Concepts of Probability and Statistics Statistical Intervals

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Concepts of Probability and StatisticsStatistic
al Intervals
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Probability and Statistics - Best Illustrated
with an example

• Suppose I flip a coin 10 times and get 8H, 2T,
  but I dont know the coin is really fair. What
  is my chance of tossing heads on the next throw?
• A) If I only have sample information at my
  disposal, I estimate 80.
• B) If I consider the total population of
  possible tosses, I would say 50.
• These two cases illustrate the basic difference
  between probability and statistics.

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• Probability deals with the likelihood of
  observing an event arising from a known process
  (model).
• i) Based on deductive reasoning
• ii) Begin with knowing something about the
  population (e.g. that the coin is fair)
• iii) Reason from population to the sample (e.g.
  how likely will the next toss be H?)
• Statistics approaches the problem backwards
  given a collection of observed data (the sample),
  one of many possible subsets of data, what can be
  said about the process (the entire population of
  data)?
• i) Based on inductive reasoning
• ii) Begin with knowing only about the sample
  results, not the population (e.g. 8H and 2T)
• iii) Reason from sample to population (e.g. is
  the coin fair based on sample results.

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• Since we dont usually know the behaviour of the
  overall population, but only about the sample
  results we collect, we are forced to make
  statistical inferences in order to estimate
  typical behaviour in the population from the
  sample results.
• As the coin tossing example illustrates, even if
  the average behaviour of the coin is to land on H
  50 of the time, specific sample results will
  vary from experiment to experiment.
• So to adequately estimate or predict the coins
  true average behaviour, one must carefully
  account for between-sample variability.

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• In an environmental pollution setting, the sample
  results will vary from period to period even if
  no contamination has occurred. Why?
• Variation in lab measurements of concentration of
  individual samples.
• Sampling variability from field collection and
  handling.
• Natural variation in background levels of
  pollutants.
• These factors combine to give random variation in
  sample results that will be seen whether or not
  contamination has occurred.
• Despite the sample fluctuations due to random
  variation, we still want to know for example if
  compliance concentrations are significantly
  higher than background concentrations on average.

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• Note that the degree of sample fluctuation in
  background and compliance point data relative to
  the difference in average background and
  compliance point concentration levels plays a
  crucial role in distinguishing background
  behaviour from compliance point behaviour.
• Only by careful measurement of sample variability
  can we accurately make statistical inferences
  about the behaviour of the overall population.
• For example, consider the question
• Is the long-term average concentration level at
  the compliance point greater than the
  background levels?

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• One way to answer the above question is to set
  up a hypothesis test using the results of sample
  groundwater analyses.
• A hypothesis test makes a decision as to which of
  two competing notions is closer to reality or
  truth.
• It is a type of statistical inference that reason
  from sample results back to the population .
• It is used in environmental monitoring settings
  since samples are costly to analyze only limited
  data are typically available for statistical
  purposes.

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• Example Flip a coin 100 times and get all
  heads.
• What do we decide about the coin? What is the
  chance of getting heads on the next toss?
• Answer Chance is 100. Why? Because coin is
  almost certainly two-headed!
• Notion being tested is whether the coin is fair
  or not.
• If we say it is fair, what evidence can we use to
  support our claim?
• Prob (100H in 100 tosses of fair coin) (1/2)100
  approx.. zero.
• However, alternative notion is that coin is
  2-headed is well supported by evidence at hand.
• The key is to determine which notion is best
  supported by the evidence.

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• In an environmental setting (monitoring and
  remediation), first make sure that the hypothesis
  being tested is appropriate.
• In detection monitoring, this becomes Ho No
  contamination versus Ha evidence of
  contamination. E.g. innocent until proven
  guilty
• In remediation, the null hypothesis changes to
  Ho guilty until proven innocent or dirty
  until proven clean.
• Then choose a statistical test that measures
  whether the sample data side better with Ho or Ha

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Point Estimation

• The sample median and sample mean estimate the
  corresponding center points of a population.
  Such estimates are called point estimates.
• For example, point estimators for the 100-year
  flood might be
• a) the largest flood which occurred during 100
  years or record.
• b) Q0.99 meanQ stdQ x Z0.99, using the
  mean and standard deviation of the flood record
  (assumes a normal distribution of the Qs).
• c) Q0.99 expmeanlogQ stdlogQ x K0.99,
  where K0.99 is the P3 distribution frequency
  factor for a skewness g.
• d) Q0.99 from regional equations.

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Things to Consider in Selecting an Estimator

• It should have little or no BIAS
• It should have low MEAN SQUARE ERROR.
• It should be RESISTANT. I.e. not affected by a
  few unusual values.
• It should be ROBUST. I.e. Its MSE should
  compare favourably with wide range of assumptions
  (e.g. distribution).
• It should be REPRODUCIBLE. Others should be able
  to repeat the calculation with no difference in
  results.

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Interval Estimation

• We want to estimate a statistical interval
  because a point estimate tells us nothing about
  the variability of the statistic. Since any
  statistic is itself a random variable, it is thus
  very important to know how it might fluctuate.
• E.g. There is a big difference between
• 20 ppm ? 10 ppm and 20 ppm ? 2 ppm.
• Interval estimates are intervals which have a
  stated probability of containing the true
  population value.
• The intervals are wider for data sets having
  greater variability.

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• Interval estimates can provide three pieces of
  information which point estimates cannot
• 1. A statement of the probability or likelihood
  that the interval contains the true population
  value (its reliability).
• - confidence intervals.
• 2. A statement of the likelihood that a single
  data point with specified magnitude comes from
  the population under study.
• - prediction intervals.
• 3. A statement of the likelihood that the
  interval contains a certain proportion of all
  population values.
• - tolerance intervals.

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Differences Between the Interval Types

• Statistical intervals have different uses
  depending on the purpose in mind
• Astronaut example
• An astronaut awaiting his tour of duty on the
  space shuttle is not concerned about what happens
  on average during such flights (confidence
  interval), nor what happens 95 of all flights
  (tolerance interval), but rather with what will
  happen on his or her specific flights (prediction
  interval).

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• Casino example
• A player is concerned with what he or she will
  win on the next few bets (prediction interval)
  the casino owners care about their average
  winnings in order to make a profit (confidence
  interval) while the roulette operator who makes
  a commission on each bet lost by a player is
  concerned about long-run proportion of lost bets
  (tolerance interval).
• Remember, the type of interval used can make a
  huge difference in the resulting decision -- in
  general, the widths of confidence, tolerance, and
  prediction intervals will be very different on
  the same sample data.

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• The width of the interval indicates the amount of
  potential error or variability associated with
  the sample average.
• The width depends on three factors
• i. Estimated standard deviation of sample data,
  s.
• ii. Level of confidence chosen beforehand.
• iii. Sample size, n.
• To reduce the width of a random interval, either
• i. Increase the sample size, or
• ii. Lower the acceptable confidence level (e.g.
  from 95 to 90).

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Parametric, Non-parametric, Symmetric and
Asymmetric Intervals

• Parametric
• assumes data (or transformed data) are normally
  distributed.
• Non-parametric
• distribution free (based on ranks).
• Symmetric
• interval divided equally on either side of true
  value. Usually used with parametric intervals.
• Asymmetric
• Interval not divided equally on either side of
  true value. It is used with skewed data (e.g.
  lognormal data).

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Sampling Distributions

• Probability of statistics e.g. mean, median,
  stdev, etc. Used for constructing statistical
  intervals.

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• While the previous statement is true, when
  dealing with environmental data however, the
  distribution of data are usually positively
  skewed. In these cases, use of the
  t-distribution may not be appropriate.
• Sometimes the t-distribution can be used after a
  suitable transformation of the data is made. E.g.
  after a log transform.