Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern MathematicsSixth Edition

• Peter Tannenbaum

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Chapter 13Collecting Statistical Data

• Censuses, Surveys, and Clinical Studies

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Collecting Statistical DataOutline/learning
Objectives

• To identify whether a given survey or poll is
  biased.
• To list and discuss the quality of several
  sampling methods.
• To identify components of a well-constructed
  clinical study.

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Collecting Statistical DataOutline/learning
Objectives

• To define key terminology in the data collection
  process.
• To estimate the size of a population using the
  capture-recapture method.

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Collecting Statistical Data

• 13.1 The Population

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Collecting Statistical Data

• Population
• Every statistical statement refers, directly or
  indirectly, to some group of individuals or
  objects.
• N-value
• Given a specific population, an obviously
  relevant question is, How many individuals or
  objects are there in that population?

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Collecting Statistical Data

• Census
• The process of collecting data by going through
  every member of the population.

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Collecting Statistical Data
Over the last 45 years, the United States Fish
and Wildlife Service has been able to keep a
remarkably accurate tally of the number of bald
eagle breeding pairs in the lower 48 states.
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Collecting Statistical Data
A tremendous amount of effort has gone into
collecting and verifying these N-values, which,
for a wildlife population, are of remarkable
accuracy. The above figure summarizes the
population numbers over the period 1963-2000.
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Collecting Statistical Data

• 13.2 Sampling

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Collecting Statistical Data

• Survey
• The practical alternative to a census is to
  collect data only from some members of the
  population and use that data to draw conclusions
  and make inferences about the entire population.
• Poll
• When the data collection is done by asking
  questions.

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Collecting Statistical Data

• Sample
• The subgroup chosen to provide the data.
• Sampling
• The act of selecting a sample.

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Collecting Statistical Data

• Target population
• The most important step in a survey is to
  distinguish the population for which the survey
  applies.
• Sampling frame
• The actual subset of the population from which
  the sample will be drawn.

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Collecting Statistical Data

• Public Opinion Polls
• Selection bias
• When the choice of the sample has a built-in
  tendency to exclude a particular group or
  characteristics within the population.
• Response rate
• The percentage of respondents out of the total
  sample.
• Nonresponse bias
• When the response rate to a survey is low.

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Collecting Statistical Data

• Convenience Sampling
• In convenience sampling the selection of which
  individuals are in the sample is dictated by what
  is easiest for the data collector.
• A classic example is when interviewers set up at
  a fixed location such as a mall or outside a
  supermarket and ask passersby to be a part of a
  public opinion poll.

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Collecting Statistical Data

• Quota Sampling
• Quota sampling is a systematic effort to force
  the sample to be representative of a given
  population through the use of quotas the sample
  should have so many women, so many men, so many
  blacks, so many whites, so many people living in
  urban areas, so many people living in rural
  areas, and so on.

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Collecting Statistical Data

• 13.3 Random Sampling

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Collecting Statistical Data

• Random sampling
• Sampling methods that use randomness as part of
  their design.
• Random sample
• Any sample obtained through random sampling.

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Collecting Statistical Data

• Simple Random Sampling
• It is based on the same principle a lottery is.
  Any set of numbers of a given size has an equal
  chance of being chosen as any other set of
  numbers of that size.

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Collecting Statistical Data

• Stratified Sampling
• The alternative to simple random sampling used
  nowadays for national surveys and public opinion
  polls. The basic idea of stratified sampling is
  to break the sampling frame into categories,
  called strata, and then randomly choose a sample
  from these strata.

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Collecting Statistical Data

• 13.4 Sampling Terminology and Key Concepts

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Collecting Statistical Data

• Statistic
• To describe any kind of numerical information
  drawn from a sample.
• Parameter
• An estimate for some unknown measure of the
  population.
• Sampling error
• To describe the difference between a parameter
  and a statistic used to estimate that parameter.

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Collecting Statistical Data

• Chance error
• The result of the basic fact that a sample,
  being just a sample, can only give us approximate
  information about the population.
• Sampling variability
• Different samples are likely to produce
  different statistics for the same population,
  even when the samples are chosen in exactly the
  same way.

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Collecting Statistical Data

• Sample bias
• The result of choosing a bad sample and is a
  much more serious problem than chance error.
• Sample proportion
• The size of the sample, denoted by n (to
  contrast with N, the size of the population).
• The ratio n/N is the sample proportion.

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Collecting Statistical Data

• 13.5 The Capture-Recapture Method

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Collecting Statistical Data

• The Capture-Recapture Method
• Step 1. Capture (sample) Capture (choose) a
  sample of size n1, tag (mark, identify) the
  animals (objects, people), and release them back
  into the general population.
• Step 2. Recapture (resample) After a certain
  period of time, capture a new sample of size n2,
  and take an exact head count of the tagged
  individuals. Lets call this number k.

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Collecting Statistical Data

• Small Fish in a Big Pond
• A large pond is stocked with catfish. As part
  of a research project we need to estimate the
  number of catfish in the pond.
• Step 1. For our first sample we capture a
  predetermined number n1 of catfish, say
  n1 200. The fish are tagged and released
  unharmed back in the pond.

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Collecting Statistical Data

• Small Fish in a Big Pond
• Step 2. After giving enough time for the released
  fish to mingle and disperse throughout the pond,
  we capture a second sample of n2 catfish. While
  n2 does not have to equal n1, it is a good idea
  for the two samples to be of approximately the
  same order of magnitude. Lets say that n2
  250.
• Of the 250 catfish in the second sample, 35 have
  tags (were part of the original sample).

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Collecting Statistical Data

• Small Fish in a Big Pond
• The ratio of tagged fish in the second sample is
  the same as the ratio of tagged fish in the pond.
• 35/250 ? 200/N
• which in turn gives
• N ? 200 X 250/35 ? 1428.57
• A sensible conclusion is that there are
  approximately N 1400 catfish in the pond.

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Collecting Statistical Data

• Clinical Studies Terminology
• Clinical study (trial). Studies concerned with
  determining whether a single variable or
  treatment can cause a certain effect.
• Confounding variables. All other possible
  contributing causes that could produce the same
  effect in a clinical study.

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Collecting Statistical Data

• Clinical Studies Terminology
• Controlled study. The subjects are divided into
  two different groups.
• Treatment group. Subjects receiving the actual
  treatment.
• Control group. Subjects that are not receiving
  any treatment.

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Collecting Statistical Data

• Clinical Studies Terminology
• Randomized controlled study. The subjects are
  assigned to the treatment group or the control
  group randomly.
• Placebo effect. A critical confounding variable
  from the generally accepted principle that just
  the idea that one is getting a treatment, can
  produce positive results.

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Collecting Statistical Data

• Clinical Studies Terminology
• Placebo. A make-believe form of treatment a
  harmless pill, an injection of saline solution,
  or any other fake type of treatment intended to
  look like the real treatment.
• Controlled placebo study. A controlled study in
  which the subjects in the control group are given
  a placebo.

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Collecting Statistical Data

• Clinical Studies Terminology
• Blind. A study in which neither the members of
  the treatment group nor the members of the
  control group know to which of the two groups
  they belong.
• Double-blind study. A controlled placebo study
  in which neither the subjects nor the scientist
  conducting the experiment know which subjects are
  in the treatment group and which are in the
  control group.

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Collecting Statistical Data
Conclusion

• Census
• Sample/ Survey/ Sample Bias
• Simple Random/Stratified Sampling
• Confounding Variables
• Controlled Study