Chapter 08

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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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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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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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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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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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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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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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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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Properties of

Point Estimators

Of all the unbiasedestimators, we prefer

the estimator whose sampling distribution

has the smallest spreador variability.

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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.

Chapter 08

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