Raoul LePage

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Raoul LePage Professor STATISTICS AND
PROBABILITY www.stt.msu.edu/lepage click on
STT315_Sp06
Week 4
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WEEK PLAN Normal Approx of Binomial Poisson and
its normal Approx Exponential Selected material
from Ch. 1.
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Normal Approx of Binomial
n 10, p 0.4 mean n p 4 sd root(n p q)
1.55
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Normal Approx of Binomial
n 30, p 0.4 mean n p 12 sd root(n p q)
2.683
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Normal Approx of Binomial
n 100, p 0.4 mean n p 40 sd root(n p q)
4.89898
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Poisson Distribution Governing Counts of Rare
Events
p(x) e-mean meanx / x! for x 0, 1, 2, ..ad
infinitum
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Poisson
e..g. X number of times ace of spades turns up
in 104 tries X Poisson with mean 2 p(x)
e-mean meanx / x! e.g. p(3) e-2 23 / 3! 0.18
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Poisson
e.g. X number of raisins in MY cookie. Batter
has 400 raisins and makes 144 cookies. E X
400/144 2.78 per cookie p(x) e-mean meanx /
x! e.g. p(2) e-2.78 2.782 / 2! 0.24 (around
24 of cookies have 2 raisins)
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Poisson
THE FIRST BEST THING ABOUT THE POISSON IS THAT
THE MEAN ALONE TELLS US THE ENTIRE
DISTRIBUTION! note Poisson sd root(mean)
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400 raisins 144 COOKIES
E X 400/144 2.78 raisins per cookie sd
root(mean) 1.67 (for Poisson)
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Poisson
THE SECOND BEST THING ABOUT THE POISSON IS THAT
FOR A MEAN AS SMALL AS 3 THE NORMAL APPROXIMATION
WORKS WELL.
1.67 sd root(mean) Special to Poisson
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mean 2.78
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WE AVERAGE 127.8 ACCIDENTS PER MO.
E X 127.8 accidents If Poisson then sd
root(127.8) 11.3049 and the approx dist is
sd root(mean) 11.3 Special to Poisson

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mean 127.8 accidents
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Exponential
The lifetime distribution when death comes by
rare event.
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Exponential density for mean life 57.3 years
Beware, the text uses notation E(57.3) to denote
exponential distribution having mean 57.3.
We will NOT do so!
mean 57.3 years
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Exponential tail areas for mean life 57.3 years
P(X gt x) e-x/mean P(X gt 100) e-100/57.3
0.1746
mean 57.3 years
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Selected material from Chapter 1
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The overwhelming majority of samples of n from a
population of N can stand-in for the population.
THE GREAT TRICK OF STATISTICS
ATT Sysco Pepsico GM Dow
population of N 5
sample of n 2
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The overwhelming majority of samples of n from a
population of N can stand-in for the population.
THE GREAT TRICK OF STATISTICS
ATT Sysco Pepsico GM Dow
ATT Pepsico
population of N 5
sample of n 2
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Sample size n must be large. For only a few
characteristics at atime, such as profit, sales,
dividend.SPECTACULAR FAILURES MAY OCCUR!
GREAT TRICK SOME CAVEATS
ATT 12 Sysco 21 Pepsi 42 GM 8 Dow 9
population of N 5
sample of n 2
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GREAT TRICK SOME CAVEATS
This sample is obviously not
representative.
ATT 12 Sysco 21 Pepsi 42 GM 8 Dow 9
Sysco 21 Pepsi 42
population of N 5
sample of n 2
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With-replacementvs without replacement.
HOW ARE WE SAMPLING ?
ATT 12 Sysco 21 Pepsi 42 GM 8 Dow 9
population of N 5
sample of n 2
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With-replacement
HOW ARE WE SAMPLING ?
ATT 12 Sysco 21 Pepsi 42 GM 8 Dow 9
Pepsi 42 Pepsi 42
population of N 5
sample of n 2
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Rule of thumb With and without replacement are
about the same ifroot (N-n) /(N-1) 1.
DOES IT MAKE A DIFFERENCE ?
with vs without
SAME ?
population of N
sample of n
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WITH-replacement samples have no limit to the
sample size n.
UNLIMITED SAMPLING
ATT 12 Sysco 21 Pepsi 42 GM 8 Dow 9
population of N 5
sample of n 6
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WITH-replacement samples have no limit to the
sample size n.
UNLIMITED SAMPLING
Dow 9 Pepsi 42 Dow 9 Pepsi 42 ATT 12 GM 8
ATT 12 Sysco 21 Pepsi 42 GM 8 Dow 9
repeats allowed
population of N 5
sample of n 6
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TOSS COIN 100 TIMES
H
T
toss coin 100 times
population of N 2
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TOSS COIN 100 TIMES
H , T , H , T , T , H , H , H , H , T , T , H , H
, H , H , H , H , H , T , T , H , H , H , H , H
, H , H , H , H , T , H , T , H , H , H , H , T ,
T , H , H , T , T , T , T ,T , T , T , H , H , T
, T , H , T , T , H , H , H , T , H , T , H , T
, T , H , H , T ,T , T , H , T , T , T , T , T ,
H , H , T , H , T , T , T , T , H , H , T , T ,
H , T , T , T , T , H , T , H , H , T , T , T ,
T , T
H
T
sample of n 100 with-replacement 49 H and 51 T
population of N 2
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POPULATION CLONING, MANY COINS SAME AS 2
TOSS COIN 100 TIMES
H , T , H , T , T , H , H , H , H , T , T , H , H
, H , H , H , H , H , T , T , H , H , H , H , H
, H , H , H , H , T , H , T , H , H , H , H , T ,
T , H , H , T , T , T , T ,T , T , T , H , H , T
, T , H , T , T , H , H , H , T , H , T , H , T
, T , H , H , T ,T , T , H , T , T , T , T , T ,
H , H , T , H , T , T , T , T , H , H , T , T ,
H , T , T , T , T , H , T , H , H , T , T , T ,
T , T
H
H
T
sample of n 100 with-replacement 49 H and 51 T
T
population of N
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HOW MANY SAMPLES ARE THERE ?
2 from 5
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HOW MANY SAMPLES ARE THERE ?
30 from 500
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HOW MANY SAMPLES ARE THERE ?
there are far more samples with-replacement
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They would have you believe the population is
8, 9, 12, 42 and the sample is 42. A SET
is a collection of distinct entities.
CORRECTION TO PAGE 25 OF TEXT
ATT 12 IBM 42 AAA 9 Pepsi 42 GM 8 Dow 9
WE SAMPLE COMPANIES NUMBERS COME WITH THEM
Pepsi 42 Pepsi 42
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IF THE OVERWHELMING MAJORITY OF SAMPLES ARE GOOD
SAMPLES THEN WE CAN OBTAIN A GOOD SAMPLE BY
RANDOM SELECTION.
THE ROLE OF RANDOM SAMPLING
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HOW TO SAMPLE RANDOMLY ?
SELECTING A LETTER AT RANDOM
Digits are made to correspond to letters. a
00-02 b 03-05 . z 75-77 Random digits
then give random letters. 1559 9068
(Table 14, pg. 809) 15 59 90 68 etc (split
into pairs) f t w etc (take
chosen letters) For samples without replacement
just pass over any duplicates.
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The Great Trick is far more powerful than we have
seen.A typical sample closely estimates such
things as a population mean or the shape of a
population density.But it goes beyond this to
reveal how much variation there is among sample
means and sample densities. A typical sample
not only estimates population quantities. It
estimates the sample-to-sample variations of its
own estimates.
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EXAMPLE ESTIMATING A MEAN

• The average account balance is 421.34 for a
  random with-replacement sample of 50 accounts.
• We estimate from this sample that the average
  balance is 421.34 for all accounts.
• From this sample we also estimate
• and display a margin of error
• 421.34 /- 65.22 .
  

s denotes "sample standard deviation"
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SAMPLE STANDARD DEVIATION
NOTE Sample standard deviation s may be
calculated in several equivalent ways, some
sensitive to rounding errors, even for n 2.
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EXAMPLE MARGIN OF ERROR CALCULATION
The following margin of error calculation for n
4 is only an illustration. A sample of four
would not be regarded as large enough. Profits
per sale 12.2, 15.3, 16.2, 12.8. Mean
14.125, s 1.92765, root(4) 2. Margin of error
/- 1.96 (1.92765 / 2) Report 14.125 /-
1.8891. A precise interpretation of margin of
error will be given later in the course,
including the role of 1.96. The interval 14.125
/- 1.8891 is called a 95 confidence interval
for the population mean. We used
(12.2-14.125)2 (15.3-14.125)2 (16.2-14.125)2
(12.8-14.125)2 11.1475.
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EXAMPLE ESTIMATING A PERCENTAGE

• A random with-replacement sample of 50 stores
  participated in a test marketing. In 39 of these
  50 stores (i.e. 78) the new package design
  outsold the old package design.
• We estimate from this sample that 78 of all
  stores will sell more of new vs old.
• We also estimate a margin of error /- 11.5

Figured 1.96 root(pHAT qHAT)/root(n)
1.96 root(.78 .22)/root(50)
0.114823 in Binomial setup
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Plot the average heights of tents placed at 10,
14. Each tent has integral 1, as does their
average.
TENTING TONIGHT
tent
density "average tent"
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THE MEAN OF A DENSITY IS TYPICALLY THE SAME AS
THE MEAN OF THE DATA FROM WHICH IT IS MADE.
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Plot the average heights of tents placed at 10,
14. Each tent has integral 1, as does their
average.
SMOOTHING DATA SO YOU CAN SEE IT
tent
density "average tent"
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Making the tents narrower isolates different
parts of the data and reveals more detail.
NARROWER TENTS MORE DETAIL
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With narrow tents.
THE DENSITY BY ITSELF
density
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Histograms lump data into categories (the black
boxes), not as good for continuous data.
DENSITY OR HISTOGRAM ?
density histogram
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Plot of average heights of 5 tents placed at data
12, 21, 42, 8, 9.
DENSITY FOR 12, 21, 42, 8, 9
tent
density
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Narrower tents operate at higher resolution but
they may bring out features that are illusory.
IS DETAIL ILLUSORY ?
which do we trust ?
kinkier
smoother
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Population of N 500 compared with two samples
of n 30 each.
BEWARE OVER-FINE RESOLUTION
POP mean 32.02
population of N 500
with 2 samples of n 30
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Population of N 500 compared with two samples
of n 30 each.
BEWARE OVER-FINE RESOLUTION
sample means are close
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
densities not good at fine resolution
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
WE DO BETTER AT COARSE RESOLUTION
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
how about coarse resolution ?
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
WE DO BETTER AT COARSE RESOLUTION
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
good at coarse resolution
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
HOW ABOUT MEDIUM RESOLUTION ?
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
medium resolution ?
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
HOW ABOUT MEDIUM RESOLUTION ?
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
not good at medium resolution
population of N 500
with 2 samples of n 30
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A sample of only n 600 from a population of N
500 million.(medium resolution)
SAMPLING ONLY 600 FROM 500 MILLION ?
large sample of n 600 ?
POP mean 32.02
medium resolution ?
population of N 500,000
with a sample of n 600
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A sample of only n 600 from a population of N
500 million.(MEDIUM resolution)
SAMPLING ONLY 600 FROM 500 MILLION ?
sample of n 600 sample mean 32.84
mean very close
POP mean 32.02
densities are close
population of N 500,000
with a sample of n 600
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A sample of only n 600 from a population of N
500 million.(FINE resolution)
SAMPLING ONLY 600 FROM 500 MILLION ?
sample of n 600 sample mean 32.84
POP mean 32.02
FINE resolution
densities very close
population of N 500,000
with a sample of n 600
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TALKING POINTS

• 1. The Great Trick of Statistics.
• 1a. The overwhelming majority of all
  samples of n can stand-in for the population to
  a remarkable degree.
• 1b. Large n helps.
• 1c. Do not expect a given sample to
  accurately reflect the population in many
  respects, it asks too much of a sample.
• 2. The Law of Averages is one aspect of The
  Great Trick.
• 2a. Samples typically have a mean that is
  close to the mean of the population.
• 2b. Random samples are nearly certain to
  have this property since the overwhelming
  majority of samples do.
• 3. A density is controlled by the width of the
  tents used.
• 3a. Small samples zero-in on coarse
  densities fairly well .
• 3b. Samples in hundreds can perform
  remarkably well.
• 3c. Histograms are notoriously unstable but
  remain popular.
• Making a density from two to four values issue
  of resolution.
• With-replacement vs without unlimited samples.
• Using Table 14 to obtain a random sample.

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