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STT 1810

by: Cale Steuber

STT 1810 STT 1810

Cale Steuber
GPA 3.81

Ross Gosky

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Ross Gosky
Class Notes
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This 15 page Class Notes was uploaded by Cale Steuber on Friday October 2, 2015. The Class Notes belongs to STT 1810 at Appalachian State University taught by Ross Gosky in Fall. Since its upload, it has received 38 views. For similar materials see /class/217698/stt-1810-appalachian-state-university in Statistics at Appalachian State University.


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Date Created: 10/02/15
Chapter 3 Sampling Designs Steps to Design a Statistical Sample 1 Determine the study s objectives Identify the Population De ne the variables to be measured Determine the statistical design required to sample from the population Collect the data khth 0 We focus on step 4 in this chapter 0 Steps 1 to 3 are often surprisingly difficult 0 Step 5 often requires lots of effort and followup especially in surveys Def The population is the entire group of individuals we want to obtain information about A sample is the part of the population we actually examine to gather information Think of the population as the group about whom you want to draw conclusions If it is feasible to obtain information from your entire population by all means do so But in almost all cases sampling will be needed to obtain a representative group of the population at large Def The design of a sample refers to the method used to choose the sample from a population 0 design of the sample is important 0 if some aspect of the design favors individuals of a particular type the results can be non representative or biased Bad Sampling Designs 1 Voluntary response samples 0 allowing anyone who chooses to respond to a survey 0 Examples website polls 0 Why are these a bad idea 2 Convenience Sampling 9 Sampling those easiest to reach 0 Examples 0 standing outside Walker Hall and asking the rst 200 people I meet 0 Surveys at the mall 0 making phone calls until you achieve a certain number of responses 0 Mass emails 0 Why are these a bad idea STT1810 Course Notes ChaDters 3 and 4 Page 1 Def A sampling design that systematically favors certain outcomes is said to be biased All of the above methods of sampling are biased Our goal is to eliminate or at least reduce any bias in our sample We will discuss how to do this neXt Valid Sampling Methods Using one of these sampling designs you can form a valid sample from any population that will be free as far as you can tell of any design bias The 4 valid methods of sampling we ll discuss in this chapter Simple Random Sampling Strati ed Random Sampling ClusterMultistage Sampling Systematic Sampling JBWN The rst method Sim le Random Sam lin SRS Steps Get a complete list of the population Randomly choose namesIDs from this list without replacement until you achieve your desired sample size Analogy drawing names from a hat This avoids bias in the selection process because selection is purely random Note The de ning characteristic of simple random sampling is The de ning characteristic is not Although this is a feature of SRS sampling there are other sampling methods with this characteristic Example Consider a manufacturing company with 5 sites called A B C D and E The company president wants to randomly choose 2 sites to visit for a surprise quality inspection The list of possible samples of size 2 from this population are AB AC AD AE BC BD BE CD CE DE So there are 10 possible samples Under Simple Random Sampling all 10 of these samples are equally likely If that is true then we can see that each site appears in exactly 4 of the 10 samples meaning each site is also equally likely to appear in the sample If a population is large a computer package can be used to select an SRS from almost any sized population STT1810 Course Notes Chapters 3 and 4 Page 2 The 2 d Method Strati ed Random Sampling Strati ed random sampling is very common The term strata refers to groups within your population that you believe will respond similarly to the questions or measurements For example men and women may inherently respond differently to a question or measurement for example assessing whether women receive equal pay for equal work or in something simpler like height measurements To avoid a SRS having too many or too few women we stratify the population by gender and select a SRS from each strata separately This would minimize the chances of getting a bad sample and give us more con dence in the accuracy of the information in our sample To Select a Strati ed Random Sample 1 Divide the population into groups of similar individuals called strata 2 Choose a separate SRS from each stratum 3 Combine each of these separate SRS s to form the overall strati ed random sample M The proportion of each group in your population should drive your decision as to how large to make each SRS within each strata For example let s say your population has 10 men and 10 women and you stratify by gender If your total combined sample will have 4 individuals you d probably take an SRS of size 2 from each strata because men and women are equally represented in your population However if your population had 30 women and 10 men you d probably take an SRS of size 3 from the women but an SRS of size 1 from the men Thus the concept is you should try to keep the percentage of each strata in your combined sample as close as possible to their allocation in the overall population The 3m Method Cluster Sampling Cluster sampling is a technique mostly used for convenience A cluster sample takes an SRS of larger groups within the population rst this is stage 1 Then an SRS at minimum or a census is taken from within each cluster stage 2 Picturewise we can represent this as STT1810 Course Notes Chapters 3 and 4 Page 3 Example Consider sampling students who live on campus at ASU for a survey that required you to administer the survey in person If you took a standard SRS you would likely have several residence halls to travel to for only 12 people in your SRS quite a waste of time What might be easier is to rst take an SRS of say 5 residence halls Then within those residence halls take an SRS of many students or a census This may save you time and effort because most individuals in your sample will live close to each other We hope that each cluster is similar to the population at large In our example this assumption could be violated if some halls have only rstyear students for example and if we suspect the survey results may differ by class Example 2 Suppose we wanted to know how all waiterswaitresses in Boone feel about customer s tipping habits No simple population list of waiterswaitresses exists But a list of restaurants does exist Perhaps an SRS of restaurants stage 1 followed by a census of all waiterswaitresses at the chosen restaurants would work well STT1810 Course Notes Chapters 3 and 4 Page 4 The 4amp method Systematic Sampling Systematic sampling refers to choosing every kth member of the population after some random starting point These are common when the other methods are not feasible and there is a sequential list of the population available Example 1 If you visit the Hershey s factory where Hershey s kisses are made and want to choose a random sample of Hershey s kisses how can you do it Example 2 Exit polls on voting day Example 3 Suppose you have a printed nonelectronic student directory You want to choose 100 of them at random Randomly selecting 100 numbers from 1 to 15000 might do the job but you might also decide that after a random starting point you will choose every 150 h student name in the list This might make compiling the sample easier than the SRS method STT1810 Course Notes Chapters 3 and 4 Page 5 Other Problems that Can Occur in Sample Surveys There are a few other problems which can occur when doing sample surveys These are not problems with the sampling methods These are additional complications that can cause bias even when the sampling method like SRS or Strati ed Sampling is unbiased Some of these problems are almost impossible to avoid You should do your best to minimize their effect Others can be successfully avoided by carefully designing your survey A list of the possible problems is 1 Having an incomplete population list This phenomenon is called undercoverage Undercoverage is hard to avoid Sampling by telephone underrepresents people Without telephones in our survey Sampling by going doortodoor to households underrepresents the homeless population Undercoverage should be minimized by l obtaining a population list speci cally for the study if possible or 2 using the most current list available 3 crosschecking the population with multiple lists eg in a small town in a cityblock structure you could use the phone book but crosscheck it with a city map to see which households are omitted 4 random digit dialing for phone surveys but again for large populations it is hard to completely avoid it STT1810 Course Notes Chapters 3 and 4 Page 6 N V Nonresponse failing to receive a response from a chosen individual in the survey either by failure to contact the person or their refusal to participate It is tempting to just randomly choose a replacement person for a nonresponder Is this a good idea As with undercoverage nonresponse is also hard to avoid A survey eventually must end Would you call someone in your sample more than 100 times to obtain a response To minimize nonresponse 0 use multiple modes of contact email phone inperson etc 0 keep the survey as short as possible and tell the respondent how long it will take 0 use name recognition ifpossible Gallup AC Nielsen ASU etc 0 thank respondent for their time 0 advance letters 0 small incentives for completing the survey Every possible effort should be made to get a response from individuals in your survey This effort will at least minimize the effect of the nonresponse Nonresponse really becomes an issue when a signi cant percentage of your sample does not respond 3 Response bias Other sources of bias include response bias Response bias occurs when respondents do not answer truthfully This can happen for a number of reasons 0 the survey contains a sensitive or personal question 0 Asking the respondent to recall past events 0 Wording of questions can cause people to answer incorrectly o Confusing Wording Do you think that students who abuse illegal drugs should not be prohibited from obtaining nancial aid for college 0 Biased Wording The rearm crime rate victims per 1000 residents has dropped from 59 in the 1990 s to 20 in 2005 Do you feel rearms should be banned 0 Assuming a motive for answering Do you support the president s health care plan because it would ensure all Americans would receive health care coverage 0 Order of questions can make an impact An experiment was conducted with the following two questions 0 A Do you think the US should let Communist newspaper reporters from other countries come in here and send back to their papers the news as they see it o B Do you think a Communist country like Russia should let American newspaper reporters come in and send back to America the news as they see it STT1810 Course Notes Chapters 3 and 4 Page 7 Some respondents were given the questions in the order AB Others were given the questions in the reverse order Which order do you think led to a higher of Yes responses to question A A good survey design can minimize the effect of response bias Bottom line Before you accept the results of any survey you should read the exact question posed Open vs Closed Questions What do you think are the pros and cons of open versus closed questions Good things about open questions Good things about closed questions Using Samples to make Conclusions about the population Ultimately we want to use the results from our sample to make conclusions about our population However it is easy to see that if you and I take two different samples we will get different results even if we use the same sampling design So we need some ways to measure how accurate our sample results are We can use this estimate of our accuracy and build them into our conclusions about the population This is why you often see a percentage reported with political polls We will be studying this concept in detail in the coming chapters of the teXt For now we will state one fact This fact is that larger samples give more accurate results than smaller samples If someone wanted to estimate the average exam 1 score in Stt1810 they d get more accurate results with a sample of 50 students than with a sample of 5 students The margin of error in sample surveys is 1sqrtn regardless of the population size So if 1000 adults are surveyed and asked whether they are in favor of the No Child Left Behind and 651 respond yes then we think the true value in the population is between 651 plus or minus 1sqrt1000 or 0031 or 31 So 651 i 31 9 62 682 These results are often given with 95 confidence which will be discussed later in the course but can be thought to represent our certainty that the interval above captured the true percentage in the population STT1810 Course Notes Chapters 3 and 4 Page 8 Chapter 4 Principles of Data Gathering Observational studies vs experiments What is the difference between an observational study and an experiment Cause and effect determined only by experiments Association can be determined by observational studies Sometimes an observational study is the best you can do Randomized Experiments 0 Often volunteers participate in these studies We need to assume these volunteers are representative of the larger population Often this is reasonable but sometimes it s not true For example cancer clinical trials often involve patients who are not responsive to standard therapies These patients may not be representative of all cancer patients 0 Participants experimental units subjects if people 0 Treatments a distinct experimental condition applied to a subject might be combinations of factors 0 Typically multiple variables measured Response variable explanatory variable or factors STT1810 Course Notes ChaDters 3 and 4 Page 9 Example 1 Which headache treatment is best 200 mg advil 200 mg aspirin 200 mg Tylenol Response time until relief from symptoms achieved Can we let people pick their treatment Why not Randomizing means that if 90 people available we randomly choose 30 for Advil 30 for Aspirin and the rest get Tylenol Example 2 Does liquid consumption with the medicine make a difference We might suppose that drinking 12 oz liquid is better than drinking say 2 oz So we might construct treatments as Drug Advil Aspirin Tylenol L I 2 oz Trt 1 Trt 2 Trt 3 Q U I 12 oz Trt 4 Trt 5 Trt 6 d The role of randomization Randomness decides who gets which treatment Randomize order of treatments if administered in sequence eg a person takes a test on a computer on paper and on a computer with help menus Make each person take each test Why And randomize the order of the tests per person STT1810 Course Notes Chapters 3 and 4 Page 10 Other features of experiments 1 Control Group a group who is treated in the same way except that they don t receive the active treatment Example You have a bench of 12 plants of same variety say Does talking to them make a difference 2 Placebos a dummy treatment with no bene t often in health studies to screen out psychological effects confounded with a treatment 3 Blinding not telling the subject which treatment they received Double Blinding not telling the subject nor the experimenter which treatment was received A separate experimenter who does not take the measurements knows who received which treatment 4 Double dummy t w u eg 1 vs 39 39 l 39 for back pain Patients know which they re getting which can be bad but we could have each patient receive both with 1 being a dummy placebo treatment Example August 2002 Journal of Human Hypertension patients with high bp given 2 week washout period 2 week placebo period of both to test for true hypertension and then double dummy treatments with 10 mg lercanidipine or 50 mg losartan STT1810 Course Notes Chapters 3 and 4 Page 11 Dealing with extraneous variables Ar1 extraneous variable during an experiment is one that may have some effect on response but is not in and of itself of interest in the study Ex Studying 3 types of paint to see which dries fastest Humidity is an extraneous variable here So is temperature Generally there are 3 approaches to handling extraneous variables in a study Try to keep them constant during all measurements ex in the previous study measure all paints the same day in the same relative location If this can t be done randomize them over the subjects if possible example studying effects of lycopene supplementation on cancer rates Extraneous variable eating habits So we randomize who gets which treatment 1ycopene vs placebo say so that the extraneous variable eating habits is at least handled randomly We can also possibly include extraneous variables as a blocking variable in a study which allows us to estimate its effects More on this later N V W V STT1810 Course Notes Chapters 3 and 4 Page 12 Valid Experimental Designs 1 Completely Randomized Design CRD Each treatment is randomly assigned to a certain number of experimental units eg 20 people get diet 1 20 get diet2 20 get diet 3 chosen randomly 2 Matched pairs designs Subjects are paired so that they re similar in every known way except for which treatment is given Often this pairing is on the same person Ex testing 2 types of sunscreen Some people burn easily others do not Have each person put sunscreen l on one arm sunscreen 2 on the other Now this person has served as their own control group screening out a lot of potential variability unrelated to sunscreen Ex surgery vs drug therapy for a disease You can t easily placebo a surgery Technically you can but it s probably not ethical So each person gets one treatment type But you match patients between the two treatments by nding 2 patients with similar age weight disease severity all of which could affect treatment and then randomize who gets which treatment Ex does church attendance prolong life This can t be done as an experiment Who wants to be randomly assigned to attend or not to attend church But people who are ill probably don t go to church and live shorter lives How to handle this Other habits of churchattendees lower alcohol consumption less tobacco use social support etc may bene t someone not necessarily the Sunday Sermon So if we nd 2 people who have relatively similar health habits age overall health and where one goes to church and the other does not we may suitably match them and note any differences in lifespan This is still observational but it s a good approach given the limitations of observational studies 3 Block Designs Idea Assign each experimental unit once within each block a group of units thought to respond similarly regardless of treatment Ex greenhouse bench plant assignment example In some matched pairs designs people serve as a block when they receive multiple treatments each person ls like a greenhouse bench Read examples on page 73 of text which are good case studies STT1810 Course Notes Chapters 3 and 4 Page 13 Valid Observational Study Approaches Sometimes an observational study is the best you can do Health studies for example 1 Retrospective and Prospective Studies Retrospective studies ask a respondent to categorize themselves or provide measurements of past behavior eg have you had a heart attack Have you smoked as an adult for at least 2 years Prospective studies follow individuals for a period of time and measure at least some outcome variables based on this tracking period eg do you smoke now YN follow them for 5 yrs and see if they have a heart attack Which are better Which are easier 2 Case Control Studies Cases of an outcome variable are compared with cases without the outcome variable and researchers look to see if any explanatory factors differ with regard to the two groups Example in text Baldness and heart attacks Hospitalizations for heart attacks are measured are male patients balding or not Then hospitalizations for other cases not heart attacks are also measured for baldness Are there differences among the two groups Case control studies are relatively easy to do compared to experiments when a control group is easy to nd and measure There is some thinking that case control studies may reduce the effect of potential confounding variables See pages 7677 in the text for an example STT1810 Course Notes Chapters 3 and 4 Page 14 Dif culties in Experiments and Observational Studies 1 Confounding Variables affecting observational studies Note cause and effect cannot be determined from observational studies A confounding variable is one that is closely associated with the explanatory variable in an observational study and which has effects that cannot be easily separated from the explanatory variable in the study Examples churchgoing and lifespan Number of re ghters sent to a re and re damage Taking vitamin supplements and being healthy Remedies none work perfectly I Put the confounding variable into the analysis to see if its effects can also be measured This doesn t work very well 2 Do a casecontrol study where individuals are matched so that they only differ with regard to the explanatory variable of interest eg match a healthy social churchgoing person with a healthy social nonchurchgoing person eg match a healthy person who eats a good diet and takes vitamin supplements with a healthy person who eats a good diet but does not take vitamin supplements 2 Extending Results Inappropriately Results can only be extended to a larger group if the small group from whom data were gathered are good representatives of the larger group Most experiments clinical trials for instance rely on volunteers Can these volunteers be considered good representatives of the larger population If not don t draw conclusions based on the experiment Example If I perform an experiment on this class I d assume you re representative students of ASU in most things But not all things you are more interested in political science for example 3 Interacting Variables An interacting variable affects the magnitude of the association between two variables What does this mean Example Effectiveness of exercise on weight loss Suppose we test exercise regiments ABC Even if Regimen B say causes people to lose the most weight age may be a factor STT1810 Course Notes Chapters 3 and 4 Page 15


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