© 2002 Prentice-Hall, Inc.Chap 6-1 Statistics for Managers using Microsoft Excel 3 rd Edition...
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Transcript of © 2002 Prentice-Hall, Inc.Chap 6-1 Statistics for Managers using Microsoft Excel 3 rd Edition...
© 2002 Prentice-Hall, Inc. Chap 6-1
Statistics for Managers using Microsoft Excel
3rd Edition
Chapter 6Confidence Interval
Estimation
© 2002 Prentice-Hall, Inc. Chap 6-2
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) Confidence interval estimation for the
proportion
© 2002 Prentice-Hall, Inc. Chap 6-3
Chapter Topics
Applications of confidence interval estimation in auditing 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 issues
(continued)
© 2002 Prentice-Hall, Inc. Chap 6-4
Estimation Process
Mean, , is unknown
Population Random Sample
Mean X = 50
Sample
I am 95% confident that is between 40 & 60.
© 2002 Prentice-Hall, Inc. Chap 6-5
Point Estimates
Estimate Population
Parameters …
with SampleStatistics
Mean
Proportion
Variance
Difference
p
2
1 2
X
SP
2S
1 2X X
© 2002 Prentice-Hall, Inc. Chap 6-6
Interval Estimates
Provides range of values Take into consideration variation in
sample statistics from sample to sample
Based on observation from 1 sample Give information about closeness to
unknown population parameters Stated in terms of level of confidence
Never 100% sure
© 2002 Prentice-Hall, Inc. Chap 6-7
Confidence Interval Estimates
Mean
Unknown
ConfidenceIntervals
Proportion
Known
© 2002 Prentice-Hall, Inc. Chap 6-8
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 / 2X Z X Zn n
© 2002 Prentice-Hall, Inc. Chap 6-9
Elements of Confidence 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
© 2002 Prentice-Hall, Inc. Chap 6-10
Level of Confidence
Denoted by A relative frequency interpretation
In the long run, of all the confidence intervals that can be constructed will contain the unknown parameter
A specific interval will either contain or not contain the parameter No probability involved in a specific interval
100 1 %
100 1 %
© 2002 Prentice-Hall, Inc. Chap 6-11
Interval and Level of Confidence
Confidence Intervals
Intervals extend from
to
of intervals constructed contain ;
do not.
_Sampling Distribution of the Mean
XX Z
X/ 2
/ 2
XX
1
XX Z
100 1 %
100 %
/ 2 XZ / 2 XZ
© 2002 Prentice-Hall, Inc. Chap 6-12
Factors Affecting Interval Width (Precision)
Data variation Measured by
Sample size
Level of confidence
Intervals Extend from
© 1984-1994 T/Maker Co.
X - Z to X + Z xx
Xn
100 1 %
© 2002 Prentice-Hall, Inc. Chap 6-13
Determining Sample Size (Cost)
Too Big:
• Requires too much resources
Too small:
• Won’t do the job
© 2002 Prentice-Hall, Inc. Chap 6-14
Determining Sample Size 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 2
2 2
1.645 45219.2 220
Error 5
Zn
© 2002 Prentice-Hall, Inc. Chap 6-15
Determining Sample Size for Mean in PHStat
PHStat | sample size | determination for the mean …
Example in excel spreadsheet
Microsoft Excel Worksheet
© 2002 Prentice-Hall, Inc. Chap 6-16
Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample
Use Student’s t Distribution Confidence Interval Estimate
Confidence Interval for( Unknown)
/ 2, 1 / 2, 1n n
S SX t X t
n n
© 2002 Prentice-Hall, Inc. Chap 6-17
Student’s t Distribution
Zt
0
t (df = 5)
t (df = 13)Bell-ShapedSymmetric
‘Fatter’ Tails
Standard Normal
© 2002 Prentice-Hall, Inc. Chap 6-18
Degrees of Freedom (df )
Number of observations that are free to vary 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)
X
X
X
© 2002 Prentice-Hall, Inc. Chap 6-19
Student’s t Table
Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
t0 2.920t Values
Let: n = 3 df = n - 1 = 2 = .10 /2 =.05
/ 2 = .05
© 2002 Prentice-Hall, Inc. Chap 6-20
Example
A random sample of 25 has 50 and 8.
Set up a 95% confidence interval estimate for
n X S
/ 2, 1 / 2, 1
8 850 2.0639 50 2.0639
25 2546.69 53.30
n n
S SX t X t
n n
© 2002 Prentice-Hall, Inc. Chap 6-21
PHStat | confidence interval | estimate for the mean, sigma unknown
Example in excel spreadsheet
Confidence Interval for( Unknown) in PHStat
Microsoft Excel Worksheet
© 2002 Prentice-Hall, Inc. Chap 6-22
Confidence Interval Estimate 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 pp Z p p Z
n n
© 2002 Prentice-Hall, Inc. Chap 6-23
ExampleA 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 1.96
400 400.053 .107
s s s ss s
p p p pp Z p p Z
n n
p
p
© 2002 Prentice-Hall, Inc. Chap 6-24
Confidence Interval Estimate for Proportion in PHStat
PHStat | confidence interval | estimate for the proportion …
Example in excel spreadsheet
Microsoft Excel Worksheet
© 2002 Prentice-Hall, Inc. Chap 6-25
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
© 2002 Prentice-Hall, Inc. Chap 6-26
Determining Sample Size for Proportion in PHStat
PHStat | sample size | determination for the proportion …
Example in excel spreadsheet
Microsoft Excel Worksheet
© 2002 Prentice-Hall, Inc. Chap 6-27
Applications in Auditing
Six advantages of statistical sampling in auditing Sample result is objective and defensible
Based on demonstrable statistical principles Provides sample size estimation in advance
on an objective basis Provides an estimate of the sampling error
© 2002 Prentice-Hall, Inc. Chap 6-28
Applications in Auditing Can provide more accurate conclusions on the
population than the actual investigation of the population
Examination of the population can be time consuming and subject to more nonsampling error
Samples can be combined and evaluated by different auditors
Samples are based on scientific approach Samples can be treated as if they have been
done by a single auditor Objective evaluation of the results is possible
Based on known sampling error
© 2002 Prentice-Hall, Inc. Chap 6-29
Confidence Interval for Population Total Amount
Point estimate
Confidence interval estimate
NX
/ 2, 1 1n
N nSNX N t
Nn
© 2002 Prentice-Hall, Inc. Chap 6-30
Confidence Interval for Population Total: Example
An auditor is faced with a population of 1000 vouchers and wishes to estimate the total value of the population of vouchers. A sample of 50 vouchers is selected with average voucher amount of $1076.39, standard deviation of $273.62. Set up the 95% confidence interval estimate of the total amount for the population of vouchers.
© 2002 Prentice-Hall, Inc. Chap 6-31
Example Solution
/ 2, 1
1000 50 $1076.39 $273.62
1
273.62 1000 501000 1076.39 1000 2.0096
1000 11001,076,390 75,830.85
n
N n X S
N nSNX N t
Nn
The 95% confidence interval for the population total amount of the vouchers is between 1,000,559.15, and 1,152,220.85
© 2002 Prentice-Hall, Inc. Chap 6-32
Example Solution in PHStat
PHStat | confidence intervals | estimate for the population total
Excel spreadsheet for the voucher example
Microsoft Excel Worksheet
© 2002 Prentice-Hall, Inc. Chap 6-33
Confidence Interval for Total Difference in the Population
Point estimate Where is the sample
average difference
Confidence interval estimate
Where
ND 1
n
ii
DD
n
/ 2, 1 1
Dn
N nSND N t
Nn
2
1
1
n
ii
D
D DS
n
© 2002 Prentice-Hall, Inc. Chap 6-34
Estimation for Finite Population
Samples are selected without replacement Confidence interval for the mean
( unknown)
Confidence interval for proportion
/ 2, 1 1n
N nSX t
Nn
/ 2
1
1S S
S
p p N np Z
n N
© 2002 Prentice-Hall, Inc. Chap 6-35
Sample Size Determination for Finite Population
Samples are selected without replacement When estimating the mean
When estimating the proportion
2 2/ 2
0 2
Zn
e
2/ 2
0 2
1Z p pn
e
© 2002 Prentice-Hall, Inc. Chap 6-36
Ethical Issues
Confidence interval (reflects sampling error) should always be reported along with the point estimate
The level of confidence should always be reported
The sample size should be reported An interpretation of the confidence
interval estimate should also be provided
© 2002 Prentice-Hall, Inc. Chap 6-37
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) Discussed confidence interval estimation for
the proportion
© 2002 Prentice-Hall, Inc. Chap 6-38
Chapter Summary
Addressed applications of confidence interval estimation in auditing Confidence interval estimation for population
total 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)