1. Slides to accompany Weathington, Cunningham & Pittenger (2010), Chapter 7: Sampling 1

2. Objectives Samples, in general Probability sampling Probability sampling methods Nonprobability sampling Central Limit Theorem Applications of CLT Sources of bias and error 2

3. Why Worry about Sampling? Don’t worry, just appreciate it Objective sampling helps us avoid the Idols of the Cave –Improving external validity of our conclusions “Good” sampling allows us to make comparisons and predictions from our data 3

4. Samples... …are (hopefully) valid representatives of the population you are studying …can grant you better (more objective, empirical) data than you will find in anecdotes …allow you to avoid reliance on one person’s opinions, perspectives, and biases 4

5. Probability Examples Probability of Heads in one flip of a fair coin: p(H) = 1/2; p(T)=1/2=.5 p(H and T) in two flips = 2/4=.5 p(correct answer on 4-option mc question) =.25 Pr. of choosing a woman in a single random selection from a class of 223 students with 150 women: p(w)=150/223=.673 5

6. Probability Sampling Random: each outcome has an equal probability of occurring, every time –Every time I flip a coin, the probability is.5 that it will be H or T Random sampling depends on this independence of outcomes Law of large numbers: On average, a large selection of items will have the same characteristics as those in the population 6

7. Populations and Samples Target vs. sampling population –Target: (universe) e.g. all depressed persons –Sampling: (accessible) all diagnosed as depressed Sampling Frame (all who can be reached) Subject (participant pool) – (willing to participate) Descriptive data helps us compare our sample against the population External validity depends largely on representativeness in sampling 7

8. Probability Sampling Characteristics Each population member has an equal chance of being a potential sample member –No systematic exclusions Sampling procedures are based on a protocol –Prevents bias effects on sample selection Probability of any specific sample can be calculated –Helps connect results with population 8

9. Simple Random Sampling Each population member has equal probability of selection to the sample –If selection is random, the sample of any size should represent the population from which it was chosen Random numbers are in tables and Excel- type computer programs 9

10. Simple Random Sampling: How-To Generate a list of possible participants (population) in Microsoft Excel In the next column insert the function “=RAND()” –Creates a random number between 0 and 1 Sort both columns by the random numbers Select the first N individuals for your sample 10

11. Sequential/Systematic Sampling Random is not always practical All sampling population members are listed and each k th member is selected to the sample k = sampling interval =Population size desired sample N 11

12. Stratified Sampling Good option when sample needs to include subgroups from a population –Based on gender, age, education, etc. Size of subgroups in final sample must be equivalent to size in population Can use simple random or sequential sampling to fill each relative subgroup 12

13. Cluster Sampling Good option when participants are already in groups that cannot be easily separated –e.g., Study of coaching’s impact on different sports teams Instead of randomly selecting team members, you randomly select teams If need certain subgroup representation, this may limit your option of teams 13

14. Nonprobability Sampling Sampling based on some other factor besides probability –May be more convenient –May not be as representative Can’t establish probabilities associated with sample membership –Can still be useful if treated with caution 14

15. Convenience Sampling “Person” on the street approach Sampling from easy to find population members (a “special” subset) Sample determined in part by researcher’s sampling method –Not by probability Can bias/distort results Sometimes the only option 15

16. Snowball Sampling Good for cohort studies or when trying to reach a dispersed population Using one cohort member to find others, and so on... Pros: Good for research on difficult populations to reach (e.g., homeless) Cons: No representative sample guarantee 16

17. Central Limit Theorem Refers to distribution of characteristics within the probability samples 1.As N (sample size) increases, the shape of the sampling distribution of means will approach a normal distribution 2. µ M = µ (mean of sample means =pop mean) 3. σ M = σ/√n (SEM) 17

18. CLT Sampling Distribution Shape –Figure 7.4  Note how the M becomes closer to µ as N increases µ M = mean of means = (sum of all sample means)/(number of samples) –M = unbiased estimate of µ σ M = std. dev. of the sampling distribution of M –As n increases, distribution of sample means will cluster closer to µ  more accurate estimate 18

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20. CLT If we use probability sampling, M = unbiased estimate of µ M becomes a better estimate of µ when n increases We can determine the probability of obtaining various M 20

21. Standard Error of the Mean Represents uncertainty of how well M represents µ SEM = SD of sampling distribution of means σ / √n (n = sample size) http://www.miniwebtool.com/standard-error- calculator/ SEM is affected by: – σ  as this decreases, SEM decreases –n  as this increases, SEM decreases (1/√n) M is best estimate of µ when SEM is low 21

22. Applying CLT Reliability of a sample mean (M) –Use SEM to calculate confidence intervals around M (see Fig 7.4, p 212) –There will be variability among sample M, but a CI can help you determine the expected range Adequacy of a sample size (n) 22

23. Confidence Intervals In a normal distribution, 68% of M within 1 SEM of µ, 95% within 1.96 SEM of, 99% within 2.58 SEM Can use CI to predict other M –95% CI = 95% of future sample M should fall within this range 23

24. Sources of Bias and Error Bias: nonrandom, systematic factors that may make M differ from µ –Could be controlled Error: random events that have the same effect, but cannot be controlled Figure 7.7 is a good illustration –Ideally, µ’ = µ, but not in these examples –Possible nonsampling biases at work 24

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26. Bias and Error If the sampling is random, then even if there is a nonsampling bias present, µ M = µ’ Sampling bias: systematic selection bias while sampling Total error = M - µ –Sum of effects from nonsampling bias, sampling bias, and sampling error 26

27. What is Next? **instructor to provide details 27