AP Statistics

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AP Statistics

• Chapter 5

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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?

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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

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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

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Chow do I collect data?

• OBSERVATIONAL STUDY
• EXPERIMENT

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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

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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

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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.

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CAUSE EFFECT

• The best way to determine this is aWELL
  DESIGNED EXPERIMENT

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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!

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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.

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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

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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

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Sampling vs. Census

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

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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

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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

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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.

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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

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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

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SRS - Simple Random Sample

• A statisticians 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

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Simple Random Samples Contd

• 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.

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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

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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

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Choosing a Client

• Listed in Book 30 clients numbered from 01
  through 30
• Example 01 A-1 Plumbing 16 JL Records 30
  Vons 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.

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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.

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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.

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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.

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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 cant be contacted or does not
  cooperate

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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 surveys outcome

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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 dont 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

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Inference About the Population Contd

• One point we should consider larger random
  samples give more accurate 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.