AP Statistics

1. AP Statistics Chapter 5

2. Class Survey • 1. Are you male or female? • 2. How many brothers or sisters do you have? • 3. How tall are you in inches to the nearest inch? • 4. Estimate the number of pairs of shoes you have. • 5. How much money in COINS are you carrying right now? • 6. On a typical school night, how much time do you spend doing HW? • 7. On a typical school night, how much time do you spend watching TV?

3. Can we know it all? • We have a “POPULATION”? • Want information about it! • We cannot “get at it” all! Why not? • So … • We gotta “Sample”! • To represent the entire Population

4. Careful how you choose! • Choosing a sample is not that simple. • Randomness is A MUST! • You want accurate representation • You want to make decisions … often very important … based on the information • Life & Death • Business – millions of $$$ - jobs • Scientific Research

5. Chow do I collect data? • OBSERVATIONAL STUDY • EXPERIMENT

6. Example of an Observational Study • Sample Survey • Reaches only a subset of a larger population of interest • Relatively easy to do • Quick • Does not disturb the population much at all in gathering the information – YOU, the observer, are not imposing a “treatment” on the subjects • Can gain information in several variables … or just one quick yes/no question

7. TREATMENT! • Experimental Design • Control the variation of confounding variables • USE OF RANDOMNESS • DO SOMETHING TO ONE GROUP and not the other • That thing you “DO”: “TREATMENT”

8. Observational Study vs. Experiment • An observational study : observes individuals and measures variables of interest but does not attempt to influence the responses. • An experiment : (on the other hand) deliberately imposes some treatment on individuals in order to observe their responses.

9. CAUSE & EFFECT • The best way to determine this is a:WELL DESIGNED EXPERIMENT

10. WELFARE • Why can we not conclude a cause and effect here? • Observational studies show job-training & job-search programs correlate to leaving the welfare system • CONFOUNDING: Education, Values, Motivation • To establish that the programs WORK (CAUSE) - need and EXPERIMENT!

11. SIMULATION • In many situations … • it may be impossible to observe individuals directly • it may be impossible to perform an experiment • it may be logistically difficult / inconvenient to sample • it may be unethical/costly to impose a treatment • Simulations provide an alterative method for producing data in such circumstances.

12. STATISTICAL INFERENCE • Statistical techniques for producing data open the door to formal branch of statistics • Statistical Inference • Making judgments about a unknown population • Conclusions are only true “with a known degree of confidence”

13. Population and Sample • POPULATIONThe entire group of individuals that we want information about • SAMPLEPart of the population that we actually examine in order to gather information

14. Sampling vs. Census • Samplinginvolves studying a part in order to gain information about the whole • Censusan attempt to contact every individual in the entire population.

15. Sampling vs. Census Accuracy • How could a sample be actually MORE ACCURATE? • The census will take too long, and things change IN THAT TIME • The census is impractical to really rely on • People get bore, tired and produce inaccurate results • It is too hard ot organize and maintain the volume of data

16. Sample Designs • SAMPLE DESIGN • How a sample is chosen - the method used to choose the sample from the population. • If conclusions based on a sample are to be valid - a sound design for selecting the sample is required

17. Voluntary Response Sample • BAD DESIGN • consists of people who choose themselves by responding to a general appeal • these samples are nearly always very biased because people with strong opinions, especially negative opinions, are most likely to respond.

18. Convenience Sampling • BAD DESIGN • another sampling design - which chooses individuals that are the easiest to reach • Both sample designs choose a sample that is almost guaranteed not to represent the entire population • These sampling methods display bias

19. BIAS • systematic error - in favoring some parts of the population over others • The design of a study is biased if it systematically favors certain outcomes

20. SRS - Simple Random Sample • A statistician’s remedy to BIAS • Allow impersonal chance to choose the sample • A sample chosen by chance allows neither favoritism by the sampler nor self-selection by respondents • Choosing a sample by chance attacks bias by giving all individuals an equal chance to be chosen …

21. Simple Random Samples Cont’d • A simple random sample (SRS) of size n consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected • A SRS not only gives each individual an equal chance to be chosen but also gives every possible sample an equal chance to be chosen.

22. Random Digits • A table of random digits is a long string of digits chosen from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} with these two properties: • Each entry in the table is equally likely to be any of the 10 digits 0 through 9. • The entries are independent of each other • Table of Random Digits

23. Choosing an SRS • Choose an SRS in two steps: • Label. Assign a numerical label to every individual in the population. • Table. Use table B to select labels at random. • 69051 64817 87174 09517 84534 06489 87201 97245

24. Choosing a Client • Listed in Book: 30 clients numbered from 01 through 30 • Example: 01 = A-1 Plumbing; 16 = JL Records; 30 = Von’s Video Store • We want to select 5 clients RANDOMLY from the list • Line 130: 69051 64817 87174 09517 84534 06489 87201 97245 • Chunk in sets of 2: 69 05 16 48 17 87 17 40 95 17 • Ignore those above 30 and repeats: • 69 05 16 48 17 87 17 40 95 17 … • Bailey Trucking … JL Records; Johnson Commodities, etc.

25. Other Sampling Designs • A probability sample – not really one we will use • Some probability sampling designs (such as SRS) give each member of the population an equal chance to be selected. This may not be true in more elaborate sampling designs. In every case, however, the use of chance to select a sample is the essential principle of statistical sampling.

26. Other Sampling Designs –Stratified Random Sample • First - divide the population into groups of similar individuals, called strata. • Choose a separate SRS in each stratum and combine these SRSs to form the full sample. • The strata is based on facts known before the sample is taken. • Can produce more exact information than an SRS of the same size by taking advantage of the fact that individuals in the same stratum are similar to one another. • If all individuals in each stratum are identical, just one individual from each stratum is enough to completely describe the population.

27. Other Sampling Designs Cont. • Another common means of restricting random selection is to choose the sample in stages. • Multistage samples select successively smaller groups with the population in stages, resulting in a sample consisting of clusters of individuals. • Ex: White … Males … Age 30 – 45 … Income between 50 & 100 K … who do not smoke.

28. Cautions About Sample Surveys • Undercoverage occurs when some groups in the population are left out of the process of choosing the sample • Non-response occurs when an individual chosen for the sample can’t be contacted or does not cooperate

29. More Cautions About Sample Surveys • Response Bias: The behavior of the respondent or of the interviewer can cause response bias in sample results. The respondent might lie in a face to face situation (shame). The interviewer my prod or imply a response • The wording of questions is the most important influence on the answers given to a sample survey. Confusing or leading questions can introduce strong bias, or even minor changes in wording can change a survey’s outcome

30. Inference About the Population • If we select two samples at random from the same population, we will draw different individuals. So the sample results will almost certain differ somewhat • Properly designed samples avoid systematic bias but their results are rarely exactly “correct” and they vary from sample to sample • The results from random sampling don’t change haphazardly from sample to sample • The results obey the laws of probability that govern chance behavior. We can say how large an error we are likely to make in drawing conclusions about the population from a sample

31. Inference About the Population Cont’d • One point we should consider: larger random samples give more PRECISE results that smaller samples – as it leads to smoothing out the variability of the extremes • Using a probability sampling design and taking care to deal with practical difficulties reduce bias in a sample. The size of the sample then determines how close to the population truth the sample result is likely to fall.