Survey'Sampling'and'Inference'The'world'through'surveys'Chapter'7.1''''Population''à'a'group'of'objects'of'people'we'wish'to'study''à'Typically'this'is'large('all'registered'voters'in'the'US,'all'babies'born'in'China'in'2012'ect)'BUT'it'can'be'a'small'('all'students'in'stats'10)'they'key'is'the'word'ALL.'''Parameter''à'a'parameter'is'a'numerical'valye'which'describes'the'population'• The'percentage'of'registered'voters'who'will'vote'or'Obama''• The'percentage'of'female'babies'born'in'china'in'2012'''• The'text'book'focuses'the'following'parameters:'the'population'mean,'the'population'standard'deviation'and'the'population'proportion'''Finding'Parameters:'''à'if'the'population'is'small('like'stats'10'lecture'5)'we'can'find'the'exact'value'of'a'parameter'by'conducting'a'census'('a'survey'in'which'every'member'of'the'population'in'measured)'''à'however'most'populations'are'too'large'for'a'census.'So'we'have'to'estimate'parameters'using'a'smaller'proportion'known'as'a'SAMPLE.'A'sample'is'a'collection'of'objects'or'people'taken'from'a'population.'(remember'this?)'''Statistic'à'after'a'sample'has'been'collected'we'measure'the'characteristic'of'interest'(gender,'birth'weight'age)''àa'STATISTIC'is'a'numerical'characteristic'of'a'sample'of'data''à'we'use'statistics'to'estimate'parameters'ß'key!''Example''à'suppose'we'make'and'sell'wedding'cakes''à'we'might'be'interested'in'measuring'the'proportion'of'our'cakes'which'might'be'considered'defective.'This'proportion'is'a'parameter'''If'we'cut'into'a'check'every'cake,'we'will'be'out'of'business,'so'we'might'have'to'take'a'smaller'sample.'The'proportion'of'a'defective'cake'in'out'sample'is'a'statistic.'''Another'Example''à'suppose'a'company'has'1.5'million'current'and'former'female'employes'à'suppose'the'company'is'being'sued'for'past'and'current'gender'discrimination'and'the'court'has'given'the'plaintiff'six'weeks'to'determine'what'proportion'of'current'employees'experienced'discrimination''à'the'law'firm'will'probably'need'to'sample'from'employment'records'to'determine'this'statistic''''More'about'statistics''à'statistics'are'also'known'as'ESTIMATORS'à'the'resulting'numbers'are'ESTIMATES'(this'is'comparable'to'predictor'and'predicted'variables)'à'suppose'we'sample'100'cakes.'The'estimator'is'the'proportion'of'cakes'in'the'sample'which''are'defective.'When'we'cut'the'cakes'and'taste'them,'we'find'that'5'(.05)'are'defective.'The'number.05'is'an'estimate''''Statistical'Inference''à'statistical'inference'is'the'science'of'drawing'conclusions'about'population'based'on'the'findings'from'a'smaller'subset'(sample)'of'the'population'''à'statistical'inference'has'a'certain'degree'of'uncertainty'so'a'lot'of'our'work'involves'measureing'the'amount'of'uncertainty'in'our'statistics''''''We'want'to'know'the'value'of'population'parameters,'but'given'out'limitations(money'and'time)'we'rely'upon'samples'and'their'statistics'and'make'inferences'based'upon'them'''''Differences'''STATISTICAL INFERENCEULTIMATELY WE ARE INTERESTED IN KNOWING THE VALUE OF POPULATION PARAMETERSBUT GIVEN OUR LIMITATIONS (TIME AND MONEY USUALLY), WE RELY UPON SAMPLES AND THEIR STATISTICS AND MAKE INFERENCES BASED ON THEMDifferencesUnknownKnownPopulation(all female employees)Sample(a small number of female employees)Parameter(proportion who experience discrimination)Statistic(proportion of the sample who experience discrimination)1. Notation'(vocabWish)'''''Problems'with'Bias'à'A'sampling'procedure'is'BIASED'it'produces'an'untrue'value''• Sampling'biasWwhen'the'sample'is'not'representative'of'the'population''• Measurement'biasWerrors'in'measurement'of'data,'asking'a'poorly'written'question'or'a'respondent'giving'an'incorrect'answer'• Natural'biasWa'technical'issue'caused'by'constructing'a'biased'estimator'''Measurement'bias'àthe'wording'of'a'question'can'cause'measurement'bias'''Sampling'Bias'àVoluntaryWResponse'Biasà'internet'polls'are'usually'biased'in'this'manner.'People'tend'to'respond'only'if'they'feel'strongly'about'the'question'being'asked''à'nonWresponse'Biasà'when'someone'selected'for'a'survery'or'a'poll'refuse'to'answer'it'a'larger'percentage'refused'then'a'biased'survey'could'result''''Randomness!'à'Randomness'can'help'correct/prevent'biases''• Random'samples'are'unbiased'and'representative'of'the'population'they'are'drawn'from''o a'random'sample'must'be'created'in'such'a'way'that'all'persons'or'objects'have'an'equally'likely'chance'of'being'chosen'for'the'sample'''Simple'Random'Sampling'(SRS)'à'SRS'is'a'basic'way'to'produce'random'samples''NotationStatisticsParametersSample MeanPopulation Mean !Sample Standard Deviation sPopulation Standard Deviation "Sample Variance s2Population Variance "2Sample ProportionPopulation Proportion P• subjects'drawn'from'population'randomly'without'replacement(once'one'subject'is'selected,'the'subject'cannot'be'select ed' again)''THE'RESULT'OF'THIS'METHOD'IS'UNBIASED'SAMPLES'''Sampling'with'Replacement'LAB'3'à'in'lab'3'Fathom'is'set'to'sample'with'replacement.'This'is'just'another'way'for'getting'a'random'sample''àwhen'you'want'to'simulate'a'random'process,'like'shooting'free'throws'a'1000'times'for'an'80%'shooter'the'outcomes'each'time'are'“hit”'“hit”'“hit”'“hit”''or'“miss”.''But'1000'is'more'than'5'so'we'need'to'let'the'software'“replace”'the'sample'before'it'takes'another'sample'''Sampling'with'Replacement'LAB'3'à'by'sampling'with'replacement,'we'are'also'able'to'demonstrate'independence'as'the'probabilities'for'each'trial'never'change'''Here'is'an'section'of'lab'3'in'fathom''''''Sampling with Replacement (Lab 3)By sampling with replacement, we are also able to demonstrate independence as the probabilities for each trial never change.''This'will'help'with'lab'3!!''Sampling with Replacement (Lab 3)By sampling with replacement, we are also able to demonstrate independence as the probabilities for each trial never