371chapter6f2011 - Chapter 6 Bernoulli Trials Revisited 6.1...

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Chapter 6 Bernoulli Trials Revisited 6.1 Is p Constant? Recall that there are three assumptions to BT. The frst, that each trial yields a dichotomy, is easy to veriFy or reFute. The other two assumptions can be neither verifed nor reFuted with certainty. In this section we present methods For investigating the second assumption, that the probability oF success remains constant. Our methods will consist oF inFormal descriptive techniques and Formal tests oF hypotheses. Both oF these categories oF methods are very useFul to a scientist. IbeginwithanextendedexampleoFdataIcollectedcirca1990onTetris. Tetris is a video game (I Fear that this expression—video game—is hopelessly dated; perhaps you can give me a better name to use For the next version oF thesenotes!)thatcanbeplayedona variety oF ‘systems.’ IF you have never played Tetris, then the Following description oF the game may be diFfcult to Follow. Tetris rewards spatial reasoning and, not surprisingly, lightning re±exes. Essentially, one oF seven possible geometrical shapes oF Four blocks ‘Falls’ From the top oF the screen. The player translates and/or rotates the Falling shape in an attempt to complete, with no gaps, a horizontal row oF blocks. Each completed row oF blocksd isappearsFromthescreenand blocks above, iF any, drop down into the vacated space to allowroomFormoreFa l l ingshapes .A player’s score equals the number oF rows completed beFore thescreenover±ows,whichmeansthat the current game is over. AFter every ten completed lines the speed oF the Falling shapes increases making the game more diFfcult. Each drop oF a shape by the computer can be viewed as a trial. Each trial has seven possible outcomes. ²or now, the shapes are divided into two types: a straight row (my Favorite, labeled a success) and any other shape (a Failure). Iobserved1,872trialsduringeightgamesoFTetrisonmyson’s Nintendo System. The methods IpresentbelowarebasedontakingthetotalnumberoFtrials,1,872inthecurrentcase,anddividing them into two or more segments. The choice oF segments is a matter oF taste and judgment and
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Table 6.1: A Comparison of Eight Games of Tetris. Frequencies Row Proportions Game S F Total SF T o t a l First 22 171 193 0.114 0.886 1.000 Second 33 185 218 0.151 0.849 1.000 Third 36 215 251 0.143 0.857 1.000 Fourth 25 206 231 0.108 0.892 1.000 Fifth 30 198 228 0.132 0.868 1.000 Sixth 42 220 262 0.160 0.840 1.000 Seventh 33 215 248 0.133 0.867 1.000 Eighth 33 208 241 0.137 0.863 1.000 Total 254 1618 1872 0.136 0.864 1.000 Figure 6.1: Plot of Proportion of Success for Eight Games of Tetris.
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This note was uploaded on 12/10/2011 for the course STATS 371 taught by Professor Hanlon during the Fall '11 term at University of Wisconsin.

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371chapter6f2011 - Chapter 6 Bernoulli Trials Revisited 6.1...

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