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Lecture 6 Who are we talking about? Populations and samples. Overview. Defining a populationTaking a simple random sampleHow similar is the sample to the population?. Population and Samples. Population All the cases (individuals, objects, or groups) in which the researcher is interested.Sample A relatively small subset from a population.ExampleThe US population: ~300 million peopleThe General Social Survey (GSS): a sample of the US populationabout 3,000 peopleStudent version of the GS273

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1. Assignment After this lecture, start Assignment #3. It’s in the course binder.

2. About 10 minutes short. Enough time for students to work an example.About 10 minutes short. Enough time for students to work an example.

3. Overview Defining a population Taking a simple random sample How similar is the sample to the population?

4. Population and Samples Population All the cases (individuals, objects, or groups) in which the researcher is interested. Sample A relatively small subset from a population. Example The US population: ~300 million people The General Social Survey (GSS): a sample of the US population about 3,000 people Student version of the GSS: a sample from the GSS about 1,500 people

5. The sampling problem We care about populations. We can only afford to look at samples. How do we know our sample is relevant?

6. Simple random sampling: Definition Define the population label every person Sample the labels randomly so everyone in the population has the same probability of being sampled

7. Simple random sampling: Example Define the population: this class=52 people label every person: Give everyone a playing card Sample the labels: Draw 5 cards from a second deck randomly: after shuffling so everyone in the population has the same probability of being sampled: Everyone’s card appears once in the deck.

8. Sampling: Bad examples Define the population: This class=52 people label every person: Put everyone in a seat Sample the labels: Choose 5 people from first row. Problems not random not everyone has the same probability only first row has any chance Sample the labels: Choose 5 volunteers. Problem not everyone has the same probability favors extroverts Sample the labels: Choose 5 people without a system Problem Is it random?

9. Sampling: Presidential election, 1936 Literary Digest poll Sampled 10 million names from lists of car and phone owners Mailed 10 million questionnaires Got 2.3 million responses Results: 57% favor Landon (R), 43% favor Roosevelt (D) Election result What went wrong?

10. Myth: Simple random samples are “representative” Actually can be quite different from population But we can usually place bounds on the difference

11. Notation

12. Population

13. Sampling error for a mean

14. Sampling variation

15. Conceptual definitions Sampling error – The sample mean is probably not the same as the population mean Sampling variation – Take a different sample, get a different sample mean.

16. Technical definitions Sampling error – The difference between the sample mean and the population mean. Sampling variation – The variation of the sample mean from one sample to another.

17. Repeated sampling

18. Sampling distribution Sampling distribution of the mean— The distribution of sample means over all possible samples.

19. Mean of the sampling distribution

20. Variation of the sampling distribution

21. Standard error: Definition

22. Standard error shrinks with sample size

23. Sample mean usually within 2 SE’s of pop. mean

24. Summary: Sampling distribution of the mean Across all possible samples has mean and standard deviation a.k.a. standard error Implications sample mean usually within 2 SEs of pop. mean In newspapers, +/- 2 SEs is often called “margin of error” larger samples have smaller SEs

25. Summary: More general If we take a simple random sample from a well-defined population we expect that the sample mean is “probably” “close” to the population mean By “close” we mean “within 2 standard errors” Larger samples have smaller standard errors. Next time we’ll say what me mean by “probably”

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