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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Introduction to Probability
and StatisticsTwelfth Edition
Robert J. Beaver Barbara M. Beaver William Mendenhall
Presentation designed and written by:
Barbara M. Beaver
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Introduction to Probability
and StatisticsTwelfth Edition
Chapter 8
Large-Sample Estimation
Some graphic screen captures from Seeing Statistics Some images 2001-(current year) www.arttoday.com
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Introduction Populations are described by their probability
distributions and parameters.
For quantitative populations, the location
and shape are described bymand s.
For a binomial populations, the location and
shape are determined byp.
If the values of parameters are unknown, we
make inferences about them using sample
information.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Types of Inference
Estimation:
Estimating or predicting the value of theparameter
What is (are) the most likely values of m
orp? Hypothesis Testing:
Deciding about the value of a parameter
based on some preconceived idea. Did the sample come from a population
with m = 5 orp= .2?
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Types of Inference
Examples:
A consumer wants to estimate the averageprice of similar homes in her city beforeputting her home on the market.
Estimation:Estimate m, the average home price.
Hypothesis test: Is the new average resistance, mN
equal to the old average resistance, mO?
A manufacturer wants to know if a new
type of steel is more resistant to high
temperatures than an old type was.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Types of Inference
Whether you are estimating parameters
or testing hypotheses, statistical methods
are important because they provide:Methods for making the inference
A numerical measure of the
goodness or reliability of the
inference
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Definitions
An estimatoris a rule, usually aformula, that tells you how to calculate
the estimate based on the sample.Point estimation: A single number is
calculated to estimate the parameter.
Interval estimation:Two numbers arecalculated to create an interval withinwhich the parameter is expected to lie.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Properties of
Point Estimators Since an estimator is calculated from
sample values, it varies from sample tosample according to its sampling
distribution. An estimatoris unbiasedif the mean of
its sampling distribution equals the
parameter of interest.It does not systematically overestimate
or underestimate the target parameter.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Properties of
Point Estimators
Of all the unbiasedestimators, we prefer
the estimator whose sampling distribution
has the smallest spreador variability.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Measuring the Goodness
of an Estimator
The distance between an estimate and the
true value of the parameter is the error of
estimation.The distance between the bullet and
the bulls-eye.
In this chapter, the sample sizes are large,
so that our unbiasedestimators will havenormal distributions. Because of the Central
Limit Theorem.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
The Margin of Error
For unbiasedestimators with normalsampling distributions, 95% of all pointestimates will lie within 1.96 standard
deviations of the parameter of interest.
estimatotheoferrorstd96.1
Margin of error: The maximum error of
estimation, calculated as
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Estimating Means
and ProportionsFor a quantitative population,
n
s
n
x
96.1)30 :(errorofMargin
:meanpopulationofestimatorPoint
For a binomial population,
n
qpn
x/npp
96.1)30
=
:(errorofMargin
:proportionpopulationofestimatorPoint
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example
A homeowner randomly samples 64 homes
similar to her own and finds that the average
selling price is $252,000 with a standard
deviation of $15,000. Estimate the averageselling price for all similar homes in the city.
Point estimator of : 252,000
15,000Margin of error : 1.96 1.96 3675
64
x
s
n
=
= =
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
ExampleA quality control technician wants to estimate
the proportion of soda cans that are underfilled.
He randomly samples 200 cans of soda and
finds 10 underfilled cans.
03.200
)95)(.05(.96.1
96.1
05.200/10
200
==
===
==
nqp
x/npp
pn
:errorofMargin
:ofestimatorPoint
cansdunderfilleofproportion
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Interval Estimation
Create an interval (a, b) so that you are fairlysure that the parameter lies between these two
values.
Fairly sure is means with high probability,measured using the confidence coefficient, 1
a.Usually, 1-a = .90, .95, .98, .99
Suppose 1-a= .95 andthat the estimator has a
normal distribution.Parameter 1.96SE
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Interval Estimation
Since we dont know the value of the parameter,consider which has a variable
center.
Only if the estimator falls in the tail areas will
the interval fail to enclose the parameter. This
happens only 5% of the time.
Estimator 1.96SE
WorkedWorked
Worked
Failed
APPLETMY
http://localhost/var/www/apps/conversion/tmp/scratch_4/Beaver/ciZ.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_4/Beaver/ciZ.html -
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
To Change the
Confidence Level To change to a general confidence level, 1-a,
pick a value ofzthat puts area 1-a in the center
of thezdistribution.
100(1-a)% Confidence Interval: Estimator za/2SE
Tail area za/2
.05 1.645
.025 1.96
.01 2.33
.005 2.58
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Confidence Intervals
for Means and Proportions
For a quantitative population,
n
s
zx
2/a
:meanpopulationaforintervalConfidence
For a binomial population,
n
qpzp
p
2/a
:proportionpopulationaforintervalConfidence
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example
A random sample of n= 50 males showed a
mean average daily intake of dairy products
equal to 756 grams with a standard deviation of
35 grams. Find a 95% confidence interval for thepopulation average m.
n
s
x 96.1 50
35
96.17
56 70.97 56
grams.65.70746.30or 7 m
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example
Find a 99% confidence interval for m, the
population average daily intake of dairy products
for men.
n
sx 58.2
50
3558.27 56 77.127 56
grams.7743.23or 77.68
mThe interval must be wider to provide for the
increased confidence that is does indeed
enclose the true value of m.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example
Of a random sample of n= 150 college students,104 of the students said that they had played on a
soccer team during their K-12 years. Estimate the
porportion of college students who played soccerin their youth with a 98% confidence interval.
n
qp
p
33.2
150
)31(.69.
33.2
104
150
09.. 69 .60or .78. p
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Estimating the Difference
between Two MeansSometimes we are interested in comparing themeans of two populations.
The average growth of plants fed using twodifferent nutrients.
The average scores for students taught with twodifferent teaching methods.
To make this comparison,
.varianceandmeanwith1population
fromdrawnsizeofsamplerandomA2
11
1
s
n
.varianceandmeanwith2population
fromdrawnsizeofsamplerandomA
2
22
2
s
n
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Estimating the Difference
between Two MeansWe compare the two averages by makinginferences about m1-m2, the difference in the twopopulation averages.
If the two population averages are the same,then m1-m2 = 0.
The best estimate of m1-m2is the difference
in the two sample means,
21 xx -
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
The Sampling
Distribution of 1 2x x-
.SEas
estimatedbecanSEandnormal,elyapproximatisof
ondistributisamplingthelarge,aresizessampletheIf.3
.SEisofdeviationstandardThe2.
means.populationthe
indifferencethe,isofmeanThe1.
2
2
2
1
2
1
21
2
22
1
2121
2121
n
s
n
s
xx
nnxx
xx
=
-
=-
--
ss
mm
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Estimating m1-m2
For large samples, point estimates and theirmargin of error as well as confidence intervalsare based on the standard normal (z)distribution.
2
2
2
1
2
1
2121
1.96:ErrorofMargin
:-forestimatePoint
n
s
n
sxx
-mm
2
2
2
1
2
12/21
21
)(
:-forintervalConfidence
n
s
n
szxx -a
mm
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example
Compare the average daily intake of dairy products of
men and women using a 95% confidence interval.
78.126 -
.78.618.78-or 21 - mm
Avg Daily Intakes Men Women
Sample size 50 50Sample mean 756 762
Sample Std Dev 35 30
2
2
2
1
2
121 96.1)(
n
s
n
sxx -
2 235 30(756 762) 1.9650 50
-
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example, continued
Could you conclude, based on this confidence
interval, that there is a difference in the average dailyintake of dairy products for men and women?
The confidence interval contains the value m1-m2= 0.
Therefore, it is possible thatm
1=
m
2.
You would not
want to conclude that there is a difference in average
daily intake of dairy products for men and women.
78.618.78- 21 - mm
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Estimating the Difference
between Two ProportionsSometimes we are interested in comparing theproportion of successes in two binomialpopulations.
The germination rates of untreated seeds and seedstreated with a fungicide.
The proportion of male and female voters whofavor a particular candidate for governor.
To make this comparison,.parameterwith1populationbinomialfromdrawnsizeofsamplerandomA
1
1
pn
.parameterwith2populationbinomial
fromdrawnsizeofsamplerandomA
2
2
p
n
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Estimating the Difference
between Two MeansWe compare the two proportions by makinginferences aboutp1-p2, the difference in the twopopulation proportions.
If the two population proportions are thesame, thenp1-p2 = 0.
The best estimate ofp1-p2is the differencein the two sample proportions,
2
2
1
121
n
x
n
xpp -=-
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
The Sampling
Distribution of 1 2 p p-
.
SEas
estimatedbecanSEandnormal,elyapproximatis
of
ondistributisamplingthelarge,aresizessampletheIf.3
.SEisofdeviationstandardThe2.
s.proportionpopulationthe
indifferencethe,isofmeanThe1.
2
22
1
11
21
2
22
1
1121
2121
n
qp
n
qppp
nqp
nqppp
pppp
=
-
=-
--
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Estimating p1-p2
For large samples, point estimates and theirmargin of error as well as confidence intervalsare based on the standard normal (z)distribution.
2
22
1
11
2121
1.96:ErrorofMargin
:forestimatePoint
n
qp
n
qppp-pp
-
2
22
1
112/21
21
)(
:forintervalConfidence
n
qp
n
qpzpp
pp
-
-
a
E l
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example
Compare the proportion of male and female college
students who said that they had played on a soccer team
during their K-12 years using a 99% confidence interval.
2
22
1
1121
58.2)(
n
qp
n
qppp -
70
)44(.56.
80
)19(.81.58.2)70
39
80
65( - 19.52.
.44..06or 21 - pp
Youth Soccer Male Female
Sample size 80 70
Played soccer 65 39
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Example, continued
Could you conclude, based on this confidence
interval, that there is a difference in the proportion of
male and female college students who said that theyhad playedon a soccer team during their K-12 years?
The confidence interval does not contains the value
p1-p2 = 0.Therefore, it is not likely that p1= p2. You
would conclude that there is a difference in theproportions for males and females.
44..06 21 -
pp
A higher proportion of males than
females played soccer in their youth.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
One Sided
Confidence Bounds Confidence intervals are by their nature two-
sided since they produce upper and lowerbounds for the parameter.
One-sided bounds can be constructed simplyby using a value ofzthat puts arather thana/2 in the tail of thezdistribution.
Estimator)ofErrorStd(Estimator:UCB
Estimator)ofErrorStd(Estimator:LCB
-
a
a
z
z
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Choosing the Sample Size
The total amount of relevant information in asample is controlled by two factors:
- The sampling planor experimental design:the procedure for collecting the information
- The sample size n: the amount ofinformation you collect.
In a statistical estimation problem, theaccuracy of the estimation is measured by themargin of erroror the width of theconfidence interval.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
1. Determine the size of the margin of error, B, that
you are willing to tolerate.
2. Choose the sample size by solving for nor n=n1 =
n2 in the inequality: 1.96 SE B,where SE is a
function of the sample size n.
3. For quantitative populations, estimate the population
standard deviation using a previously calculated
value of sor the range approximation Range / 4.4. For binomial populations, use the conservative
approach and approximatepusing the value p=.5.
Choosing the Sample Size
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
ExampleA producer of PVC pipe wants to survey
wholesalers who buy his product in order toestimate the proportion who plan to increase their
purchases next year. What sample size is required if
he wants his estimate to be within .04 of the actual
proportion with probability equal to .95?
04.96.1 n
pq04.
)5(.5.96.1
n
5.2404.
)5(.5.96.1= n 25.6005.24
2= n
He should survey at least 601
wholesalers.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Key ConceptsI. Types of Estimators
1. Point estimator: a single number is calculated to estimatethe population parameter.
2. Interval estimator:two numbers are calculated to form aninterval that contains the parameter.
II. Properties of Good Estimators1. Unbiased: the average value of the estimator equals the
parameter to be estimated.
2. Minimum variance: of all the unbiased estimators, the bestestimator has a sampling distribution with the smallest
standard error.3. The margin of errormeasures the maximum distance
between the estimator and the true value of the parameter.
K C t
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Key ConceptsIII. Large-Sample Point Estimators
To estimate one of four population parameters when the
sample sizes are large, use the following point estimators with
the appropriate margins of error.
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Copyright 2006 Brooks/ColeA division of Thomson Learning, Inc.
Key ConceptsIV. Large-Sample Interval Estimators
To estimate one of four population parameters when thesample sizes are large, use the following interval estimators.
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i h k / l
Key Concepts1. All values in the interval are possible values for the
unknown population parameter.2. Any values outside the interval are unlikely to be the value
of the unknown parameter.
3. To compare two population means or proportions, look forthe value 0 in the confidence interval. If 0 is in the interval,it is possible that the two population means or proportionsare equal, and you should not declare a difference. If 0 isnot in the interval, it is unlikely that the two means or
proportions are equal, and you can confidently declare adifference.
V. One-Sided Confidence Bounds
Use either the upper () or lower (-) two-sided bound, withthe critical value ofzchanged fromz
a/2toza.