Stat 1510 Statistical Thinking & Concepts

1. Stat 1510Statistical Thinking & Concepts Producing Data: Sampling

2. Data • Primary Data is the data collected by the investigator conducting the research / study with a specific purpose. • Secondary Data - is data collected by someone other than the user for the same or different purpose.

3. Population • Researchers often want to answer questions about some large group of individuals (this group is called the population) • Population is a set of units. This population is potentially infinite or even hypothetical. • If the time that unit are measured is important, then the population is often called process. • So in analysis, it is important to be clear about what is the definition of population.

4. Population & Sample We consider three type of units. • The target population is the set of units to which the investigators set out to investigate in the definition of the problem • The study population is the set of units that could have been in the sample • The sample which is the set of units actually selected for the investigation. The total number of units in the sample is called sample size and the way that the samples are selected is called sampling protocol or sampling design.

5. Population & Sample

6. Example Faculty of Science of Memorial University want to know the opinion of students on the university facilities. For this purpose, they conducted a survey by selecting a random sample of 200 students survey registered for Winter 2013. Identify target population, study population sample & sampling unit

7. Example Target Population: Faculty of Science, MUN, Students Study Population – All students registered for Winter 2013 in FoS of MUN Sample – 200 students selected for this survey Sampling Unit: Each selected student

8. Bad Sampling Designs • Voluntary response sampling • allowing individuals to choose to be in the sample • Convenience sampling • selecting individuals that are easiest to reach • Both of these techniques are biased • systematically favor certain outcomes

9. Voluntary Response • To prepare for her book Women and Love, Shere Hite sent questionnaires to 100,000 women asking about love, sex, and relationships. • 4.5% responded • Hite used those responses to write her book • Moore (Statistics: Concepts and Controversies, 1997) noted: • respondents “were fed up with men and eager to fight them…” • “the anger became the theme of the book…” • “but angry women are more likely” to respond

10. Convenience Sampling • Sampling mice from a large cage to study how a drug affects physical activity • lab assistant reaches into the cage to select the mice one at a time until 10 are chosen • Which mice will likely be chosen? • could this sample yield biased results?

11. Purposive Sampling • Consider the selection of football team or soccer team. • Consider selection of students for a math skill competition In the above sampling scheme, we select the sampling units with a well defined purpose and samples are not randomly picked.

12. Simple Random Sampling • Each individual in the population has the same chance of being chosen for the sample • Each group of individuals (in the population) of the required size (n) has the same chance of being the sample actually selected • Random selection: • “drawing names out of a hat” • table of random digits • computer software

13. Table of Random Digits • Table B on pg. 692 of text • each entry is equally likely to be any of the 10 digits 0 through 9 • entries are independent of each other (knowledge of one entry gives no information about any other entries) • each pair of entries is equally likely to be any of the 100 pairs 00, 01,…, 99 • each triple of entries is equally likely to be any of the 1000 values 000, 001, …, 999

14. Choosing a Simple Random Sample (SRS) STEP 1: Label each individual in the population STEP 2: Use Table B to select labels at random

15. Simple Random Sample with and without replacement Case 1: In without replacement, each selected sampling unit will not replaced back to the population. Case 2: In with replacement, each sampled unit will be replaced back to the population.

16. Probability Sample • a sample chosen by chance • must know what samples are possible and what chance, or probability, each possible sample has of being selected • a SRS gives each member of the population an equal chance to be selected

17. Stratified Random Sample • first divide the population into groups of similar individuals, called strata • second, choose a separate SRS in each stratum • third, combine these SRSs to form the full sample

18. Stratified Random SampleExample Suppose a university has the following student demographics: Undergraduate Graduate First Professional Special 55% 20% 5% 20% A stratified random sample of 100 students could be chosen as follows: select a SRS of 55 undergraduates, a SRS of 20 graduates, a SRS of 5 first professional students, and a SRS of 20 special students; combine these 100 students.

19. Stratified Random SampleExample We would like to take a sample to represent Canadian population We have different provinces and we wish represent the all provinces should be represented in the sample A stratified random sample of 1000 people could be chosen as follows: From each province, we select random samples. Since population in each province differ heavily, samples from each province should be proportional to its population.

20. Multistage Sample • several stages of sampling are carried out • useful for large-scale sample surveys • samples at each stage may be SRSs, but are often stratified • stages may involve other random sampling techniques as well (cluster, systematic, random digit dialing, …)

21. Cautions about Sample Surveys • Undercoverage • some individuals or groups in the population are left out of the process of choosing the sample • Nonresponse • individuals chosen for the sample cannot be contacted or refuse to cooperate/respond • Response bias • behavior of respondent or interviewer may lead to inaccurate answers or measurements • Wording of questions • confusing or leading (biased) questions; words with different meanings

22. Nonresponse • To prepare for her book Women and Love, Shere Hite sent questionnaires to 100,000 women asking about love, sex, and relationships. • 4.5% responded • Hite used those responses to write her book • angry women are more likely to respond

23. Response Bias • A door-to-door survey is being conducted to determine drug use (past or present) of members of the community. Respondents may give socially acceptable answers (maybe not the truth!) • For this survey on drug use, would it matter if a police officer is conducting the interview? (bias from interviewer)

24. Asking the UninformedWashington Post National Weekly Edition (April 10-16, 1995, p. 36) Response Bias • A 1978 poll done in Cincinnati asked people whether they “favored or opposed repealing the 1975 Public Affairs Act.” • There was no such act! • About one third of those asked expressed an opinion about it.

25. Wording of Questions A newsletter distributed by a politician to his constituents gave the results of a “nationwide survey on Americans’ attitudes about a variety of educational issues.” One of the questions asked was, “Should your legislature adopt a policy to assist children in failing schools to opt out of that school and attend an alternative school--public, private, or parochial--of the parents’ choosing?” From the wording of this question, can you speculate on what answer was desired? Explain.

26. Wording: Deliberate Bias • “If you found a wallet with $20 in it, would you return the money?” • “If you found a wallet with $20 in it, would you do the right thing of returning the money?”

27. Wording: Unintentional Bias • “I have taught several students over the past few years.” • How many students do you think I have taught? • How many years am I referring to? • “Over the past few days, how many servings of fruit have you eaten?” • How many days are you considering? • What constitutes a serving?

28. Wording: Unnecessary Complexity • “Do you sometimes find that you have arguments with your family members and co-workers?” • Arguments with family members • Arguments with co-workers

29. Wording: Ordering of Questions • “How often do you normally go out on a date? about ___ times a month.” • “How happy are you with life in general?” • Strong association between these questions. • If the ordering is reversed, then there would be no strong association between these questions

30. Inferences about the Population • Values calculated from samples areused to make conclusions (inferences) about unknown values in the population • Variability • different samples from the same population may yield different results for a particular value of interest • estimates from random samples will be closer to the true values in the population if the samples are larger • how close the estimates will likely be to the true values can be calculated -- this is called the margin of error