Lecture'11'The'normal'model''à'most'widely'used'probability'model''à'used'to'describe'continuous'numeric'variables''à'normal'probability'describes'variables'fairly'accurately''à'this'is'an'important'concept'in'Statistics'''Rationale'for'the'Normal'Model''à'Many'variables'when'graphed'are'“bell'shaped”,'however'these'variables'also'have'different'units,'means,'standard'deviations'ect'à'Because'of'all'these'different'variations'of'“bell'shaped”'graphs'it'would'be'useful'to'have'a'standard'off'of'which'you'could'make'comparisons'to'and'use'to'make'statements'about'probability'''Summary'à'normal'probability'distribution('bell'shaped'curve)'arises'naturally'due'to'small'errors(or'differences)'when'examined'as'a'while'and'they'center'around'a'single'value'à'many'important'variables'are'naturally'“normal”'à'in'order'to'make'comparisons'across'variables,'a'standard'would'be'helpful'''Visualizing'the'normal'Model''à'the'normal'model'is'appropriate'probability'for'a'continuous'random'variable'if:''• The'distribution'is'unimodal,'approx.'symmetric'and'bell'shaped.''à'a'normal'distribution'is'also'called'a'normal'curve'of'a'Gaussian'distribution.'''Examples'''While'these'are'not'perfectly'normal,'normal'model'describes'them'reasonably'well'''''Biological'variables'are'frequently'normally'distributed''''The'center'and'spread'of'the'Normal''''• !'stands'for'the'center'or'mean'of'a'distribution''Some ExamplesWhile these are not perfectly normal, the normal model describes them reasonably well. Biological variables are frequently normally distributed• !'stands'for'the'standard'deviation'of'a'distribution'• !"#$:!!!!"#!!'are'used'for'distributions'and'x'and's'are'used'for'sample'data'• 'Notation'and'Area''à'N(6,2)'means'the'normal'distribution'with'mean'!'='6'and'standard'deviaiton'! = 2''à'the'area'under'the'normal'curve,'above'the'x'axis'and'left'of'the'x=4'represents'P('x'< 4)''à'P('x'< 4)'='P'(x'≤ 4)!'for'a'continuous'variable''''AN'EXAMPLE'FROM'THE'TEXT''• Resurch'has'shown'that'the'mean'length'of'a'newborn'Pacific'harbot'seal'is'29.5'inches'and'the'standard'deviation'! = 1.2.!'Suppose'that'the'lengths'follow'the'Normal'model.'Find'the'probability'that'a'randomly'selevted'pup'will'be'more'that'32'inches.''à'The'answer'is'P(x'> 32) ! ≈ 0.019'• There'multiple'ways'to'get'the'answer'here'are'2:'à'the'first'say'is'using'a'computer''1. Go'to'http://socr.stat.ucla.edu/htmls/SOCR'Distributions.html'and'choose'the'Normal'Distribution''2. Plug'in'the'mean'and'the'standard'deviation'values''3. Move'the'slider'accordingly'''''The'answer'is'P(x'> 32) ! ≈ 0.019''àThe''second'way'is'by'using'the'standard'normal'table''• The'Standard'Normal''o Symetric,'bell'shaped,'the'“bell'curve”'o Mean'0'(symbol'!'),'standard'deviation'1'(symbol'!)'o The'medain'is'where'half'(50%)'of'the'observations'are'on'either'side''à'IN'THIS'TYPE'OF'DISTRIBUTION,'THE'MEAN'IS'ALWAYS'EQUAL'TO'THE'MEDIAN''àthe'values'on'the'horizonal'axis'are'called'Z'SCORES'or'STANDARD'UNITS.'Values'of'Z'above'the'median'are'positive(+)','values'of'Z'below'the'mean'are'negative'(f)'o The'total'area'under'the'normal'curv'is'equal'to'100%'when'expressed'as'a'percentage.'The'shaded'area'represents'the'percentage'of'the'observationsin''your'data'to'the'left'of'the'values'of'Z'o The'curve'never'crosses'the'horozonal'axis,'only'extends'to'–'infinite'and'+'infinity''à'empirical'rule'(looking'back'to'chapter'3)'• To'refresh'your'memorry'the'empirical'rule'means'that'if'a'districution'is'APROXIMATLY'NORMAL'then,'68'%'of'the'data'will'be'within'ONE'standard'deviation,'95'%'within'2'standard'deviations,'and'99.7%'or'almost'all'within'3'''Here is a visual model of the empirical rule • These'standard'deviations'are'just''scores'or'“standard'units”''or'“standard'deviation'units'''Standard'(Deviation)'Units'''à'a'score'z'is'in'standard'unites'if'it'tells'us'how'many'standard'deviation'units'an'original'value'is'above'or'below'the'average.''• For'example:'if'zf=3.1'then'the'original'value'was'1.3'stanard'deviat ions'above'the'average''• For'example:'if'z=f.055'then'the'original'score'was'.55'standard'deviations'BELOW'the'average.''Here'is'the'formula'for'converting'data'from'original'units'to'Z'scores'(remember,'in'ch.'3'we'learned'the'z'score 'formu la'in'relation'to'the 'mean'and'standard'deviation'! = !!!!!'')'à'the'same'idea'works'again'here,'but'we'have'to'change'the'context'to'the'mean'and'standard'deviation'of'aPROBABILITY'distribution''! =! − !!''What'we'can'do'with'this'formula'à'subtracting'the'population'mean''(!'called'“mu”)'from'a'given'value'(x)'' à'this'has'the'effext'of'shifting'the'distribution'in'such'as'wy'that'any'value'equal'to'the'population'wil'have'a'z'score'of'0'' à'by'dividing'by'sigma(the'population'standard'deviation)'will'rescale'the'distributionin'such'a'sway'that'a'valu e' that'is'one'standaed'deviation'above'the'mean'will'take'the'value'of'1,'a'value'that'is'2'standard'deviations'above'the'mean'will'have'a'value'of'2'ect''Example:'àthe'LSAT'is'a'standard'ized'test''• There'are'6'sections:'4'secrions'of'scored'mutiple'choice,'1'unscored'experimental'section,'and'an'unscored'writing'seection''• Raw'test'scores'are'rescaled'to'fit'a'normal'distribution''• Normalized'scores'are'distributed'on'a'scale'with'a'low'of'120'and'a'high'of'180'with'a'median'of'150''• Assuming'stanard'deviation'is'a'10,'what'is'the'mean'here?''à'the'first'step:'convert'original'to'Z'• The'typical'law'student'ar'yale'had'an'LSAT'score'of'171.'We'could'express'the'171'in'Z'scores'to'give'us'a'sense'of'how'many'standard'deciations'171'is'above'the'mean''''! =! − !!→ ! =171 − 15010=2110= ! +2.10'' à'Now'we'know'that'the'171'Is'2.10'standard'deviations'ABOCE'the'mean''à'the'second'step:'find'the'Z'score''• Look'up'+2.10'in'the'table(which'you'will'be'given'on'exams'and'final)''See'here'it'is!'• The'value'associated'with'+2.10'Z'is'.9821''The'last'step'is'interpreting'this'data''• The'typical'Yale'student'that'took'the'LSAT'has'a'z'score'of'2.10.'the'student'is'2.10'standard'deviations'above'the'average,'according'to'their'LSAT'scores'OR'•