maths

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maths statistics from Probability to the
Normal Distribution Dr William
Megill 4E3.43 enswmm_at_bath.ac.uk
XX10118
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Quick Review

• Definition
• Statistics Analysis and interpretation of data
  with a view toward objective evaluation of the
  reliability of the conclusions based on the
  data.
• Why?
• Variability in the real world (Stochasticity)
• Issues with experimental data
• Issues with design for robustness

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Stats in experimental design

• Knowing some stats ahead of time is a good thing
• too often students/researchers attempt analysis
  of research data only to find
• they have too few data to see differences
• they have too much of the wrong data

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Types of statistics

• Descriptive stats
• Organising data to talk about them
• Inferential stats
• Inferring information about the whole (popn)
  from characteristics of its parts (sample)

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Types of data

• Ratio data
• Constant size interval between adjacent units
• Need a zero (to enable multiplication)
• e.g. a 15cm bar is half the length of a 30cm one
• Interval data
• Constant size interval
• No zero (or zero is arbitrary)
• e.g. temperature
• 25-20C (77-68F) 10-5C (50-41F)
• but 40C (104F) ltgt 2 x 20C (68F)
• NB. Kelvin temperature is Ratio scale

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Types of data

• Circular data
• Variation on interval scale
• e.g. compass degrees
• Ordinal data
• Ordered, but not constant interval
• e.g. Grades, comfort
• Nominal data
• No order
• e.g. car names, blood types

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Accuracy Precision

• Accuracy
• nearness of a measurement to the actual value
• Precision
• closeness of repeated measures to each other
• Significant figures
• refer to accuracy
• use all digits in calcs, report proper sig figs

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Frequency Distributions

• Exploring data using tables graphs

You can get graphs to say anything Be wary when
reading them Be honest when drawing them
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Grouping data

• how many groups?
• rule of thumb about 10 to span range

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Frequency Polygons

• Use instead of bar graphs/histograms
• Not for ordinal or nominal data x-axis not a
  ratio scale
• Absolute or relative frequency
• NB do not read intermediate frequencies

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Cumulative Frequency Polygons (Ogives)

• How many were better than...

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Populations Samples

• Primary objective of statistics
• to infer characteristics of a group
• by analysing characteristics of a small
  sampling of the group
• Population
• entire collection of measurements about which to
  draw conclusions
• e.g. cars in UK, lifetime of shock absorbers,
  bearing age at failure
• Sample
• subset of all of the measurements in the
  population
• from the characteristics of sample, draw
  conclusions about popn
• NB One often samples a population that does not
  exist
• e.g. fuel additive in 40 cars, measure gas
  consumption. Population is all cars which
  could have had additive.
• Such a population is called hypothetical or
  potential

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Random Sampling

• Every member of the population has an equal and
  independent chance of being selected
• each measurement in the population has an equal
  chance of being selected in the sample
• the selection of any member of the sample has no
  influence of the selection of any other member
• Usually this is obvious. Sometimes it isnt...

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Experimental Design

• Pseudoreplication
• Imagine a study of tyre wear.
• Experimenter has access to 8 cars, with 4 tyres
  each.
• How many independent measurements?

Depends on the experiment either 32 or 8
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Probability

• Definitions
• Experiment
• An activity with an observable result, or set of
  results
• e.g. tossing a coin
• Outcome
• An observable result of an experiment
• e.g. heads
• Event
• An outcome or set of outcomes of interest
• e.g. H
• Sample Space
• Set of all possible outcomes of an experiment
• e.g. H,T

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Counting outcomes

• IF
• Sample space of one experiment has k1 elements
• Sample space of another has k2
• Then
• Combined sample space has k1 x k2

duh...
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Permutations Combinations

• Permutation
• arrangement of objects in a specific sequence
• e.g. Horse (H), Cow (C), Sheep (S)
• six different arrangements
• HCS, HSC, CHS, CSH, SHC, SCH
• note once first is placed, choices diminish for
  others
• Notation nPn n(n-1)(n-2)...(3)(2)(1) n!

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Permutations

• Fewer than n positions...
• Horse, Cow, Sheep, Pig
• total if 4 positions 4P4 4! 24
• but if only two positions...
• HC,HS,HP,CH,CS,CP,SH,SC,SP,PH,PC,PS (12)
• If some objects indistinguishable...
• HHCC,CCHH,HCHC,CHCH,HCCH,CHHC

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Combinations

• group of objects where sequence doesnt matter
• HC,HS,HP,CH,CS,CP,SH,SC,SP,PH,PC,PS
• but HCCH, HSSH, etc.

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Sets

• Set Collection of elements x
• Subset
• Intersection Union
• Complement
• Venn Diagram

B
A
C
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Probability of an Event

• Likelihood of an event expressed by
• relative frequency observed from large dataset
• knowledge of the system under study

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Adding Probabilities

• Mutually exclusive?
• Add em up
• P(A or B) P(A) P(B)
• Not mutually exclusive?
• P(A or B) P(A) P(B) P(A and B)
• P(A and B) P(A)P(B)

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Distributions

• Definition
• the relative numbers of times each possible
  outcome will occur in a number of trials
• Probability function/density
• Function describing the probability that a given
  value will occur
• Distribution function
• Function describing the cumulative probability
  that a given value or any value smaller than it
  will occur

NB PDF is derivative of DF
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• Discrete Probability Distributions
• Binomial
• Poisson
• Hypergeometric
• Continuous Probability Distributions
• Normal

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Binomial Distribution

• Discrete probability distribution of obtaining
  exactly n successes out of N trials
• e.g. machine known to produce, on average, 2
  defective components. what is probability that 3
  items are defective in the next 20 produced
• Binomial coefficient number of ways of picking k
  unordered outcomes from n possibilities

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Poisson Distribution

• Used to find probability of a single event
  occurring a n times in an interval of time
• Different from Binomial, in that we dont know q
• e.g. death by horse kick
• Conditions
• Random events throughout an interval
• Interval can be subdivided such that
• Prob of gt1 event occurring in subinterval is zero
• Prob of 1 event occurring in subinterval is prop
  to length of subinterval
• Events in one subinterval independent of other
  subintervals

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Poisson Distribution

• Start with binomial distribution
• express prob as function of total obs
• rewrite
• take limit as N gets big
• voila

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Poisson Distribution

• So what?

Neat thing about Poisson distribution mean
(expectation) variance n
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Example
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Hypergeometric Distribution

• If M defective parts in a total population of N,
  then the prob of selecting r defectives in a
  sample of size n is

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Normal Distribution

• Generalisation of the binomial for continuous
  variables
• Remember
• Frequency distribution shows P(Xx)
• Std normal distn
• (m0, s21)
• Z statisticstandardised normal variable

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Normal Distribution

• Area under whole curve 1
• Probability of ?
• Area under proportion of curve between Z0 and
  Z0.54
• Use formula or lookup table
• p(0ltZlt0.54)0.2054
• linear interpolation
• Careful to read table caption
• Often given as proportion that lies beyond Z
• Important values
• p(0ltZlt0.955) 0.33
• p(0ltZlt1.96) 0.4750
• p(0ltZlt2.575) 0.495

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Normal Distribution

• Add/subtract areas to calculate other
  probabilities

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Applications of Normal Distribution

• i.e. not just nice curves...

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Probability Intervals

• 95 of observations within 1s of m
• p(0ltZlt1.96) 0.4750
• p(-1.96ltZlt0) 0.4750
• p(-1.96ltZlt1.96)0.95
• 5 of obs outsidethis range

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Probability Limits

• Z(x-m)/s -gt X m Zs
• p(-1.96ltZlt1.96)95 -gt X ? m-1.96s,m1.96s

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One-tailed probability

• p(-?ltZlt1.65)0.95

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Sums Diffs Normal Vars

• See workbook
• Basic point distribution of a sum or difference
  of normally distributed variables will itself
  also be normally distributed

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Hypothesis testing
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Verifying claims

• Advertising line
• Luxcar, makers of the best luxury cars
• Burnol, the finest fuel you can buy
• ConstructAll, designers of beautiful buildings
• the average life of these tyres is 20,000miles
• on avg, low energy bulbs will last 8000hrs
• average bottle contents 330ml

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Hypotheses

• Null alternative hypotheses
• Null nothing interesting is happening
• Alternate something interesting is happening
• e.g.
• Explosives engineers deciding how fast to run
• one says from experience, mean burn rate
  600mm/sec
• other says gt600mm/sec
• Stats test
• H0 m600
• H1 mgt600

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Types of Error

• Cant be 100 sure, so possible that
• (a) correct hypothesis rejected
• (b) false hypothesis accepted
• (a) Type I error
• (b) Type II error

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Test of proportion

• Hypotheses
• H0 99 of the control modules match the spec
• H1 lt99 of the control modules match the spec
• Distribution?
• either they match or they dont -gt binomial
• calculate m and s
• Normal approximation
• because
• we can use

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Graphically...

• Assess how close 985 is to expected 990
• Convert to standard normal curve
• H1 is directional gt one tail
• since H1 pltp0, use left tail
• Define 95 confidence
• Probability is
• Discrete binomial, not continuous normal, so
  were approximating

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Choosing a tail
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Test of a population mean

• e.g. to verify a manufacturers claim re boiling
  point of a coolant
• if we know the population variance (i.e. from
  spec sheet, or large of experiments)
• null hypothesis- H0 x m
• distribution- assume normal need mean stdev
• since were sampling, we need std error of mean
• assume large np gt
• calculate Z statistic
• alternate hypothesis

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Testing a manufacturers claim

• back to coolant example

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Experimental differences between paired
treatments can be tested by comparing sample of
differences to a population with mean of zero....
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Two-sample testing

• Two samples...
• do they represent a real (significant) treatment
  effect, or
• just two samples from the same population?

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Two-sample testing

• same as before
• H0 m1m2 H1 m1 ? m2
• Combined variance
• Z statistic
• compare to standard normal distribution
• two-tailed in this case, Z0.05(2)1.96

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Two-sample testing

• Unknown, but equal variance (usual case)
• Use pooled sample variance
• t statistic
• n1n2-2 degrees of freedom
• Unknown, unequal variance
• same statistic
• but degrees of freedom, n

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