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2002 Prentice-Hall, Inc. Chap 8-1
Basic Business Statistics(8th Edition)
Chapter 8
Confidence Interval Estimation
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2002 Prentice-Hall, Inc. Chap 8-2
Chapter Topics
Estimation process
Point estimates
Interval estimates Confidence interval estimation for the mean
( known)
Determining sample size
Confidence interval estimation for the mean( unknown)
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2002 Prentice-Hall, Inc. Chap 8-3
Chapter Topics
Confidence interval estimation for theproportion
Confidence interval estimation for
population total Confidence interval estimation for total
difference in the population
Estimation and sample size determination
for finite population Confidence interval estimation and ethical
considerations
(continued)
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2002 Prentice-Hall, Inc. Chap 8-4
Estimation Process
Mean, , isunknown
Population Random Sample
MeanX = 50
Sample
I am 95%
confident that
is between 40& 60.
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2002 Prentice-Hall, Inc. Chap 8-5
Point Estimates
Estimate PopulationParameters
with SampleStatistics
Mean
Proportion
Variance
Difference
p
2
1 2
X
SP
2S
1 2X X
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2002 Prentice-Hall, Inc. Chap 8-6
Interval Estimates
Provides range of values
Takes into consideration variation in sample
statistics from sample to sample Is based on observation from one sample
Gives information about closeness to
unknown population parameters Is stated in terms of level of confidence
Never 100% certain
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Confidence Interval Estimates
Mean
Unknown
ConfidenceIntervals
Proportion
Known
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Confidence Interval for( Known)
Assumptions
Population standard deviation is known
Population is normally distributed If population is not normal, use large
sample
Confidence interval estimate
/ 2 / 2 X Z X Z n n
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Elements ofConfidence Interval Estimation
Level of confidence
Confidence in which the interval will contain
the unknown population parameter Precision (range)
Closeness to the unknown parameter
Cost Cost required to obtain a sample of size n
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Level of Confidence
Denoted by
A relative frequency interpretation
In the long run, of all the confidenceintervals that can be constructed will contain theunknown parameter
A specific interval will either contain or not
contain the parameter No probability involved in a specific interval
100 1 %
100 1 %
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Interval and Level of Confidence
Confidence Intervals
Intervals
extend from
to of intervalsconstructedcontain ;
donot.
_Sampling Distribution of the Mean
XX Z
X
/ 2/ 2
XX
1
XX Z
1 100%
100 %
/ 2 XZ
/ 2 XZ
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Factors AffectingInterval Width (Precision)
Data variation
Measured by
Sample size
Level of confidence
Intervals Extend from
1984-1994 T/Maker Co.
X- Z to X+ Zxx
Xn
100 1 %
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Determining Sample Size (Cost)
Too Big:
Requirestoo many
resources
Too small:
Wont dothe job
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Determining SampleSize for Mean
What sample size is needed to be 90% confident of
being correct within 5? A pilot study suggested that
the standard deviation is 45.
Round Up
2 22 22 2
1.645 45219.2 220
Error 5
Zn
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Determining SampleSize for Mean in PHStat
PHStat | sample size | determination for themean
Example in excel spreadsheet
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Assumptions
Population standard deviation is unknown
Population is normally distributed If population is not normal, use large sample
Use students t distribution
Confidence interval estimate
Confidence Interval for( Unknown)
/ 2, 1 / 2, 1n n
S S X t X t
n n
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2002 Prentice-Hall, Inc. Chap 8-17
Students tDistribution
Z
t0
t(df= 5)
t(df= 13)Bell-ShapedSymmetric
Fatter
Tails
Standard
Normal
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2002 Prentice-Hall, Inc. Chap 8-18
Degrees of Freedom (df)
Number of observations that are free tovary after sample mean has been calculated
Example: Mean of 3 numbers is 2
degrees of freedom= n -1
= 3 -1
= 2
1
2
3
1 (or any number)2 (or any number)
3 (cannot vary)
XX
X
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2002 Prentice-Hall, Inc. Chap 8-19
Students t Table
Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
20.817 1.886
2.920
3 0.765 1.638 2.353
t0 2.920tValues
Let: n = 3
df = n - 1 = 2
= .10
/2 =.05
/ 2 = .05
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2002 Prentice-Hall, Inc. Chap 8-20
Example
/ 2, 1 / 2, 1
8 850 2.0639 50 2.063925 25
46.69 53.30
n n
S S X t X t
n n
A random sample of 25 has 50 and 8.
Set up a 95% confidence interval estimate for
n X S
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2002 Prentice-Hall, Inc. Chap 8-21
PHStat | confidence interval | estimate for themean, sigma unknown
Example in excel spreadsheet
Confidence Interval for( Unknown) in PHStat
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2002 Prentice-Hall, Inc. Chap 8-22
Confidence IntervalEstimate for Proportion
Assumptions
Two categorical outcomes
Population follows binomial distribution Normal approximation can be used if
and
Confidence interval estimate
5np 1 5n p
/ 2 / 2
1 1S S S SS S
p p p p p Z p p Z
n n
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2002 Prentice-Hall, Inc. Chap 8-23
Example
A random sample of 400 voters showed 32
preferred candidate A. Set up a 95% confidence
interval estimate for p.
/ /
1 1
.08 1 .08 .08 1 .08.08 1.96 .08 1.96400 400
.053 .107
s s s s
s s
p p p p p Z p p Z
n n
p
p
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2002 Prentice-Hall, Inc. Chap 8-24
Confidence Interval Estimate forProportion in PHStat
PHStat | confidence interval | estimate for theproportion
Example in excel spreadsheet
D t i i S l Si f
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2002 Prentice-Hall, Inc. Chap 8-25
Determining Sample Size for
Proportion
Out of a population of 1,000, we randomly
selected 100, of which 30 were defective. What
sample size is needed to be within 5% with
90% confidence?
Round Up
2 2
2 2
1 1.645 0.3 0.7
Error 0.05227.3 228
Z p pn
f
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2002 Prentice-Hall, Inc. Chap 8-26
Determining Sample Size forProportion in PHStat
PHStat | sample size | determination for theproportion
Example in excel spreadsheet
C fid I t l f
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2002 Prentice-Hall, Inc. Chap 8-27
Confidence Interval for
Population Total Amount
Point estimate
Confidence interval estimate
NX
/ 2, 1
1n
N nS NX N t
Nn
C fid l f
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2002 Prentice-Hall, Inc. Chap 8-28
Confidence Interval forPopulation Total: Example
An auditor is faced with apopulation of 1000 vouchersand wants to estimate thetotal value of the population.
A sample of 50 vouchers isselected with averagevoucher amount of$1076.39, standard deviation
of $273.62. Set up the 95%confidence interval estimateof the total amount for thepopulation of vouchers.
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2002 Prentice-Hall, Inc. Chap 8-29
Example Solution
/ 2, 1
1000 50 $1076.39 $273.62
1
273.62 1000 501000 1076.39 1000 2.0096
1000 1100
1,076,390 75,830.85
n
N n X S
N nS NX N t
Nn
The 95% confidence interval for the population totalamount of the vouchers is between 1,000,559.15, and
1,152,220.85
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2002 Prentice-Hall, Inc. Chap 8-30
Example Solution in PHStat
PHStat | confidence intervals | estimate forthe population total
Excel spreadsheet for the voucher example
C fid I t l f T t l
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2002 Prentice-Hall, Inc. Chap 8-31
Confidence Interval for TotalDifference in the Population
Point estimate
Where is the sample average
difference Confidence interval estimate
Where
ND 1
n
i
i
D
Dn
/ 2, 1
1
Dn
N nS ND N t
Nn
2
1
1
n
i
i
D
D D
Sn
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2002 Prentice-Hall, Inc. Chap 8-32
Estimation for Finite Population
Samples are selected without replacement
Confidence interval for the mean ( unknown)
Confidence interval for proportion
/ 2, 11
nN nSX tNn
/ 2
1
1
S S
S
p p N np Z
n N
S l Si D t i ti
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2002 Prentice-Hall, Inc. Chap 8-33
Sample Size Determinationfor Finite Population
Samples are selected without replacement
When estimating the mean
When estimating the proportion
2 2
/ 20 2
Zne
2/ 20 2
1 Z p pne
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2002 Prentice-Hall, Inc. Chap 8-34
Ethical Considerations
Report confidence interval (reflect sampling
error) along with the point estimate
Report the level of confidence Report the sample size
Provide an interpretation
of the confidence interval estimate
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2002 Prentice-Hall, Inc. Chap 8-35
Chapter Summary
Illustrated estimation process
Discussed point estimates
Addressed interval estimates Discussed confidence interval estimation
for the mean ( known)
Addressed determining sample size Discussed confidence interval estimation
for the mean ( unknown)
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Chapter Summary
Discussed confidence interval estimation forthe proportion
Addressed confidence interval estimation for
population total Discussed confidence interval estimation for
total difference in the population
Addressed estimation and sample size
determination for finite population Addressed confidence interval estimation and
ethical issues
(continued)
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