rotor nd one short-rotor helicopter. All drops of both helicopters will be performed by the sme person nd from the sme height. Flip coin to see which one will be dropped first; fter it is dropped, then drop the other one. Repet this process 3 times, lwys dropping one nd then the other. The pired t test of significnce is pproprite for this experimentl design becuse the helicopter drops re pired by order. Once the dt re collected, use t test on the pired differences to see whether there is significnt evidence ginst H : µ long short = vs. H long short. Alterntively, crete confidence intervl estimte of µ long short. In either cse, the scope of inference is ll the possible drops of these two prticulr helicopters with this prticulr dropper. (It is not possible, sttisticlly, to generlize to the long- nd short-rotor designs.) 5. Do helicopters tke different length of time, on verge, to fll indoors versus outdoors? Mke 25 long-rotor helicopters. Ech helicopter is dropped once indoors nd once outdoors from the sme fixed height. As in experiment 4, do the drops in pirs, using coin flip for ech tril to determine whether the helicopter is dropped indoors first nd then outdoors, or vice-vers. Mesure the times of descent, nd test H in out = vs. H in out. Extrpolte to the popultion of ll long-rotor helicopters dropped by this dropper. A vrition would be to hve different dropper for ech tril. Then the results could be extrpolted to drops by ny dropper. 6. Are Glori nd Floyd eqully good, on verge, t mking their helicopters lnd ner specified trget? This experiment involves two droppers nd 2 longrotor helicopters, ll dropped from the sme fixed height. Choose helicopter t rndom nd flip coin to see who drops it first. Both droppers drop the sme helicopter, then move on to nother rndomly selected helicopter, gin choosing who drops first by rndom coin flip. The response vrible is the distnce from desired point trget tht the helicopter lnded. The pired t will test the significnce of evidence for H : µ Glori Floyd = vs. H Glori Floyd. Piring controls for wer nd ter on helicopter, incresed bility over time, environmentl fctors, etc. Any significnt difference my be extrpolted to Floyd s nd Glori s bility with ll long-rotor helicopters. NCSSM pge 3

4 Two smple t: 7. Do short-rotor nd long-rotor helicopters tke different lengths of time, on verge, to fll? Construct 25 short-rotor nd 25 long-rotor helicopters. One dropper (or ll different droppers, for lrger scope of inference) drops ll 5 helicopters from the sme fixed height nd their times of descent re mesured nd recorded. The order of the 5 drops is completely rndom determined, sy, by shuffling 25 blck crds nd 25 red crds nd then drwing them one t time, dropping short-rotor for the blck crds nd long-rotor for the red crds. This completely rndom order (s opposed to dropping the helicopters in pirs s in experiment 4) is wht mkes this design require two-smple t inference procedures rther thn one-smple t on pired differences. Test H : µ short = µ long vs. H short µ long or crete confidence intervl estimte of µ short µ long. 8. Do long-rotor helicopters tke different length of time, on verge, to fll indoors compred to outdoors? 4 different long-rotor helicopters re constructed nd re rndomly llocted to two groups: indoor nd outdoor. Ech is dropped once in its pproprite loction from the sme fixed height (obviously this is crucil!) with the order of the 4 drops being completely rndom, s in experiment 7. The time of descent is mesured nd recorded for ech drop. Use 2-smple t procedures to test whether the men flling time is different for indoors nd outdoors. One smple proportion: 9. How good is Brnby t hitting trget with long-rotor helicopters? 3 longrotor helicopters re constructed. A trget is drwn on the ground below dropping pltform. (The trget should be lrge enough tht it will get hit t lest 1 times nd will lso be missed t lest 1 times in order to hve np ˆ > 1 nd ( pˆ ) n 1 > 1.) One person drops ll 3 helicopters in rndom order nd records how mny of the drops hit the trget. A confidence intervl estimte of the proportion of hits is then constructed. NCSSM pge 4

5 Two smple proportions: 1. Do Zck nd Xen both hve the sme probbility of hitting trget with this prticulr helicopter? One long-rotor helicopter is constructed. Ech of two people will drop it 25 times towrds lrge trget from fixed height, with the 5 drops being performed in completely rndom order (s in experiment 7). The response vrible recorded is whether or not the helicopter hit the trget. One my then test the hypothesis H : pzck = pxen vs. H A: pzck pxenor crete confidence intervl estimte of pzck pxen. Vritions on this experiment: Do long- nd short-rotor helicopters hve n equl chnce of hitting trget? Is it esier for person to hit trget when dropping from lower height thn higher height? (This ltter vrition would be good exmple of one-sided test.) Chi-squre goodness-of-fit test: 11. Does this helicopter, under these environmentl conditions, hve directionl bis in where it lnds? Drw verticl nd horizontl line on the ground with their intersection directly beneth the dropping point. One person drops single long-rotor helicopter 4 times from fixed height. The qudrnt tht the helicopter lnds in is the response vrible. Under the hypothesis of no directionl bis, one would expect: H : pq1 = pq2 = pq3 = pq4 =.25. A chisqure goodness-of-fit test will test tht hypothesis. Evidence ginst it would suggest either directionlly-bised helicopter or, more likely, consistent wind direction. Chi-squre test of independence: 12. Is closeness to trget (concentric circles drwn round trget on the ground like drtbord) independent of rotor length? Construct 3 long-rotor nd 3 short-rotor helicopters. One person drops them in completely rndom order towrds drtbord trget drwn on the ground: two concentric circles seprting the ground into the regions close, medium, nd fr. Count how mny of ech type lnd in ech region nd test the independence of rotor length nd closeness to trget by using chi-squre sttistic on 2 3 tble. (Note: Evidence of difference in the two does not necessrily indicte tht either one is significntly better i.e., lnds significntly closer on verge thn the other. It only indictes difference in the distribution of where they lnd.) NCSSM pge 5

6 13. Is the probbility of hitting trget relted to drop height? This experiment is similr to number 12. Use low, medium, nd high for drop heights nd hit, not hit for the response vrible, with counts recorded in 3 2 tble. Use single helicopter nd dropper, or else single dropper with multiple helicopters of the sme type for lrger scope of inference. A chi-squre test of independence will indicte whether there is significnt evidence of ssocition between drop height nd successful trget-hitting. Vrition on this experiment: Is the probbility of hitting trget relted to the weight of the helicopter? Use pper clips, smll binder clips, nd lrger binder clips to crete different weights. Slope of Regression Line: Note: both of the experiments below re best done where helicopters cn be dropped from gret height (t lest 5 meters). 14. Does dding weight to helicopter speed up its descent? Construct 3 helicopters nd rndomly llocte them to 5 groups of 6, ech group of which will correspond to prticulr dded weight. In completely rndom order, drop ech helicopter from the sme height, recording the time of descent. Construct lestsqures line using the weight s the explntory vrible nd the time of descent s the response vrible. One my either estimte β with confidence intervl or test H : β = vs. H : β <. A 15. Do helicopters dropped from higher height tke longer on verge to fll thn helicopters dropped from lower height? Or: How fst do these helicopters fll once they rech their terminl velocity? Construct 5 long-rotor helicopters nd rndomly llocte them to 5 different fixed drop heights. Then mke ll 5 drops in completely rndom order, timing the descent of ech helicopter. Construct regression line using height s the explntory vrible nd time of descent s the response vrible. A test of H : β = vs. H A : β > should produce significnt evidence for the ltter. One my lso construct confidence intervl estimte of β, which is the reciprocl of the verge flling speed of the helicopters fter reching terminl velocity. (The intercept of the regression line results from the initil period of fluttering before the helicopter settles into its terminl velocity. It is likely to be negtive, since the helicopters tend to fll fster before they rech their terminl velocity, shortening the times of their flls.) NCSSM pge 6