Parametric Statistics

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Parametric Statistics

• Descriptive statistics
• Hypothesis testing

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Definitions

• Population the entire set about which info is
  needed, Greek letters are used
• Sample a subset studied random samples
• Parameter numerical characteristic
• Inferential statistics
• Discrete and continuous
• Distribution the pattern of variation of a
  variable
• One sided probability comparing two data sets
  (ie. a gt b) two-sided probability a not equal
  to b

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Detection Limit

• Action Limit 2s, 97.7 certain that signal
  observed is not random noise.
• Detection Limit 3s, 93.3 certain to detect
  signal above the 2s limit when the analyte is at
  this concentration.
• Quantitation Limit 10s signal required for 10
  RSD
• Type I Error identification of random noise as
  signal
• Type II Error not identifying signal that is
  present.

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Numerical Descriptive Statistics

• Types of numerical summary statistics
• Measures of Location
• Measures of Center
• Other Measures
• Measures of Variability

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Probability Density Function

• Probability density function f(x) probability
  of obtaining result x for the variable in your
  experiment (relative frequency for discreet
  measurements)
• The total Sf(xi)1 the total sum of all relative
  frequencies
• Distribution function probability that x is
  less than or equal to xi
• F(xi) Sf(xj) over all j such that xj xi

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Discreet Data PDFs

• Binomial distribution
• f(x) (n!/(x!(n-x)!))px(1-p)n-x
• The probability of getting the result of interest
  (success) x times out of n, if the overall
  probability of the result is p
• Note that here, x is a discrete variable
• Integer values only

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Uses of the Binomial Distribution

• Quality assurance
• Genetics
• Experimental design

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Binomial PDF Example

• n 6 number of dice rolled
• pi 1/6 probability of rolling a 2 on any die
• x 0 1 2 3 4 5 6 sample results of 2s out
  of 6
• Graph of f(x) versus x for rolling a 2

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Binomial PDF Example 2

• n 8 number of puppies in litter
• pi 1/2 probability of any pup being male
• x 0 1 2 3 4 5 6 7 8 example data for the of
  males out of 8
• Graph of f(x) versus x

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Binomial PDF Characteristics

• Shape is determined by values of n and p
  (parameters of the distribution function)
• Only truly symmetric if p 0.5
• Approaches Poissons distribution if n is very
  large and p is very small,
• Approaches the normal distribution if n is large,
  and p is not small
• Mean number of successes or the expectation
  value X np
• Variance is np(1- p)

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Poisson Distribution

• Can be derived as a limiting form of Binomial
  Distribution
• when n?8 as the mean lnp remains constant
• this means conducting a large number of trials
  with p very small
• Can be derived directly from basic assumptions
• Assumptions determine the real situations where
  Poissons distribution is useful

Simeon D. Poisson (1781-1840)
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Poissons Assumptions

• Time or other interval type study
• The time interval is small
• The probability of one success is proportional to
  the time interval
• The number of successes in one time interval is
  independent of the number of successes in another
  time interval

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Derive Poisson from basic assumptions

• Derivation by Induction
• To find an expression for p(x), first find p(0),
  then p(k) and p(k1) then generalize for p(x).
• Basic properties used

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Poissons Assumptions Example

• The probability of one photon arriving in the
  time interval Dt is proportional to Dt when Dt is
  small
• The probability that more than one photon arrives
  in Dt is negligible for small Dt
• The number of photons that arrive in one time
  interval is independent of the number of photons
  that arrive in any other non-overlatping interval

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Normal Approximation to Poissons Distribution

• http//www.stat.ucla.edu/dinov/courses_students.d
  ir/Applets.dir/NormalApprox2PoissonApplet.html

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Measures of Center

• Mode
• Median
• Population Mean (µ) and Sample Average (x)

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Measures of Spread

• Variance square of standard deviation
• Standard deviation
• Population standard deviation s large sample
  sets, the population mean (µ) is known.
• Sample standard deviation (s) small sample sets,
  sample average (x) is used.
• Pooled standard deviation (s ). When several
  small sets have the same sources of indeterminate
  error (ie the same type of measurement but
  different samples)

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Standard Error of the Mean

• uncertainty in the average(sm) different from
  the standard deviation s (variation for each
  measurement) if n1, sm s
• i. If s is known, the uncertainty in the mean is
• ii. If s is unknown, use the t-score to
  compensate for the uncertainty in s.
• t - from a table for confidence level and n-1
  degrees of freedom. (one degree of freedom is
  used to calculate the mean.)

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Chebychev and Empirical Rules

• 's Rule The proportion of observations within k
  standard deviations of the mean, where , is at
  least , i.e., at least 75, 89, and 94 of the
  data are within 2, 3, and 4 standard deviations
  of the mean, respectively.
• Empirical Rule If data follow a bell-shaped
  curve, then approximately 68, 95, and 99.7 of
  the data are within 1, 2, and 3 standard
  deviations of the mean, respectively.

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Z-score

• -scores are a means of answering the question
  how many standard deviations away from the mean
  is this observation?''

z 0 1 2 3
P 1sided 0.5 .84 .98 .999
P 2sided 0 .68 .95 .99
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Confidence Interval

• The range of uncertainty in a value at a stated
  percent confidence
• Percent confidence that the value is within the
  stated range
• s is known
• s is unknown

Look up the appropriate z or t values to use
x /- ts/sqrt(N)
http//math.uc.edu/brycw/classes/148/tables.htm
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Inferential Statistics

• Comparing two sample means

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T-test (Student's t)

• Used to calculate the confidence intervals of a
  measurement when the population standard
  deviation s is not known
• Used to compare two averages
• corrects for the uncertainty of the sample
  standard deviation (s) caused by taking a small
  number of samples.

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Comparison Tests

• Comparing the sample to the true value.
• Comparing two experimental averages

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Significance Testing

• Confidence interval
• Statistical Hypotheses
• Ho and H1

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Comparison Test Comparing the sample to the true
value

• Method 1.
• If the difference between the measured value and
  the true value (µ) is greater than the
  uncertainty in the measurement, then there is a
  significant difference between the two values at
  that confidence level.
• Method 2.
• experimental t-score (t ) is compared to
  t-critical (found in a table)
• There is a significant difference if experimental
  t is greater than critical t .
• t is chosen for N-1 degrees of freedom at the
  desired percent confidence interval.
• If the experimental value may be greater or less
  than the true value, use a two sided t-score. If

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Comparison Tests Comparing two experimental
averages.

• t-test
• use the pooled standard deviation and calculate t
  as experimental
• If experimental t is greater than critical t then
  there is a significant difference between the two
  means.
• t is determined at the appropriate confidence
  level from a table
• the t-statistic for N N - 2 degrees of freedom.

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T-table