https://drive.google.com/file/d/0B z5oZTdyBW1NUkwVnJoMUVkU0k/view?pli=1

5-1

SAMPLING
&
DIST.SAMPLING

5-2

Sampling and Sampling
Distributions

5-3

Sampling dan Dist. Sampling
Statistik (dari sampel) sebagai Estimator dari
Parameter (dari populasi)

Dist.Sampling

Estimator dan sifat-2 nya


Degrees of Freedom (derajat bebas)


Setelah mempelajari bagian ini
diharapkan dapat









Memilih sampel acak
Membedakan statistik dan parameter
Menerapkan the central limit theorem (dalil limit pusat)
Membangun distribusi sampling rata-rata dan proporsi
Menjelaskan kenapa statistik adalah estimators yang terbaik
untuk parameter

Menerapkan konsep derajat bebas
Menggunakan metode sampling yang tepat

5-4

5-5

Kegunaan


Statistical Inference:
 Predict and forecast values of
population parameters...
 Test hypotheses about values
of population parameters...
 Make decisions...

Make
Make
generalizationsabout

about
generalizations
thecharacteristics
characteristicsofof
the
population...
aapopulation...

On basis of sample statistics
derived from limited and
incomplete sample
information

Onthe
thebasis
basisof
of
On
observationsofofaa
observations

sample,aapart
partofofaa
sample,
population
population

5-6

The Literary Digest Poll (1936)
Unbiased
Sample

Democrats

Population

People who have
phones and/or cars
and/or are Digest
readers.


Democrats

Population

Republicans

Biased
Sample

Republicans

Unbiased,
representative sample
drawn at random from
the entire population.
Biased,
unrepresentative
sample drawn from
people who have cars

and/or telephones
and/or read the Digest.

Sample Statistics as Estimators of
Population Parameters


A sample statistic is a
numerical measure of a
summary characteristic
of a sample.

A population parameter
is a numerical measure of
a summary characteristic
of a population.

Anestimator
estimatorof
ofaapopulation

populationparameter
parameterisisaasample
samplestatistic
statistic
•• An
usedto
toestimate
estimateor
orpredict
predictthe
thepopulation
populationparameter.
parameter.
used
Anestimate
estimateof
ofaaparameter
parameterisisaaparticular
particularnumerical
numericalvalue

value
•• An
ofaasample
samplestatistic
statisticobtained
obtainedthrough
throughsampling.
sampling.
of
pointestimate
estimateisisaasingle
singlevalue
valueused
usedas
asan
anestimate
estimateof
ofaa
•• AApoint
populationparameter.

parameter.
population

5-7

5-8

Estimators
The sample
sample mean,
mean,X ,, isis the
the most
most common
common
•• The
estimator of
of the
the population
population mean,
mean, 

estimator
The sample
sample variance,
variance, ss22,, isis the
the most
most common
common
•• The
22.
estimator
of
the
population
variance,

estimator of the population variance,  .
The sample
sample standard
standard deviation,
deviation, s,s, isis the

the most
most
•• The
common estimator
estimator of
of the
the population
population standard
standard
common
deviation, ..
deviation,
The sample
sample proportion,
proportion,p̂,, isis the
the most
most common
common
•• The
estimator of
of the
the population
population proportion,
proportion, p.
p.
estimator

5-9

Population and Sample Proportions
• The population proportion is equal to the number of
elements in the population belonging to the category of
interest, divided by the total number of elements in the
population:
X

p
N



The sample proportion is the number of elements in the
sample belonging to the category of interest, divided by
the sample size:
x
p 
n

5-10

A Population Distribution, a Sample from a
Population, and the Population and Sample Means
Population mean ()
Frequency distribution
of the population

X
X
X

X
X

X
X
X

X
X

X
X

X
X
X

Sample points
Sample mean X
( )

X
X
X

5-11

Other Sampling Methods
• Stratified sampling: in stratified sampling, the

population is partitioned into two or more
subpopulation called strata, and from each stratum a
desired sample size is selected at random.
• Cluster sampling: in cluster sampling, a random
sample of the strata is selected and then samples
from these selected strata are obtained.
• Systemic sampling: in systemic sampling, we start
at a random point in the sampling frame, and from
this point selected every kth, say, value in the frame
to formulate the sample.

5-12

Sampling Distributions

• The sampling distribution of a statistic is the
probability distribution of all possible values the
statistic may assume, when computed from
random samples of the same size, drawn from a
specified population.
• The sampling distribution of X is the
probability distribution of all possible values the
X
random variable
may assume when a sample
of size n is taken from a specified population.

5-13

Sampling Distributions (Continued)
Uniform population of integers from 1 to 8:
P(X)
P(X)

XP(X)
XP(X)

(X-)x)
(X-
x

(X-)x2)2
(X-
x

P(X)(X-)x2)2
P(X)(X-
x

11
22
3
3
44
55
66
77
88

0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125
0.125

0.125
0.125
0.250
0.250
0.375
0.375
0.500
0.500
0.625
0.625
0.750
0.750
0.875
0.875
1.000
1.000

-3.5
-3.5
-2.5
-2.5
-1.5
-1.5
-0.5
-0.5
0.5
0.5
1.5
1.5
2.5
2.5
3.5
3.5

12.25
12.25
6.25
6.25
2.25
2.25
0.25
0.25
0.25
0.25
2.25
2.25
6.25
6.25
12.25
12.25

1.53125
1.53125
0.78125
0.78125
0.28125
0.28125
0.03125
0.03125
0.03125
0.03125
0.28125
0.28125
0.78125
0.78125
1.53125
1.53125

1.000
1.000

4.500
4.500

5.25000
5.25000

Uniform Distribution (1,8)
0.2

P(X)

XX

0.1

0.0
1

2

3

4

5

X

E(X)== ==4.5
4.5
E(X)
22 = 5.25
V(X)
=

V(X) =  = 5.25
SD(X)== ==2.2913
2.2913
SD(X)

6

7

8

5-14

Sampling Distributions (Continued)


1
2
3
4
5
6
7
8

There are 8*8 = 64 different but
equally-likely samples of size 2
that can be drawn (with
replacement) from a uniform
population of the integers from 1
to 8: of Size 2 from Uniform (1,8)
Samples
1
1,1
2,1
3,1
4,1
5,1
6,1
7,1
8,1

2
1,2
2,2
3,2
4,2
5,2
6,2
7,2
8,2

3
1,3
2,3
3,3
4,3
5,3
6,3
7,3
8,3

4
1,4
2,4
3,4
4,4
5,4
6,4
7,4
8,4

5
1,5
2,5
3,5
4,5
5,5
6,5
7,5
8,5

6
1,6
2,6
3,6
4,6
5,6
6,6
7,6
8,6

7
1,7
2,7
3,7
4,7
5,7
6,7
7,7
8,7

8
1,8
2,8
3,8
4,8
5,8
6,8
7,8
8,8

Each of these samples has a sample
mean. For example, the mean of the
sample (1,4) is 2.5, and the mean of
the sample (8,4) is 6.

1
2
3
4
5
6
7
8

Sample Means from Uniform (1,8), n =
1 2
3
4
5
6 7
8
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

5-15

Sampling Distributions (Continued)
The probability distribution of the sample mean is called the
sampling distribution of the the sample mean.
mean
Sampling Distribution of the Mean
XP(X)

X- X

(X- X)

P(X)(X- X)

0.015625
0.031250
0.046875
0.062500
0.078125
0.093750
0.109375
0.125000
0.109375
0.093750
0.078125
0.062500
0.046875
0.031250
0.015625

0.015625
0.046875
0.093750
0.156250
0.234375
0.328125
0.437500
0.562500
0.546875
0.515625
0.468750
0.406250
0.328125
0.234375
0.125000

-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5

12.25
9.00
6.25
4.00
2.25
1.00
0.25
0.00
0.25
1.00
2.25
4.00
6.25
9.00
12.25

0.191406
0.281250
0.292969
0.250000
0.175781
0.093750
0.027344
0.000000
0.027344
0.093750
0.175781
0.250000
0.292969
0.281250
0.191406

1.000000

4.500000

2.625000

Sampling Distribution of the Mean

2

0.10

P(X)

P(X)

2

0.05

0.00
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

X

E ( X )   X  4.5
V ( X )   2X  2.625
SD( X )   X  1.6202

5-16

Properties of the Sampling Distribution
of the Sample Mean
Comparingthe
thepopulation
population
•• Comparing

P(X)

0.2

0.1

0.0
1

2

3

4

5

6

7

8

X

Sampling Distribution of the Mean

0.10

P(X)

distributionand
andthe
thesampling
sampling
distribution
distributionof
ofthe
themean:
mean:
distribution
The
Thesampling
samplingdistribution
distributionisis

morebell-shaped
bell-shapedand
and
more
symmetric.
symmetric.
Both
Bothhave
havethe
thesame
samecenter.
center.

The
Thesampling
samplingdistribution
distributionof
of

themean
meanisismore
morecompact,
compact,with
with
the
smallervariance.
variance.
aasmaller

Uniform Distribution (1,8)

0.05

0.00
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

X

5-17

Relationships between Population Parameters and
the Sampling Distribution of the Sample Mean
The expected value of the sample mean is equal to the population mean:

E ( X )   
X

X

The variance of the sample mean is equal to the population variance divided by
the sample size:

V ( X ) 

2
X





2
X

n

The standard deviation of the sample mean, known as the standard error of
the mean,
mean is equal to the population standard deviation divided by the square
root of the sample size:

SD( X )  
X



X

n

5-18

Sampling from a Normal Population
Whensampling
samplingfrom
fromaanormal
normalpopulation
populationwith
withmean
meanand
andstandard
standard
When
deviation,
,the
thesample
samplemean,
mean,X,
X,has
hasaanormal
normalsampling
samplingdistribution:
distribution:
distribution
deviation
distribution
2


X ~ N (, )
n
Sampling Distribution of the Sample Mean
0.4

Sampling Distribution: n =16
0.3

f(X)

Thismeans
meansthat,
that,as
asthe
the
This
samplesize
sizeincreases,
increases,the
the
sample
samplingdistribution
distributionof
ofthe
the
sampling
samplemean
meanremains
remains
sample
centeredon
onthe
thepopulation
population
centered
mean,but
butbecomes
becomesmore
more
mean,
compactlydistributed
distributedaround
around
compactly
thatpopulation
populationmean
mean
that

Sampling Distribution: n = 4

0.2

Sampling Distribution: n = 2

0.1

Normal population

Normal population
0.0



5-19

The Central Limit Theorem
n=5
0.25
0.20

P(X)

0.15
0.10
0.05
0.00

X

n = 20
P(X)

0.2

0.1

0.0

X

When sampling
sampling from
from aa population
population
When
with mean
mean  and
and finite
finite standard
standard
with
deviation ,
, the
the sampling
sampling
deviation
distribution of
of the
the sample
sample mean
mean will
will
distribution
tend to
to aa normal
normal distribution
distribution with
with
tend

mean

and
standard
deviation
as
mean  and standard deviation n as
the sample
sample size
size becomes
becomes large
large
the
(n >30).
>30).
(n

Large n
0.4

0.2
0.1
0.0

-



X

For “large
“large enough”
enough” n:
n: X ~ N ( , / n)
For
2

f(X)

0.3

5-20

The Central Limit Theorem Applies to
Sampling Distributions from Any Population
Normal

Uniform

Skewed

General

Population

n=2

n = 30



X



X



X



X

The Central Limit Theorem

5-21

Mercurymakes
makesaa2.4
2.4liter
literV-6
V-6engine,
engine,the
theLaser
LaserXRi,
XRi,used
usedin
inspeedboats.
speedboats.
Mercury
Thecompany’s
company’sengineers
engineersbelieve
believethe
theengine
enginedelivers
deliversan
anaverage
averagepower
powerof
of
The
220horsepower
horsepowerand
andthat
thatthe
thestandard
standarddeviation
deviationof
ofpower
powerdelivered
deliveredisis15
15
220
HP. AApotential
potentialbuyer
buyerintends
intendsto
tosample
sample100
100engines
engines(each
(eachengine
engineisisto
tobe
be
HP.
runaasingle
singletime).
time). What
Whatisisthe
theprobability
probabilitythat
thatthe
thesample
samplemean
meanwill
willbe
beless
less
run
than217HP?
217HP?
than

 X   217  
P ( X  217)  P




 n
n














217  220
217  220
 P Z 
  P Z 

15
15








10
100
 P ( Z   2) 0.0228

5-22

Example 5-2
EPS Mean Distribution
2.00 - 2.49
2.50 - 2.99
3.00 - 3.49
3.50 - 3.99

25

Frequency

20

4.00 - 4.49
4.50 - 4.99
5.00 - 5.49
5.50 - 5.99

15
10
5
0
Range

6.00 - 6.49
6.50 - 6.99
7.00 - 7.49
7.50 - 7.99

5-23

Student’s t Distribution
thepopulation
populationstandard
standarddeviation,
deviation,,
,isisunknown,
unknown,
replacewith
with
unknown replace
IfIfthe
unknown
thesample
samplestandard
standarddeviation,
deviation,s.s. IfIfthe
thepopulation
populationisisnormal,
normal,the
the
the
resultingstatistic:
statistic: t X  
resulting
s/ n

hasaattdistribution
distributionwith
with(n
(n--1)
1)degrees
degreesof
offreedom.
freedom.
freedom
has
freedom
Thet tisisaafamily
familyofofbell-shaped
bell-shapedand
andsymmetric
symmetric
•• The

••
••

••

distributions,one
onefor
foreach
eachnumber
numberofofdegree
degreeofof
distributions,
freedom.
freedom.
Theexpected
expectedvalue
valueofoft tisis0.0.
The
Thevariance
varianceofoft tisisgreater
greaterthan
than1,1,but
butapproaches
approaches
The
thenumber
numberofofdegrees
degreesofoffreedom
freedomincreases.
increases.
11asasthe
Thet tisisflatter
flatterand
andhas
hasfatter
fattertails
tailsthan
thandoes
doesthe
the
The
standardnormal.
normal.
standard
Thet tdistribution
distributionapproaches
approachesaastandard
standardnormal
normal
The
thenumber
numberofofdegrees
degreesofoffreedom
freedomincreases.
increases.
asasthe

Standard normal
t, df=20
t, df=10




5-24

The Sampling Distribution of the
Sample Proportion, p
0 .4

P(X)

Thesample
sampleproportion
proportionisisthe
thepercentage
percentageof
of
The
successesininnnbinomial
binomialtrials.
trials. ItItisisthe
the
successes
numberof
ofsuccesses,
successes,X,
X,divided
dividedby
bythe
the
number
numberof
oftrials,
trials,n.n.
number

n=2, p = 0.3
0 .5

0 .3
0 .2
0 .1
0 .0
0

1

2

X
n=10,p=0.3

X
n

0.2

P(X)

Sample proportion: p 

0.3

0.1

0.0
1

2

3

4

5

6

7

8

9

10

X

n=15, p = 0.3
0.2

P(X)

Asthe
thesample
samplesize,
size,n,n,increases,
increases,the
thesampling
sampling
As
distributionof
of p approaches
approachesaanormal
normal
distribution
distributionwith
withmean
meanppand
andstandard
standard
distribution
deviation p(1  p)
deviation
n

0

0.1

0.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

X

0 1 2 3 4 5 6 7 8 9 10 1112 13 1415
15 1515 15 15 15 15 15 151515 1515 15 1515

^
p

5-25

Sample Proportion
recentyears,
years,convertible
convertiblesports
sportscoupes
coupeshave
havebecome
becomevery
verypopular
popularininJapan.
Japan. Toyota
Toyota
InInrecent
currentlyshipping
shippingCelicas
CelicastotoLos
LosAngeles,
Angeles,where
whereaacustomizer
customizerdoes
doesaaroof
rooflift
liftand
and
isiscurrently
shipsthem
themback
backtotoJapan.
Japan. Suppose
Supposethat
that25%
25%ofofall
allJapanese
Japaneseininaagiven
givenincome
incomeand
and
ships
lifestylecategory
categoryare
areinterested
interestedininbuying
buyingCelica
Celicaconvertibles.
convertibles. AArandom
randomsample
sampleofof100
100
lifestyle
Japaneseconsumers
consumersininthe
thecategory
categoryof
ofinterest
interestisistotobe
beselected.
selected. What
Whatisisthe
theprobability
probability
Japanese
thatatatleast
least20%
20%ofofthose
thoseininthe
thesample
samplewill
willexpress
expressan
aninterest
interestininaaCelica
Celicaconvertible?
convertible?
that
n  100
p  0.25
np  (100 )( 0.25)  25  E ( p )
p (1  p )



(.25)(.75)

n

100

p (1  p )
n



 0.001875  V ( p )

0.001875  0.04330127  SD ( p )



 p  p
.20  p 
P ( p  0.20 )  P
 p (1  p )  p (1  p ) 


n
n



 .05 
.20  .25 

 P z 
 P z 



.0433 
(.25)(.75)




100
 P ( z   1.15)  0.8749

5-26

Estimators and Their Properties
Anestimator
estimatorof
ofaapopulation
populationparameter
parameterisisaasample
samplestatistic
statisticused
usedto
to
An
estimatethe
theparameter.
parameter. The
Themost
mostcommonly-used
commonly-usedestimator
estimatorof
ofthe:
the:
estimate
PopulationParameter
Parameter
SampleStatistic
Statistic
Population
Sample
Mean()
()
the
Mean(X)
(X)
Mean
isisthe
Mean
Variance(
(22))
the
Variance(s(s22))
Variance
isisthe
Variance
StandardDeviation
Deviation()
()
the
StandardDeviation
Deviation(s)
(s)
Standard
isisthe
Standard
Proportion(p)
(p)
the
Proportion((p ))
Proportion
isisthe
Proportion

Desirable properties
properties of
of estimators
estimators include:
include:
••Desirable
Unbiasedness
Unbiasedness

Efficiency
Efficiency

Consistency
Consistency

Sufficiency
Sufficiency


5-27

Unbiasedness
Anestimator
estimatorisissaid
saidto
tobe
beunbiased
unbiasedififits
itsexpected
expectedvalue
valueisisequal
equalto
to
An
thepopulation
populationparameter
parameterititestimates.
estimates.
the
Forexample,
example,E(X)=so
E(X)=sothe
thesample
samplemean
meanisisan
anunbiased
unbiasedestimator
estimator
For
ofthe
thepopulation
populationmean.
mean. Unbiasedness
Unbiasednessisisan
anaverage
averageor
orlong-run
long-run
of
property. The
Themean
meanof
ofany
anysingle
singlesample
samplewill
willprobably
probablynot
notequal
equalthe
the
property.
populationmean,
mean,but
butthe
theaverage
averageof
ofthe
themeans
meansof
ofrepeated
repeated
population
independentsamples
samplesfrom
fromaapopulation
populationwill
willequal
equalthe
thepopulation
population
independent
mean.
mean.
Anysystematic
systematicdeviation
deviationof
ofthe
theestimator
estimatorfrom
fromthe
thepopulation
population
Any
parameterof
ofinterest
interestisiscalled
calledaabias.
bias.
bias
parameter
bias

5-28

{

Unbiased and Biased Estimators

Bias

An unbiased estimator is on
target on average.

A biased estimator is
off target on average.

5-29

Efficiency
Anestimator
estimatorisisefficient
efficientififitithas
hasaarelatively
relativelysmall
smallvariance
variance(and
(and
An
standarddeviation).
deviation).
standard

An efficient estimator is,
on average, closer to the
parameter being estimated..

An inefficient estimator is, on
average, farther from the
parameter being estimated.

5-30

Consistency and Sufficiency
Anestimator
estimatorisissaid
saidto
tobe
beconsistent
consistentififits
itsprobability
probabilityof
ofbeing
beingclose
close
An
tothe
theparameter
parameterititestimates
estimatesincreases
increasesas
asthe
thesample
samplesize
sizeincreases.
increases.
to

Consistency
n = 100
n = 10
Anestimator
estimatorisissaid
saidto
tobe
besufficient
sufficientififititcontains
containsall
allthe
theinformation
information
An
inthe
thedata
dataabout
aboutthe
theparameter
parameterititestimates.
estimates.
in

5-31

Properties of the Sample Mean
For a normal population, both the sample mean and
sample median are unbiased estimators of the
population mean, but the sample mean is both more
efficient (because it has a smaller variance), and
sufficient.
sufficient Every observation in the sample is used in
the calculation of the sample mean, but only the middle
value is used to find the sample median.
In general, the sample mean is the best estimator of the
population mean. The sample mean is the most
efficient unbiased estimator of the population mean. It
is also a consistent estimator.

5-32

Properties of the Sample Variance
Thesample
samplevariance
variance(the
(thesum
sumof
ofthe
thesquared
squareddeviations
deviationsfrom
fromthe
the
The
samplemean
meandivided
dividedby
by(n-1)
(n-1)isisan
anunbiased
unbiasedestimator
estimatorof
ofthe
the
sample
populationvariance.
variance. In
Incontrast,
contrast,the
theaverage
averagesquared
squareddeviation
deviation
population
fromthe
thesample
samplemean
meanisisaabiased
biased(though
(thoughconsistent)
consistent)estimator
estimatorof
ofthe
the
from
populationvariance.
variance.
population
2



(
x
x
)

2
 2
E (s )  E 

 (n  1) 

  ( x  x )2 
2
E
 
n



5-33

Degrees of Freedom
Consider a sample of size n=4 containing the following data points:
x1=10

x2=12

and for which the sample mean is:

x3=16

x4=?

x

x
14
n

Given the values of three data points and the sample mean, the
value of the fourth data point can be determined:
 x 12  14  16  x4
x=

14
n
4
12  14  16  x 56
4

x4  56  12  14  16
56
xx 44 = 14

5-34

Degrees of Freedom (Continued)
If only two data points and the sample mean are known:
x1=10

x2=12

x3=?

x4=?

x 14

The values of the remaining two data points cannot be uniquely
determined:
12  14  x  x4
3
x=

14
n
4
x

12  14  x  x4  56
3

5-35

Degrees of Freedom (Continued)
Thenumber
numberof
ofdegrees
degreesof
offreedom
freedomisisequal
equalto
tothe
thetotal
totalnumber
numberof
of
The
measurements(these
(theseare
arenot
notalways
alwaysraw
rawdata
datapoints),
points),less
lessthe
thetotal
total
measurements
numberof
ofrestrictions
restrictionson
onthe
themeasurements.
measurements. AArestriction
restrictionisisaa
number
quantitycomputed
computedfrom
fromthe
themeasurements.
measurements.
quantity
Thesample
samplemean
meanisisaarestriction
restrictionon
onthe
thesample
samplemeasurements,
measurements,so
so
The
aftercalculating
calculatingthe
thesample
samplemean
meanthere
thereare
areonly
only(n-1)
(n-1)degrees
degreesof
of
after
freedomremaining
remainingwith
withwhich
whichto
tocalculate
calculatethe
thesample
samplevariance.
variance.
freedom
Thesample
samplevariance
varianceisisbased
basedon
ononly
only(n-1)
(n-1)free
freedata
datapoints:
points:
The
2

(
x
x
)

2

s
(n  1)

5-36

Example
sampleof
ofsize
size10
10isisgiven
givenbelow.
below. We
Weare
aretotochoose
choosethree
threedifferent
differentnumbers
numbersfrom
from
AAsample
whichthe
thedeviations
deviationsare
aretotobe
betaken.
taken. The
Thefirst
firstnumber
numberisistotobe
beused
usedfor
forthe
thefirst
firstfive
five
which
samplepoints;
points;the
thesecond
secondnumber
numberisistotobe
beused
usedfor
forthe
thenext
nextthree
threesample
samplepoints;
points;and
and
sample
thethird
thirdnumber
numberisistotobe
beused
usedfor
forthe
thelast
lasttwo
twosample
samplepoints.
points.
the
Sample #

1

2

3

4

5

6

7

8

9

10

Sample
Point

93

97

60

72

96

83

59

66

88

53

i.

What three numbers should we choose in order to minimize the SSD
(sum of squared deviations from the mean).?



Note: SSD   x  x 

2

5-37

Example 5-4 (continued)
Solution:Choose
Choosethe
themeans
meansof
ofthe
thecorresponding
correspondingsample
samplepoints.
points. These
Theseare:
are:83.6,
83.6,
Solution:
69.33,and
and70.5.
70.5.
69.33,
ii. Calculate
Calculatethe
theSSD
SSDwith
withchosen
chosennumbers.
numbers.
ii.
Solution:SSD
SSD==2030.367.
2030.367.See
Seetable
tableon
onnext
nextslide
slidefor
forcalculations.
calculations.
Solution:
iii. What
Whatisisthe
thedfdffor
forthe
thecalculated
calculatedSSD?
SSD?
iii.
Solution: dfdf==10
10––33==7.7.
Solution:
iv. Calculate
Calculatean
anunbiased
unbiasedestimate
estimateof
ofthe
thepopulation
populationvariance.
variance.
iv.
Solution: An
Anunbiased
unbiasedestimate
estimateof
ofthe
thepopulation
populationvariance
varianceisisSSD/df
SSD/df==2030.367/7
2030.367/7
Solution:
290.05.
==290.05.

5-38

Example (continued)
Sample #

Sample Point

Mean

Deviations

Deviation
Squared

1

93

83.6

9.4

88.36

2

97

83.6

13.4

179.56

3

60

83.6

-23.6

556.96

4

72

83.6

-11.6

134.56

5

96

83.6

12.4

153.76

6

83

69.33

13.6667

186.7778

7

59

69.33

-10.3333

106.7778

8

66

69.33

-3.3333

11.1111

9

88

70.5

17.5

306.25

10

53

70.5

-17.5

306.25

SSD

2030.367

SSD/df

290.0524

5-39

Using the Template

SamplingDistribution
Distributionof
ofaaSample
SampleMean
Mean
Sampling

5-40

Using the Template

SamplingDistribution
Distributionof
ofaaSample
SampleMean
Mean(continued)
(continued)
Sampling

5-41

Using the Template

SamplingDistribution
Distributionof
ofaaSample
SampleProportion
Proportion
Sampling

5-42

Using the Template

SamplingDistribution
Distributionof
ofaaSample
SampleProportion
Proportion(continued)
(continued)
Sampling

5-43

Using the Computer
Constructing a sampling distribution of the mean from a uniform
population (n=10) using EXCEL (use RANDBETWEEN(0, 1)
command to generate values to graph):

200

Frequency

CLASS MIDPOINT FREQUENCY
0.15
0
0.2
0
0.25
3
0.3
26
0.35
64
0.4
113
0.45
183
0.5
213
0.55
178
0.6
128
0.65
65
0.7
20
0.75
3
0.8
3
0.85
0
999

Histogram of Sample Means

250

150

100

50

0
1

2

3

4

5

6

7

8

9

10

11

12

Sample Means (Class Midpoints)

13

14

15