7 - 1 © 1998 Prentice-Hall, Inc. Chapter 7 Inferences Based on a Single Sample: Estimation with Confidence Intervals.

7 - 7 - 11

© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.

Chapter 7Chapter 7

Inferences Based on a Single Sample: Inferences Based on a Single Sample: Estimation with Confidence IntervalsEstimation with Confidence Intervals

7 - 7 - 22


Learning ObjectivesLearning Objectives

1.1. State what is estimatedState what is estimated

2.2. Distinguish point & interval estimatesDistinguish point & interval estimates

3.3. Explain interval estimatesExplain interval estimates

4.4. Compute confidence interval estimates Compute confidence interval estimates for population mean & proportionfor population mean & proportion

5.5. Compute sample sizeCompute sample size

7 - 7 - 33


Thinking ChallengeThinking Challenge

Suppose you’re Suppose you’re interested in estimating interested in estimating the average amount of the average amount of money that second-year money that second-year business students business students (population) have on (population) have on them. How would you them. How would you find out?find out?

AloneAlone GroupGroup Class Class

7 - 7 - 44


Introduction Introduction to Estimationto Estimation

7 - 7 - 55


Types of Types of Statistical Statistical

ApplicationsApplications

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

7 - 7 - 66


Estimation ProcessEstimation Process

7 - 7 - 77



Mean, , is unknown

PopulationPopulation

7 - 7 - 88



Mean, , is unknown

PopulationPopulation Random SampleRandom Sample

Mean X= 50

Sample

7 - 7 - 99



Mean, , is unknown

PopulationPopulation Random SampleRandom SampleI am 95%

confident that is between

42 & 58.

Mean X= 50

Sample

7 - 7 - 1010


Unknown Population Unknown Population Parameters Are Parameters Are

Estimated Estimated

7 - 7 - 1111



Estimated Estimated

Estimate populationEstimate populationparameter...parameter...

with samplewith samplestatisticstatistic

7 - 7 - 1212



Estimated Estimated



MeanMean xx

7 - 7 - 1313



Estimated Estimated



MeanMean xx

ProportionProportion pp pp̂̂

7 - 7 - 1414



Estimated Estimated



MeanMean xx


VarianceVariance 22 ss 22

7 - 7 - 1515



Estimated Estimated



MeanMean xx


VarianceVariance 22 ss 22

DifferencesDifferences 11 - - 22 xx11 --xx22

7 - 7 - 1616


Estimation MethodsEstimation Methods

7 - 7 - 1717



Estimation

7 - 7 - 1818



Estimation

PointEstimation

7 - 7 - 1919



Estimation

PointEstimation

IntervalEstimation

7 - 7 - 2020


Point EstimationPoint Estimation

7 - 7 - 2121



Estimation

PointEstimation

IntervalEstimation

7 - 7 - 2222


Point EstimationPoint Estimation

1.1. Provides single valueProvides single value Based on observations from 1 sampleBased on observations from 1 sample

2.2. Gives no information about how close Gives no information about how close value is to the unknown population value is to the unknown population parameterparameter

3.3. Example: Sample meanExample: Sample meanx x = 3 is point = 3 is point estimate of unknown population meanestimate of unknown population mean

7 - 7 - 2323


Interval EstimationInterval Estimation

7 - 7 - 2424



Estimation

PointEstimation

IntervalEstimation

7 - 7 - 2525


Interval EstimationInterval Estimation

1.1. Provides range of values Provides range of values Based on observations from 1 sampleBased on observations from 1 sample

2.2. Gives information about closeness to Gives information about closeness to unknown population parameterunknown population parameter Stated in terms of probabilityStated in terms of probability

Knowing exact closeness requires knowing unknown Knowing exact closeness requires knowing unknown population parameterpopulation parameter

3.3. Example: unknown population mean lies Example: unknown population mean lies between 50 & 70 with 95% confidencebetween 50 & 70 with 95% confidence

7 - 7 - 2626


Key Elements of Key Elements of Interval EstimationInterval Estimation

7 - 7 - 2727



Sample statistic Sample statistic

(point estimate)(point estimate)

7 - 7 - 2828



Confidence Confidence intervalinterval



7 - 7 - 2929






Confidence Confidence limit (lower)limit (lower)

Confidence Confidence limit (upper)limit (upper)

7 - 7 - 3030






Confidence Confidence limit (lower)limit (lower)

Confidence Confidence limit (upper)limit (upper)

A A probabilityprobability that the population parameter that the population parameter falls somewhere within the interval.falls somewhere within the interval.

7 - 7 - 3131


Confidence Limits Confidence Limits for Population Meanfor Population Mean

( )

( )

1

5

X Error

Error X X

ZX Error

Error Z

X Z

x x

x

x

(2) or

(3)

(4)

( )

( )

1

5

X Error

Error X X

ZX Error

Error Z

X Z

x x

x

x

(2) or

(3)

(4)

Parameter = Statistic ± Error

© 1984-1994 T/Maker Co.

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Many Samples Have Many Samples Have Same IntervalSame Interval

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xx__

XX

7 - 7 - 3434



xx__

XX

XX= = ± Z ± Zxx

7 - 7 - 3535



90% Samples90% Samples

xx__

XX

XX= = ± Z ± Zxx

+1.65+1.65xx-1.65-1.65xx

7 - 7 - 3636





+1.65+1.65xx

xx__

XX

+1.96+1.96xx

-1.65-1.65xx-1.96-1.96xx

XX= = ± Z ± Zxx

7 - 7 - 3737






+1.645+1.645x x +2.58+2.58xx

xx__

XX

+1.96+1.96xx

-2.58-2.58xx -1.645-1.645xx

-1.96-1.96xx

XX= = ± Z ± Zxx

7 - 7 - 3838


1.1. Probability that the unknown population Probability that the unknown population parameter falls within intervalparameter falls within interval

2.2. Denoted (1 - Denoted (1 - is probability that parameter is is probability that parameter is notnot

within intervalwithin interval

3.3. Typical values are 99%, 95%, 90%Typical values are 99%, 95%, 90%

Confidence Level Confidence Level

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Intervals & Intervals & Confidence Level Confidence Level

7 - 7 - 4040



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling

Distribution Distribution of Meanof Mean

7 - 7 - 4141



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling


Confidence Confidence IntervalInterval

7 - 7 - 4242



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling



7 - 7 - 4343



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling


Intervals Intervals extend from extend from X - ZX - ZXX to to

X + ZX + ZXX


7 - 7 - 4444



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling



X + ZX + ZXX


7 - 7 - 4545



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling



X + ZX + ZXX


7 - 7 - 4646



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling


Large number of intervalsLarge number of intervals


X + ZX + ZXX


7 - 7 - 4747



x =

1 - /2/2

X_

x_

x =

1 - /2/2

X_

x_Sampling Sampling


Large number of intervalsLarge number of intervals


X + ZX + ZXX

(1 - (1 - )100 % )100 % of intervals of intervals contain contain . .

% do % do not.not.

7 - 7 - 4848


Factors Affecting Factors Affecting Interval WidthInterval Width

1.1. Data dispersionData dispersion Measured by Measured by

2.2. Sample sizeSample size X X = = / / nn

3.3. Level of confidence Level of confidence (1 - (1 - )) Affects ZAffects Z

Intervals extend fromIntervals extend from

X - ZX - ZXX to toX + ZX + ZXX

© 1984-1994 T/Maker Co.

7 - 7 - 4949


Confidence Interval Confidence Interval EstimatesEstimates

7 - 7 - 5050



OnePopulation

7 - 7 - 5151



OnePopulation

Mean

7 - 7 - 5252



OnePopulation

Mean Proportion

7 - 7 - 5353



OnePopulation

ZDistribution

LargeLargeSampleSample

Mean Proportion

7 - 7 - 5454



OnePopulation

ZDistribution

tDistribution


Mean ProportionSmallSmall

SampleSample

7 - 7 - 5555



OnePopulation

ZDistribution

tDistribution

LargeSample

ZDistribution

Mean ProportionSmallSample

OnePopulation

ZDistribution

tDistribution

LargeSample

ZDistribution

Mean ProportionSmallSample

7 - 7 - 5656


Confidence Interval Confidence Interval Estimate Mean (Large Estimate Mean (Large

Sample)Sample)

7 - 7 - 5757



OnePopulation

ZDistribution

tDistribution


ZDistribution


SampleSample

7 - 7 - 5858


Confidence Interval Confidence Interval Mean (Large Sample)Mean (Large Sample)

1.1. AssumptionsAssumptions Sample size at least 30 (Sample size at least 30 (nn 30) 30) Random sample drawnRandom sample drawn If population standard deviation unknown, If population standard deviation unknown,

use sample standard deviationuse sample standard deviation

7 - 7 - 5959


Confidence Interval Confidence Interval Mean (Large Sample)Mean (Large Sample)

1.1. AssumptionsAssumptions Sample size at least 30 (Sample size at least 30 (nn 30) 30) Random sample drawnRandom sample drawn If population standard deviation unknown, If population standard deviation unknown,

use sample standard deviationuse sample standard deviation

2.2. Confidence interval estimateConfidence interval estimate

X Zn

X Zn

/ /2 2X Zn

X Zn

/ /2 2

7 - 7 - 6060


Estimation Example Estimation Example Mean (Large Mean (Large

Sample)Sample)The mean of a random sample of The mean of a random sample of nn = 36 = 36 isisX = 50. Set up a 95% confidence X = 50. Set up a 95% confidence interval estimate for interval estimate for if if = 12. = 12.

7 - 7 - 6161


Estimation Example Estimation Example Mean (Large Mean (Large

Sample)Sample)The mean of a random sample of The mean of a random sample of nn = 36 = 36 isisX = 50. Set up a 95% confidence X = 50. Set up a 95% confidence interval estimate for interval estimate for if if = 12. = 12.

X Zn

X Zn

/ /

. .

. .

2 2

50 1961236

50 1961236

46 08 53 92

X Zn

X Zn

/ /

. .

. .

2 2

50 1961236

50 1961236

46 08 53 92

7 - 7 - 6262



You’re a Q/C inspector for You’re a Q/C inspector for Gallo. The Gallo. The for 2-liter for 2-liter bottles is bottles is .05.05 liters. A liters. A random sample of random sample of 100100 bottles showedbottles showedX = 1.99 X = 1.99 liters. What is the liters. What is the 90%90% confidence interval confidence interval estimate of the true estimate of the true meanmean amount in 2-liter bottles? amount in 2-liter bottles? 2

liter© 1984-1994 T/Maker Co.

2 liter


7 - 7 - 6363


Confidence Interval Confidence Interval Solution*Solution*

X Zn

X Zn

/ /

. ..

. ..

. .

2 2

199 164505100

199 164505100

1982 1998

X Zn

X Zn

/ /

. ..

. ..

. .

2 2

199 164505100

199 164505100

1982 1998

7 - 7 - 6464


Confidence Interval Confidence Interval Estimate Mean (Small Estimate Mean (Small

Sample)Sample)

7 - 7 - 6565



OnePopulation

ZDistribution

tDistribution


ZDistribution


SampleSample

7 - 7 - 6666


Confidence Interval Confidence Interval Mean (Small Sample) Mean (Small Sample)

1.1. AssumptionsAssumptions Sample size less than 30 (Sample size less than 30 (nn < 30) < 30) Population normally distributedPopulation normally distributed Population standard deviation unknownPopulation standard deviation unknown

2.2. Use Student’s t distributionUse Student’s t distribution

7 - 7 - 6767


Confidence Interval Confidence Interval Mean (Small Sample) Mean (Small Sample)

1.1. AssumptionsAssumptions Sample size less than 30 (Sample size less than 30 (nn < 30) < 30) Population normally distributedPopulation normally distributed Population standard deviation unknownPopulation standard deviation unknown

2.2. Use Student’s t distributionUse Student’s t distribution


X tSn

X tSnn n / , / ,2 1 2 1X t

Sn

X tSnn n / , / ,2 1 2 1

7 - 7 - 6868


Student’s t Student’s t DistributionDistribution

7 - 7 - 6969


ZZ


00

Standard Standard NormalNormal

7 - 7 - 7070


ZZtt


00


t (t (dfdf = 13) = 13)Bell-ShapedBell-Shaped

SymmetricSymmetric

‘‘Fatter’ TailsFatter’ Tails

7 - 7 - 7171


ZZtt


00

t (t (dfdf = 5) = 5)


t (t (dfdf = 13) = 13)Bell-ShapedBell-Shaped

SymmetricSymmetric

‘‘Fatter’ TailsFatter’ Tails

7 - 7 - 7272


Student’s Student’s tt Table Table

7 - 7 - 7373


v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182


7 - 7 - 7474



t valuest values

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

7 - 7 - 7575



t valuest valuest0 t0

/ 2/ 2

/ 2/ 2

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

7 - 7 - 7676




/ 2/ 2

/ 2/ 2

Assume:Assume:nn = 3 = 3dfdf = = nn - 1 = 2 - 1 = 2 = .10= .10/2 =.05/2 =.05

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

7 - 7 - 7777


v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182



/ 2/ 2

/ 2/ 2


7 - 7 - 7878


v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182



/ 2/ 2Assume:Assume:nn = 3 = 3dfdf = = nn - 1 = 2 - 1 = 2 = .10= .10/2 =.05/2 =.05

.05.05

7 - 7 - 7979


v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

v t.10 t.05 t.025

1 3.078 6.314 12.706

2 1.886 2.920 4.303

3 1.638 2.353 3.182

t0 t0



2.9202.920t valuest values

/ 2/ 2

.05.05

7 - 7 - 8080


Degrees of Freedom Degrees of Freedom ((dfdf))

1.1. Number of observations that are free to Number of observations that are free to vary after sample statistic has been vary after sample statistic has been calculatedcalculated

7 - 7 - 8181




2.2. Example:Example:Sum of 3 numbers is 6Sum of 3 numbers is 6

XX1 1 = =

XX2 2 = =

XX3 3 ==

Sum = 6Sum = 6

7 - 7 - 8282





XX1 1 = 1 (Or any number)= 1 (Or any number)

XX2 2 = =

XX3 3 ==

Sum = 6Sum = 6

7 - 7 - 8383







XX3 3 ==

Sum = 6Sum = 6

7 - 7 - 8484







XX3 3 = = 33 (Cannot vary) (Cannot vary)

Sum = 6Sum = 6

7 - 7 - 8585







XX3 3 = = 33 (Cannot vary) (Cannot vary)

Sum = 6Sum = 6

degrees of freedom degrees of freedom = = nn -1 -1 = 3 -1= 3 -1= 2= 2

7 - 7 - 8686


Estimation Example Estimation Example Mean (Small Mean (Small

Sample)Sample)A random sample of A random sample of nn = 25 has = 25 hasxx = 50 & = 50 & ss = 8. Set up a 95% confidence interval = 8. Set up a 95% confidence interval estimate for estimate for ..

7 - 7 - 8787


Estimation Example Estimation Example Mean (Small Mean (Small

Sample)Sample)A random sample of A random sample of nn = 25 has = 25 hasxx = 50 & = 50 & ss = 8. Set up a 95% confidence interval = 8. Set up a 95% confidence interval estimate for estimate for ..

X tSn

X tSnn n

/ , / ,

. .

. .

2 1 2 1

50 2 0639825

50 2 0639825

46 69 53 30

X tSn

X tSnn n

/ , / ,

. .

. .

2 1 2 1

50 2 0639825

50 2 0639825

46 69 53 30

7 - 7 - 8888



You’re a time study You’re a time study analyst in manufacturing. analyst in manufacturing. You’ve recorded the You’ve recorded the following task times (min.): following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.13.6, 4.2, 4.0, 3.5, 3.8, 3.1..

What is the What is the 90%90% confidence interval confidence interval estimate of the population estimate of the population meanmean task time? task time?


7 - 7 - 8989



X = 3.7X = 3.7

SS = 3.8987 = 3.8987

nn = 6, df = = 6, df = nn - 1 = 6 - 1 = 5 - 1 = 6 - 1 = 5

SS / / nn = 3.8987 / = 3.8987 / 6 = 1.5926 = 1.592

tt.05,5.05,5 = 2.0150 = 2.0150

3.7 - (2.015)(1.592) 3.7 - (2.015)(1.592) 3.7 + (2.015)3.7 + (2.015)(1.592) (1.592)

0.492 0.492 6.908 6.908

7 - 7 - 9090


Confidence Interval Confidence Interval Estimate of ProportionEstimate of Proportion

7 - 7 - 9191


Data TypesData Types

Data

Quantitative Qualitative

Discrete Continuous

Data

Quantitative Qualitative

Discrete Continuous

7 - 7 - 9292


Qualitative DataQualitative Data

1.1. Qualitative random variables yield Qualitative random variables yield responses that classifyresponses that classify e.g., gender (male, female)e.g., gender (male, female)

2.2. Measurement reflects # in categoryMeasurement reflects # in category

3.3. Nominal or ordinal scaleNominal or ordinal scale

4.4. ExamplesExamples Do you own savings bonds? Do you own savings bonds? Do you live on-campus or off-campus?Do you live on-campus or off-campus?

7 - 7 - 9393


ProportionsProportions

1.1. Involve qualitative variablesInvolve qualitative variables

2.2. Fraction or % of population in a categoryFraction or % of population in a category

3.3. If two qualitative outcomes, binomial If two qualitative outcomes, binomial distributiondistribution Possess or don’t possess characteristicPossess or don’t possess characteristic

7 - 7 - 9494


ProportionsProportions

1.1. Involve qualitative variablesInvolve qualitative variables

2.2. Fraction or % of population in a categoryFraction or % of population in a category

3.3. If two qualitative outcomes, binomial If two qualitative outcomes, binomial distributiondistribution Possess or don’t possess characteristicPossess or don’t possess characteristic

4.4. Sample proportion (Sample proportion (pp))

pxn

number of successes

sample sizep

xn

number of successes

sample size

^̂

7 - 7 - 9595


Sampling Sampling Distribution Distribution of Proportionof Proportion

7 - 7 - 9696



Sampling DistributionSampling Distribution

.0.0

.1.1

.2.2

.3.3

.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0

PP^̂

P(PP(P^̂ ))

7 - 7 - 9797


1.1. Approximated by Approximated by normal distributionnormal distribution

excludes 0 or nexcludes 0 or n



.0.0

.1.1

.2.2

.3.3

.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0

PP^̂

P(PP(P^̂ ))np np p 3 1b g np np p 3 1b g

7 - 7 - 9898




2.2. MeanMean


Pp Pp


.0.0

.1.1

.2.2

.3.3

.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0

PP^̂


Pp Pp

7 - 7 - 9999


ppaa



2.2. MeanMean

3.3. Standard errorStandard error


Pp Pp


pp̂̂pp

nn

11 ff

.0.0

.1.1

.2.2

.3.3

.0.0 .2.2 .4.4 .6.6 .8.8 1.01.0

PP^̂


Pp Pp

7 - 7 - 100100



OnePopulation

ZDistribution

tDistribution


ZDistribution


SampleSample

7 - 7 - 101101


Confidence Interval Confidence Interval Proportion Proportion

7 - 7 - 102102



1.1. AssumptionsAssumptions Two categorical outcomesTwo categorical outcomes Population follows binomial distributionPopulation follows binomial distribution Normal approximation can be usedNormal approximation can be used

does not include 0 or 1does not include 0 or 1np np p 3 1b g np np p 3 1b g

7 - 7 - 103103



1.1. AssumptionsAssumptions Two categorical outcomesTwo categorical outcomes Population follows binomial distributionPopulation follows binomial distribution Normal approximation can be usedNormal approximation can be used

does not include 0 or 1does not include 0 or 1


( )

( )p z

p pn

p p zp p

n

2 2

1 1 ( )

( )p z

p pn

p p zp p

n

2 2

1 1

np np p 3 1b g np np p 3 1b g

7 - 7 - 104104


Estimation Example Estimation Example ProportionProportion

A random sample of 400 graduates A random sample of 400 graduates showed 32 went to grad school. Set up a showed 32 went to grad school. Set up a 95% confidence interval estimate for 95% confidence interval estimate for pp..

7 - 7 - 105105


Estimation Example Estimation Example ProportionProportion

A random sample of 400 graduates A random sample of 400 graduates showed 32 went to grad school. Set up a showed 32 went to grad school. Set up a 95% confidence interval estimate for 95% confidence interval estimate for pp..

( )

( )

. .. ( . )

. .. ( . )

. .

/ /p Zp p

np p Z

p pn

p

p

2 21 1

08 19608 1 08

40008 196

08 1 08400

053 107

( )

( )

. .. ( . )

. .. ( . )

. .

/ /p Zp p

np p Z

p pn

p

p

2 21 1

08 19608 1 08

40008 196

08 1 08400

053 107

7 - 7 - 106106



You’re a production You’re a production manager for a newspaper. manager for a newspaper. You want to find the % You want to find the % defective. Of defective. Of 200200 newspapers, newspapers, 3535 had had defects. What is the defects. What is the 90%90% confidence interval confidence interval estimate of the population estimate of the population proportionproportion defective? defective?


7 - 7 - 107107



( )

( )

. .. (. )

. .. (. )

. .

/ /p zp p

np p z

p pn

p

p

2 21 1

175 1645175 825

200175 1645

175 825200

1308 2192

( )

( )

. .. (. )

. .. (. )

. .

/ /p zp p

np p z

p pn

p

p

2 21 1

175 1645175 825

200175 1645

175 825200

1308 2192

7 - 7 - 108108


Finding Sample SizesFinding Sample Sizes

7 - 7 - 109109


Finding Sample Finding Sample Sizes Sizes

for Estimating for Estimating

7 - 7 - 110110




7 - 7 - 111111




7 - 7 - 112112




(1)

(2)

ZX Error

Error Z Zn

nZ

Error

x x

x

( )3

2 2

2

(1)

(2)

ZX Error

Error Z Zn

nZ

Error

x x

x

( )3

2 2

2

Error is also called bound, Error is also called bound, BB

7 - 7 - 113113



for Estimating for Estimating I don’t want to sample too much or too little!

(1)

(2)

ZX Error

Error Z Zn

nZ

Error

x x

x

( )3

2 2

2

(1)

(2)

ZX Error

Error Z Zn

nZ

Error

x x

x

( )3

2 2

2

Error is also called bound, Error is also called bound, BB

7 - 7 - 114114


Sample Size Sample Size ExampleExample

What sample size is needed to be What sample size is needed to be 90%90% confident of being correct within confident of being correct within 5 5? A ? A pilot study suggested that the standard pilot study suggested that the standard deviation is deviation is 4545..

7 - 7 - 115115


Sample Size Sample Size ExampleExample

What sample size is needed to be What sample size is needed to be 90%90% confident of being correct within confident of being correct within 5 5? A ? A pilot study suggested that the standard pilot study suggested that the standard deviation is deviation is 4545..

nZ

Error

2 2

2

2 2

2

1645 45

5219 2 220

..

a fa fafn

Z

Error

2 2

2

2 2

2

1645 45

5219 2 220

..

a fa faf

7 - 7 - 116116



You work in Human You work in Human Resources at Merrill Lynch. Resources at Merrill Lynch. You plan to survey employees You plan to survey employees to find their average medical to find their average medical expenses. You want to be expenses. You want to be 95%95% confident that the sample confident that the sample meanmean is within is within ± $50± $50. . A pilot study showed that A pilot study showed that was about was about $400$400. What . What sample sizesample size do you use? do you use?


7 - 7 - 117117


Sample Size Sample Size Solution*Solution*

nZ

Error

2 2

2

2 2

2

196 400

50

245 86 246

.

.

a fa faf

nZ

Error

2 2

2

2 2

2

196 400

50

245 86 246

.

.

a fa faf

7 - 7 - 118118


ConclusionConclusion

1.1. Stated what is estimatedStated what is estimated

2.2. Distinguished point & interval estimatesDistinguished point & interval estimates

3.3. Explained interval estimatesExplained interval estimates

4.4. Computed confidence interval estimates Computed confidence interval estimates for population mean & proportionfor population mean & proportion

5.5. Computed sample sizeComputed sample size

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This Class...This Class...

1.1. What was the most important thing you What was the most important thing you learned in class today?learned in class today?

2.2. What do you still have questions about?What do you still have questions about?

3.3. How can today’s class be improved?How can today’s class be improved?

Please take a moment to answer the following questions in writing:

End of Chapter

Any blank slides that follow are blank intentionally.

7 - 1 © 1998 Prentice-Hall, Inc. Chapter 7 Inferences Based on a Single Sample: Estimation with Confidence Intervals.

Documents

sample slide

estimation process slide

x slide

estimated slide

sample statistic slide

alonegroupclass slide

estimate of unknown

unknown population parameters