Computation Details Confidence Intervals for the Center of Data

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Computation Details -Confidence Intervals for
the Center of Data

• Interest in median (typical value)
  non-parametric interval or parametric interval.
• A NP interval estimate for the true population
  median is computed using the cumulative
  probabilities from the Binomial distribution.
• 1. The desired ? is stated, acceptable risk of
  not including the true median.
• 2. ?/2 of this risk is assigned to each end of
  the interval,

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• 3. Table A of the cdf of the binomial
  distribution with parameters n and p 0.5
  provides the lower and upper critical values x
  and x at one-half the desired ? level. This
  table is identical to the one used for the Sign
  Test to be discussed later.
• 4. These critical values are transformed to ranks
  R1 and Ru corresponding to data points Cl and Cu
  at the ends of the confidence interval.
• Rl x 1
• Ru n - x x

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• The resulting confidence interval will reflect
  the shape (skewed or symmetric) of the original
  data.
• NP interval cannot always exactly produce the
  desired confidence level when the sample sizes
  are small.

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• Minitab (SINTERVAL) uses non-linear interpolation
  to get exact (1 - ?) levels.
• For n gt 20, a normal approximation can be used to
  compute the intervals.
• Computed ranks are rounded to the nearest integer
  when necessary.

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Parametric Interval Estimate for the Median

• As mentioned in FIRST LECTURE, the geometric mean
  of X (GMX) is an estimate of the median of X when
  yln(X) is normal or fairly symmetric.
• The mean of y and confidence interval on the mean
  of y become the geometric mean with its
  (asymmetric) confidence interval after being
  transformed back to original units by
  exponentiation.

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Confidence Intervals for the Mean

• Intervals may also be computed for the true
  population mean ?. These are appropriate if the
  center of mass of the data is the statistic of
  interest.
• Symmetric intervals around the sample mean are
  computed most often. For large sample sizes a
  symmetric interval adequately describes the
  variation of the mean regardless of the shape of
  the data distribution.
• Note C.I. on the geometric mean is not an
  interval estimate of the mean.

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Prediction Intervals to Evaluate a Future New
Observation

• The question is often asked whether a new
  observation is likely to have come from the same
  distribution as previously collected data.
• This can be evaluated by determining whether the
  new observation is outside the prediction
  interval computed from existing data.
• E.g. Calculate prediction interval from
  background data (or same well data), check new
  compliance data (or new observation in same well)
  with prediction interval.

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• PI are wider that CI because an individual
  observation is more variable than is a summary
  statistic.
• NP prediction interval - valid for all data.
• Symmetric prediction interval - valid only for
  symmetric data
• Asymmetric prediction interval - valid only when
  logs are symmetric.
• For a normal population, table is available for
  prediction interval of k new observations.

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Two - Sided NP PI

• The NP PI of confidence level (1 - ?) is simply
  the interval between ?/2 and (1- ?/2) percentile
  of the distribution.

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• The interval contains 100(1- ?) percent of the
  data.
• Therefore if the new observation comes from the
  same distribution as the previously measured
  data, there is a 100 ? chance that is will lie
  outside the previously measured data.
• to

One-Sided NP PI
One-sided PI are appropriate if the interest is
in whether a new observation is larger than
existing data, or smaller than existing data, but
not both.
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• One-sided PI
• New lt X?(n1) or New gt X(1- ?)(n1) (But not
  either or).

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Parametric PI - Symmetric PI

• Assumes data follow a normal distribution. PI
  are considered to be symmetric around the sample
  mean and wider than CI on the mean.
• The equation for PI differs from that for the CI
  by adding a term s, the std. dev. of an
  individual observation around their mean.
• PI to

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• Difference in length between the PI and CI is
• There PI can be computed from CI and sample size
  n. This is useful when using Minitab or other
  software which do not have a routine to compute
  the PI.
• One sided PI are computed as before using ?
  rather than ?/2 and comparing new data to only
  one end of the PI.

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Asymmetric PI

• This is based on the log of the skewed data.
• PI
• to
• where y ln(X)
• Use parametric intervals when data are normal or
  lognormal only.

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Confidence Intervals for Quantiles (Percentiles)

• E.g. Whats the CI of the 100 yr. flood?
• The 100 yr. flood is the 99th percentile (0.99
  quantile) of the distribution of annual flood
  data.
• Similarly, the 2 yr. flood is the median or 50th
  percentile of annual floods.
• In environmental monitoring, the median, 95th, or
  some other percentile should not exceed (or be
  below) a standard (e.g. water quality standard.

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Valid for All Data - NP Interval

• Similar procedure as for median, except binomial
  probability is with parameters n and p quantile
  of interest.
• For n gt 20, the normal approximation can be used.
• The 0.5 is a continuity correction term.
• The computed ranks Rl and Ru are rounded to the
  nearest integer.

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NP Test Whether a Percentile Differs From Xo
(2-sided test)
NP Tests for Percentiles

• E.g. A water quality standard Xo could be set
  such that the median of daily concentrations
  should not exceed Xo ppb.

• Compute interval for percentile, if Xo falls
  within this interval, the percentile is not
  significantly different from Xo at the ? level

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NP Test for Whether a Percentile Exceeds Xo
(One-sided Test)

• Compute one-sided CI for the percentile.
  Remember that the entire error level ? is placed
  on the side below the percentile point estimate.

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NP Test for Whether a Percentile is Less than Xo
(One-sided Test)

• Compute one-sided CI for the percentile, place
  all error ? on one side above the estimated
  percentile.

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Intervals for Normal Population

• Factors for calculating two-sided 95 intervals
  for a normal distribution (or transformed to
  normal) are available in a table.
• For the 95 CI to contain the true mean, it is
  given by
• where the factor cM(n) is obtained from the
  table, and s is the standard deviation.
• The factor is basically the 97.5th percentile of
  the t-distribution with n-1 degrees of freedom.
• E.g. If n5, calculated mean 50.1 and
  standard deviation 1.31, factor is 1.24.

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• To compute the 95 tolerance intervals to contain
  a specific percentage e.g. 90 of a normal
  population, we use
• where, the factor is cT,90 (n) for the given
  sample size n, and s is the standard deviation.
• E.g. We have 5 observations and the sample mean
  is 50.1, and the standard deviation is 1.31.
  From the table, the factor to contain 90 of a
  normal distribution with 95 confidence when n5
  is 4.28. Hence, the tolerance interval is from
  44.49 to 55.71.
• Other percentages 95 and 99 are also given in
  the table.

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• Prediction intervals for future observations (PI
  to contain all k future observations) are
  computed similarly using the factors from the
  table. With 95 confidence that all k future
  observations from a previously sampled normal
  population will be located in the interval given
  by
• for k 1, 2, 5, 10, and 20.
• E.g. Random sample of n5 observations, a 95 PI
  to contain the values of k2 further randomly
  selected observations from the same population is
  50.1 3.70 (1.31).

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Comparison of length of intervals
Comparison of lengths of statistical intervals
for examples used.
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Bootstrap Methods for Standard Errors and
Confidence Intervals

• The bootstrap method for estimating the standard
  error of a statistic is one of the most
  significant developments in the field of
  statistics. The method has also been called the
  Computer Intensive method or a Resampling method.
  It can be used for hypothesis testing and
  probability evaluation. The method is completely
  nonparametric. The essential features of the
  bootstrap approach are best illustrated by and
  example.

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• Suppose we wish to estimate the median m from a
  sample of 13 data points.
• 19.2 16.2 10.7 16.6 3.6 18.1 8.6
• 15.3 14.0 14.2 16.9 13.4 5.7
• and that we want the standard error of estimate,
  or CI for m. The sample median of these data is
  m 14.0.
• Step 1
• Draw a random sample of size 13, with
  replacement, from the original sample, e.g.
• 3.6 19.2 8.6 15.3 8.6 3.6 14.2 10.7 10.7
  10.7 16.9 14.2

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• This is the first bootstrap sample. Note that
  some of the original sample values occur more
  than once, and others not at all. The keyword
  here is with replacement.
• Step 2
• Calculate the median for the bootstrap sample
• m 10.7
• Step 3
• Carry out Steps 2 and 3 a large number B (say
  200) times, to obtain 200 bootstrap estimates
  m1, ... , m200.

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• Step 4
• The standard deviation of the 200 m values gives
  the standard error of m.
• To obtain the confidence interval, first sort the
  200ms in ascending order (lowest to highest).
  For a 90 confidence interval for m, choose the
  rank 10 and 191 bootstrap values for the lower
  and upper confidence limits.
• The above steps can be easily implemented in
  MINITAB by writing a short simple macro. The
  macro can be written during the MINITAB session
  or can be written using a text editor.

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• Assume data is in column C1. Use Notepad or text
  editor type
• gmacro
• bootmed
• do k11200
• sample 13 c1 c2
• replace.
• let c3(k1) median(c2)
• enddo
• endmacro
• save file as bootmed.mtb in macro directory of
  Minitab
• to run, type during Minitab session
• MTBgt bootmed
• the 200 values of the bootstrap medians will be
  in C3.
• DESCRIBE C3 will give stats of the 200 medians.

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Summary

• 1. Probability vs. Statistics (Deductive vs.
  Inductive)
• 2. Things to consider in an estimator.
• 3. Parametric and Non-parametric interval
  estimates.
• 4. Types of interval estimates
• confidence - mean or median
• prediction - one or more future values
• confidence -on percentile (tolerance)