Elementary Statistics

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Elementary Statistics

• Sampling

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Where are we?

• Scientific Method
• Formulate a theory -- hypothesis
• Collect data to test the theory--??
• Analyze the results
• Interpret the results and make a decision
  --p-value approach

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Collect data Sample--- Good data

• Sample
• A part of the population that is actually used to
  get information
• To test a theory
• Collect data sample
• Collect good data

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What is good data?

• Biased??
• Suppose a instructor would like to learn about
  how the class is going. He might ask students in
  class to raise their hands if they are not dong
  well in the class.
• Data 0/30 students indicate that they are not
  doing well.
• Conclusion Everyone is doing great!
• Biased too high
• Response biase

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Biased data

• A sampling method is biased if it produces
  results that systematically differ from the truth
  about the population.
• A convenience sample is a sample consisting of
  units of the population that are easily
  accessible.
• Conduct a survey by phone at daytime
• A volunteer sample is a sample consisting of
  units of the population, which chose to respond.
• Conduct a survey by on-line fill-in form

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Other type of bias Selection bias

• Selection Bias is the systematic tendency on the
  part of the sampling procedure to exclude or
  include a certain type of unit.
• Nonresponse Bias is the distortion that can arise
  because a large number of units selected for the
  sample do not respond or refuse to respond, and
  these no responders have a tendency to be
  different from the responders.
• Response Bias is the distortion that can arise
  because the wording of a question and the
  behavior of the interviewer can affect the
  responses received.

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Examples

• Do you agree or disagree President Bushs
  proposal .. response bias
• Comments regarding the instructor (organized,
  enthusiastic ,etc)-response bias
• Mail survey
• A self-addressed stamped envelope reduce non
  response bias

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Fair selection method?

• A sampling method that gives each unit in the
  population a known, non-zero chance of being
  selected is called a probability sampling method.
  
• Simple Random Sampling
• Stratified Random Sampling
• Systematic Sampling
• Cluster Sampling
• Multistage Sampling.

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Simple Random Sampling

• Random--- Choose indiscriminately

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Example Choose two students at random in class
(n26).

• Step 1 assign labels
• Step 2
• Use random table
• Row 16 Column 16
• 51259 77452 16308 60756 92144 49442 53900
• Or Use calculator or computer
• Choose seed, randInt(1,24) or randInt(1,24,2)

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rand ---- press MATH button, select
PRB randInt ----, press MATH button, select PRB,5
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SRS Simple Random Sampling

• SRS unbiased and objective
• May not be efficient for large population.
• Note we need a list of all units of the
  population.

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Stratified Random Sampling

• Idea dividing into subgroups, then take a SRS.
• To study the average cost of day care
• Licensed daycare center
• Home daycare provider
• A stratified random sample is selected by
  dividing up or stratifying the population into
  mutually exclusive subgroups (strata) and taking
  a simple random sample of units from each
  stratum. The units sampled from each stratum are
  combined to form the complete sample.
• mutually exclusive each unit belongs to only one
  stratum.

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Example 2.16 Weighted Average

• Suppose that a population consists of four
  females with response values 1,2,3,4, and three
  males whose response values are 5,6,7. Object
  population average
• Step 1 divide into two stratum female male
• Step 2
• Pick a SRS from female stratum
• Pick a SRS from male stratum
• Step 3
• Combine samples to calculate the weighted average

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• Let's Do It! 2.8 Faculty Salaries
• A study is being conducted of the faculty
  salaries for a public
• university. Of the 1200 tenure-track faculty, 480
  are full
• professors, 336 are associate professors and the
  remaining are
• assistant professors. A simple random sample of
  100 full
• professors, 50 associate professors and 50
  assistant professors will
• be taken and information about salary will be
  obtained.
• (a) What type of sampling method is used to
  obtain the
• 200 selected faculty members?
• (b) Using your calculator (with a seed value of
  18) or the
• random number table (Row 14, Column 1) to give
  the
• labels for the first 5 full professors to be
  selected.
• (c) The table summarizes the average salary for
  the sampled
• faculty members by rank.
• Give the overall estimate of the average salary
  for all faculty

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Stratified random sampling

• Note with stratified random sampling, not all
  samples of size n have the same chance.
• Note ideally, the variability of the units
  within each stratum should be small compared to
  the variability between the strata.

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Systematic Sampling

• For a 1-in-k systematic sample, you order the
  units of the population in some way, and randomly
  select one of the first k units in the ordered
  list. This selected unit is the first unit to be
  included in the sample. You continue through the
  list selecting every kth unit from then on.
• Example 2.18 -- A 1-in-4 Systematic Sample
• 19 units in population, A through S, take a
  1-in-4 systematic sample.
• A B C D E F G H I
• 1 2 3 4
• J K L M N O P Q R S

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Cluster Sampling

• In cluster sampling, the units of the population
  are grouped into clusters. One or more clusters
  are selected at random. If a cluster is
  selected, all of the units that form that cluster
  are included in the sample.
• Notice the difference between stratified sampling
  method and cluster sampling method.

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Summary
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