Introduction to Social Research
Introduction to Social Research Soc 320
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This 13 page Class Notes was uploaded by Alexzander Cole on Thursday September 17, 2015. The Class Notes belongs to Soc 320 at Washington State University taught by Staff in Fall. Since its upload, it has received 4 views. For similar materials see /class/205957/soc-320-washington-state-university in Sociology at Washington State University.
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Date Created: 09/17/15
CONFIDENCE LIMITS FOR SIMPLE INDEPENDENT RANDOM SAMPLES Fall 2005 Suppose we want to estimate our precision accuracy in a simple binary two choice situation Let s suppose we have a list of population elements eg registered voters who are expected to vote and want to estimate the proportion of the population that will select the first choice based on examining the results of a sample of N elements We can label the sample proportion of elements selecting the first alternative as p and the corresponding population proportion as n We re really interested in the possibility that n the population proportion is meaning we can t tell which alternative predominates Probability theory shows that the standard error the standard deviation of the sampling distribution of p will be given by Ml gtlt N N IN A sampling distribution is the distribution of values of p from all possible samples of size N taken from the population Probability theory also shows that 196 standard errors will contain the true population proportion n with probability 95 Since 196 is quite close to 200 we will simply use that value Thus our estimated 95 confidence interval for the population proportion is Upper Limit p Lower Limit p These terms refer to all possible samples of size N that might be drawn from our population If we were to take all possible simple independent random samples from our population and compute p for each of them then 95 of the intervals constructed by the method outlined would contain the population value a Note that our statistical approach does not say that any specific interval we calculate for a single sample necessarily contains the value of a The following figure shows the general relation between sample size and precision The table shows estimated standard errors and estimated precision for selected sample sizes from 100 to 100000 Precision by Sample Size SIRS i 60 Precision in 00 100 1100 2100 3100 4100 5100 6100 7100 8100 9100 10100 Sample Size ESTIMATED 95 PRECISION CONFIDENCE INTERVALS Simple Independent Random Sampling Sample Size E StError E Precision 100 0050 100 200 0035 71 300 0029 58 400 0025 50 500 0022 45 600 0020 41 700 0019 38 800 0018 35 900 0017 33 1000 0016 32 1100 0015 30 1200 0014 29 1300 0014 28 1400 0013 27 1500 0013 26 1600 0013 25 1700 0012 24 1800 0012 24 1900 0011 23 2000 0011 22 3000 0009 18 4000 0008 16 5000 0007 14 6000 0006 13 7000 0006 12 8000 0006 11 9000 0005 11 10000 0005 10 20000 0004 07 30000 0003 06 40000 0003 05 50000 0002 04 60000 0002 04 70000 0002 04 80000 0002 04 90000 0002 03 100000 0002 03 Deductive Theory L N Gray August 29 2004 GENERAL PRINCIPLES OF AXIOMATIC THEORY 1 Introduce Primitive Terms undefined terms Primitive Terms are those that are not themselves defined but are understood by our audience Ideally these are terms in our language for which we share definitions and which are not controversial If they are to be measured directly at some point additional clarification may be required These are our starting points and should be considered carefully Care in the choice of primitive terms is not always evident however For example a term like quotattitudequot should probably not be chosen as a primitive term since there is considerable disagreement about what it stands for 2 Develop Derived Terms defined terms Derived terms are those defined through the use of primitive terms and logical or linguistic connectives Ideally they have no empirical content though in practice derived terms and definitions often function in a way similar to axioms or postulates In some cases there is considerable disagreement over whether a given statement in a theory is a definition or derived term or something more One of the most famous examples is Newton s definition of force here abbreviated as f ma force mass X acceleration Theorists have argued about the status of this statement since its original appearance around 1687 3 Present Axioms or Postulates The axioms or postulates are statements often of relation linking primitive terms and such derived or defined terms as may have been developed Ordinarily but not always these relations are assumed to represent knowledge on which we currently agree andor take as true Often these are considered untestable for the sake of theoretical development Any derivations that follow depend on the truth of these axioms so if any are false derivations are likely to be faulty 4 Theorems or derived relations Based on the primitive terms derived terms and the axioms additional derivations using an appropriate calculus eg mathematics symbolic logic etc are developed When all aspects have been clearly specified it should be possible to follow the logic of theoretical development clearly That makes this approach useful for teaching since the structures of theoretical argument can be made apparent so that mistakes can be shown Of course in practice this approach is rarely used as generally described here EXAMPLE I Classical Mechanics Physics Primitive undefined Terms 1 Distance 2 Mass 3 Time Derived defined Terms l velocity dX 7 dt Where X represents distance velocity is measured as a unit of distance per unit of time eg meter per second or msquot 2 Acceleration ivd2X dt d12 39 Acceleration is measured as a unit of distance per unit of time per unit of time eg meter per second per second or msquotamp 3 Force dv F m 47 ma dt Force is measured as a unit of mass times a unit of distance per unit of time per unit of time eg kilogram meter per second per second or kgnrs Axiom postulate GITIITI 12 FI72 Newton s Law of Gravity r Where Rh and m2 stand for the masses of objects 1 and 2 r stands for the distance between the two objects and G is some constant number Note that the units of measurement for both sides of this equation must be the same so the units of G must be mikg39ks Substitutions Let Rh M the mass of Earth a constant Let m m2 the mass of an object on Earth Let r the distance from an object on the surface of Earth to the center of the planet a constant Deductions With substitutions definition 3 can be restated as F mg and the single axiom as Since equals can always be substituted for equals we have The n5 values cancel ie divide each side of the equation by m2 leaving a Conclusion Each component of the right side of this equation is a constant term Thus the acceleration of gravity on Earth is a constant Since the same derivation could be used for any object on Earth quotEverything on Earth falls at the same ratequot EXAMPLE II A Sociological Example liberally adapted from Bruce H Mayhew and Roger L Levinger 1976 quotSize and the Density of Interaction in Human Aggregatesquot American Journal of Sociology 8286 110 Primitive undefined Terms 1 Behavior 2 Person 3 Interaction Derived defined Terms 1 AF count of instances of a behavior within some political unit over a known period of time For example a count of instances of behavior X within a state in a given year 2 S count of persons within the same political unit over the same period of time For example the population of a state in a given year 3 D SS 1 z SA The interaction potential of a count of persons is the number of permutations of pairs of persons self excluded 4 RD interaction realization The count of pairs of persons that meet in the defined political unit during the defined period of time 5 b Intensity A function of temporal economic and social conditions that together produce increased interaction in a population Axioms postulates 1 AF p R n The count of a behavior is a constant proportion p of a function of interaction realization 2 Run z Db Interaction realization can be approximated by interaction potential exponentiated by intensity Derivations 1 Under the assumption that the interaction potential represents an upper limit for interaction realization 5 s b s l 2 Employing a statistical approach b Ami pDieiI or lnA lnpblnDilnei Fi where eiis a residual or error term Many macro studies of behavior use rates rather than frequencies or counts A rate is defined as the count of the events divided by the population size for the geographic unit and time period This number is usually multiplied by a constant c so that we can talk about the number of events per 100000 persons or any other base we desire eg C t fE Rate of EventE w x 100 000 S Given b 2b Ami pDiei Psi eil we can divide both sides of the equation by Sidivide equals by equals A A xcl R1 S i 5 D1 Si A Dquot5 x Db Dquot5 x Fi 1 C P iei 1 C so Ami lnA lncp b 5 ln Di ln e1 Ri chibquot5ei or 3 If intensity increases with the count of persons at an increasing rate b gt 5 then the rate of a behavior will increase with the count of persons 4 If intensity cannot be distinguished from the count of persons b 5 then rate is independent of the count of persons If an application is made to quotcriminal behaviorquot defined as appearing on the list of Federal quotindexquot offenses and some crimes eg quotviolentquot are identified as quotsocialquot and others eg quotpropertyquot are identified as quotnon socialquot then H1 k gt bp and H0 bV k5 Thus the rate of violent behavior should depend on the count of persons Since violent behavior is assumed to be social we cannot adjust completely for population size by dividing through by it ie taking rates The following figures suggest that this approach may have some value Crime Frequency by Population O FVi0ent I FPr0perty P0wer FOIiolent P0wer FPr0perty 10000000 1000000 I A I39lI O 5 O I 100000 I 2 2 E lquot M a O O 10000 0 U Q 1000 g 39 O 0 100000 1000000 10000000 100000000 Population Crime Rate by Population O ROIiolent I RPr0perty P0wer RVi0ent P0wer RPr0perty 10000 1000 a a N n 100 100000 1000000 10000000 100000000 Population QUALITATIVE INTERVIEW EXAMPLE Soc 320 2004 Fall Two types of justice event accounts were used A participant or self generated account participants were asked to describe just and unjust events from their lives and Real life situations described to the participants to create standardized events 1 2 Each 1 interview consisted of five sections The participant s general conception of justice and injustice The their lives a b C d e The and participants were asked to recall five events from The event types were An unjust negative bad outcome A just negative bad outcome A just positive good outcome An unjust positive good outcome and A negative bad outcome that was not relevant to justice participants were asked to describe their feelings thoughts for each of the five events Participant s thoughts on the justness or injustice of three real life events a b C The three situations were A murder by a 15 year old The American Health Care system and The case in which an elderly gentleman went to Florida because he thought he had won a publisher s sweepstakes A return to their general conceptions of justice and the scope of conditions to which justice principles apply A CRIMINAL COURT CASE EXAMPLE Please listen to the story and imagine you as a member of this community have been asked to join a group of citizens who are going to present the community s thoughts ideas and opinions on this matter to a judge What do you think the judge should consider who should he consider and what would a just outcome be A fifteen year old boy shoots and kills a 60ish year old man We will refer to him as William The boy had taken the gun from a friend s house There were no accusations that he intended robbery had threatened anyone or had committed any other criminal acts He had never been in any trouble with the law and had no serious trouble at school or home Generally he was an average 15 year old While William was at a bar after work someone came in to tell him that there was a kid getting in his truck William went out to find this young man attempting to steal some cigarettes from his truck William yelled at the young man as he headed toward his truck The young man shot and killed William
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