Quantitative Methods of Social Research
Quantitative Methods of Social Research SOCI 4880
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Chapter 10 S 1 Sampling and Sampling Distributions amp lug Population 7 A group that includes all the cases individuals objects or groups in which the researcher is interested 7 All third gmde children in the United States 7 All the Wheat crops in Nonh America 7 All college students Sample 7 A relatively small subset from a population 7 3rd grade children in Mrs Nozingo s class at Wiles Elementary 7 Southernmost acre ofwheat in each farm 7 Students taking SOCI 4880 at UNT WW4 cmmm Notation Sampling Table 1 v 1 Sampla and Papulutiun Nolulions Parameter 7 A measure for example my SumplNnm on hummumnm mean or standard deviation used to reap V m describe a population distribution mmquot p x 5mm demon 5v quotv Statistic 7 A measure for example mean Vancmte 52 5 or standard devratron used to describe a sample distribution minus cmmur Sampling Parameter amp Statistic haw H l Hie Fmponion of Whvle Rtspondenls in 1 population and in o Sample ii honor a a a k W Mi show In 5 What is the probability that I would role a 6 assuming that I have a honest die P616 or 1667 chm In t What is the probability or all pnssihle nutcnmes again zsmmingthzt 1 have zhnnest die P or 1667 total 66 or 10000 And if it was perfectly dishonest such that It 1y rolled 6s P1 06 or 0000 P 2 P 6 66 or 1000 total 66 or 10000 Hoopla l r7 Probability Sampling I Probability sampling a A method ofsampling that enables the researcher to specify for each case in the population the probability of is inclusion in the samp e I Goal 7 Representative Sample Rtproduces the important characteristics or the population Hpopulation is 40 female so should our sam chm In x Random Sampling EPSEM Simple Random Sample 7 A sample 0 Equal Probability of Selection Method designed in such a way as to ensure that 7 Fundamental Principle of probability sampling 7 1 every member of the population has an 7 Every element or case in the population must equal Chance Of being Chosen have an equal probability of being selected for 7 2 every combination of N members has an the sample equal chance of being chosen 7 EPSEM does not guarantee representativeness 7 3 Need a complete list of population This can be done using a computer calculator or a table of random numbers Chapterl 79 Chapta39l 7l by selecting a representative sample frog the p0 1011 Population infere R 039 Chapterl 7ll Chapterl 712 Random Sampling Systematic random sampling 7 A method of sampling in which every Kth member K is a ratio obtained by dividing the population size by the desired sample size in the total population is chosen for inclusion in the sample after the rst member of the sample is selected at random from among the rst K members of the population nwm m r 13 Systematic Random Sampling my v I 7 synm unnumsampnng mum in r u Strati ed Random Sampling Stratified random sample 7 A method of sampling obtained by 1 dividing the population into subgroups based on one or more variables central to our analysis 2 then drawing a simple random sample from each of the subgroups nwm m r u Strati ed Random Sampling Proportionate stratified sample 7 The size of the sample selected from each subgroup is proportional to the size of that subgroup in the entire population Disproportionate stratified sample 7 The size of the sample selected from each subgroup is disproportional to the size of that subgroup in the population mum in r m Disproportionate Strati ed Sample mm H 3 A Random Sample siratilied sy RunErhniriry em mm a 0 thw l uv f at N w it cmmiuin Sampling Distributions Sampling error 7 The discrepancy between a sample estimate ofa population parameter and the real population parameter Sampling distribution 7 A theoretical distribution of all possible sample values for the statistic in which we are interested 7 Represents every conceivable combination of cases from the population 5mm in 7 m Sampling Distributions I Sampling distribution of the mean 7 A theoretical probability distribution of sample means that would be obtained by drawing from the population all possible samples of the same size Ifwe repeater y drew mmplex 39nm a papula nn and cal ulatedme sample maanx tlmxe sample mean wmdd be nu 39 ex drawn increaxes 711 next several slillex demunfmte tlm I Standard error of the mean 7 The standard deviation of the sampling distribution of the mean It describes how much dispersion there is in the sampling distribution of the mean Cl39lmarlu719 Example We want to know age of a community of 10000 We draw and EPSEM sample of 100 residents their age 7 Mean 27 Now we toss the rst 100 back in the mix Draw another sample of 100 7 Mean 30 We continue to do this over and over again 7 Ultimately the distribution of sample means is normal cmmiuim Distribution of Sample Means with 21 Samples S D 2 72 Mean ofmeans 41 0 Number ofMeans 21 Sample Means Chapter will Distribution of Sample Means with 96 Samples Frequency Sample Means Chapter In 7 22 Distribution of Sample Means with 170 Samples The Central Limit Theorem If all possible random samples of Size N are drawn from a population with mean My and a standard deviation O39y then as N becomes larger the sampling distribution of sample means becomes approximately normal with mean My and standard deviation 0y Sample Means Chapter In 7 23 Chapter In 7 24 Why does our sampling error go down with increase in sample size What does CLT mean Basically with suf cient sample size the sampling distribution of the mean will be normal regardless of the shape of the population distribution SO We know As the sample size gets larger the mean of the sampling distribution becomes equal to the population mean As the sample size gets larger the standard error ofthe mean decreases in size Chapterl rZS Chapterl rl How does this all relate to the normal curve y do we care We said that the sampling distribution of the mean Vital to doing inferential statistics ls normally dlsmbuted Allows us to estimate population parameters We can use the properties of the normal curve to from sample Statistics gure out the probability that a sample mean will fall within a certain distance of the population mean 7 Let s us set our confidence that a statistic is accurately re ecting a parameter For example we can expect approximately 68 of all sample means to fall within plus or minus 1 USCd for everythmg We do from thlS polnt standard error of the population mean forward Chapterl r Chapterl rZB