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# Class Note for ECON 2370 at UH

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Date Created: 02/06/15

Chapter 7 Sampling Distribution In this chapter we talk about the techniques of collecting data and drawing samples that represent the distribution of values in a population Examples of data sets that use samples 1 Decennial Census 2 Current Population Survey CPS 3 Consumer Price Index CPI Decennial Census 1 Began August 1790 Conducted every 10 years 2 Motivation estimate population that can be taxed and assess the country s industrial and military potential 3 Data are kept con dential for 75 years Data items collected 1 1790 number of persons in household and counts of persons in the following categories free white males and females all other free persons slaves ethnicity 2 Added items agricultural mining government activities religious bodies 1810 taxes education wages value of property 1 1840 1850 farm and home mortgages 1880 unemployment 1980 business housing and transportation 1940 place of work and means to work 1960 occupation history 1970 Data items collected 1 1980 use and creation of TIGER les data can be mapped geographically Methodology 1 1790 1880 Marshals interview household 2 1880 Standard survey form used prior to this marshals used whatever paper was available rule it write in headings and bind the sheets together 3 1870 Rudimentary tallying device used to help clerks 4 1890 Herman Hollerith introduced punchcards and electric tabulating machines Methodology 1 1910 Census of ce organized as a permanent agency 2 1950 First full use of computer support 3 1960 Devices set up to read data on returns use of the Postal System to distribute surveys 4 1990 Counts of the homeless instituted Use of samples 1 1880 Basic counts took almost until 1890 census to tabulate and publish 2 1890 Supplemental survey some subjects were covered in more detail 3 1940 Sampling introduced 5 of the population were asked an additional set of questions Consumer Price Index CPI 1 Collected monthly 2 lnititated during World War I when rapid increases in prices particularly in shipbuilding centers made such an index essential for calculating cost of living adjustments in wages 3 Hypothetical bundle of goods is de ned for the typical family Changes to the bundle and the family composition occurred over time 4 Samples of prices collected from establishments are used to estimated value of bundles 3 Current Population Survey CPS 1 12 monthly survey topics include employment temporary workers job tenure and occupational mobility school enrollment race and ethnicity voting and voter registrations food security work schedules computer ownership fertility and marital history 2 set up in the late 1930 s to provide direct measurements of unemployment each month 3 Probability sampling as used from the beginning Early samples of 50000 households used to estimate employment activities of the general population 4 Supplements decennial census data In this chapter we will concern ourselves with samples and sampling distributions 1 Previously when we looked at distributions we examined data distributions 2 In this chapter the distributions are made of parameters from samples mean distribution p distribution 3 Before we talk about sampling distributions we need to talk about how a sample is derived 4 In this chapter certain terms that we used before are given a new name 1 Parameters of interest gt u 02 and p are values or parameters that are ones we wish to derive from the samples 2 Statistics With each sample we derive statistics or the set of parameters of interest Methods of extracting a sample 1 Data on population is available extract a subset 2 Data on population is not available determine how many people are needed to obtain a representative sample survey n individuals Types of samples 1 Random sample 2 Non random methods convenience sample judgement sample quota sampling Data sets created by any of these methods cannot be used for making inferences Random Samples 1 In the simple version each element of the population has same chance of being selected 3 2 Other versions can also provide an unbiased sample a Strati ed random sample b Cluster sample c 1 in k systematic random sample Two methods of data collecting 1 Sampling from existing database eg stock market activity price data from a random sample of grocery stores researcher making use of data collected by the government secondary source data collection 2 Retrieving data directly from the respondents using a survey designed by the researcher primary source data collection Secondary data source 1 Bene ts low cost data is typically high quality takes less time to obtain 2 Cost variables might not be a close t to the variables desired by the researcher Primary data source 1 Bene ts variables come close to tting the type of variables desired 2 Costs added cost of survey design data collection and processing the data Data collection problems that can result in an biased sample 1 Distributing surveys to a random sample and accepting a low response rate 2 Collection techniques reaches only a subset of the full sample Even with a 100 response rate the methods will produce a biased sample 3 Wording Interviewer bias The choice of words used in the survey and the choice of interviewers can bias the results Sampling Distribution Sampling distribution of a statistics is the probability distribution for all possible values of the statistics that results when random sample of size n are repeatedly drawn from the population Methods of obtaining a sampling distribution 1 Derive the distribution mathematically using the laws of probability Examples 73 and tables 75 are examples of this method 2 Approximate the distribution empirically by drawing a large number of samples 7 3 Use statistical theorems such as the Central Limit theorem to derive exact or approximate distributions Central Limit Theorem If random samples of n observations are drawn from a non normal population with nite mean u and standard deviation 0 then when n is large the sampling distribution of the same mean aquot is approximately normally distributed with mean and standard deviation also known as the standard error of the mean Conditions Under certain conditions the means of random samples drawn from a population tend to approximate a normal distribution Conditions 1 If the population can be represented by a normal distribution the sampling distribution of aquot will be normal 2 If the population can be represented by a symmetric distribution the sampling 8 distribution of 3 becomes normal for small values of n for samples that are small relative to the population If the population can be represented by a skewed distribution the sampling distribution of 3 becomes normal for large values of nfor samples that are large relative to the population Tools to assess aquot given u u 1 Compute the mean and standard deviation of the sample distribution m of Determine condition to test eg PG lt Convert aquot to a z score using the following function 35 Mr 0 Use the table in the back of the book to test the probability condition region of the distribution w Sampling Distribution of sample proportion 1 Recall from the previous chapter gt Let X be a binomial random variable with n trials and probability p of success The probability distribution of X approximates the normal with u np and 0 Mpg 2 There is a similar outcome in sampling Let s assume that the sampling distribution has the following characteristics For a sample the probability of successes is equal to the number of person with this characteristics over the total number of persons in the sample or A a P n where 25 is the probability of success derived from the sample 3 For the sampling distribution 15 2997 Where q 1 p If np gt 5 and nq gt 5 the sampling distribution can be approximated by the normal distribution Tools to assess 9 given that up p 1 Convert 25 into a z score and calculate the probability

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