Download Mean, , is unknown

Confidence Interval
& Estimation
Zulkarnain Ishak
PSIE Pasca Sarjana Unsri
2007
Topics
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•
•
•
•
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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
Topics
(continued)
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
Estimation Process
Population
Random Sample
Mean, , is
unknown
Mean
X = 50
Sample
I am 95%
confident that
 is between 40
& 60.
Points Estimation
Estimate Population
Parameters …
Mean
Proportion
Variance
Difference

p
with Sample
Statistics
X
PS

1  2
2
S
2
X1  X 2
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
Confidence Interval Estimates
Confidence
Intervals
Mean
 Known
Proportion
 Unknown

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
X  Z / 2

n
   X  Z / 2

n
Elements of
Confidence Interval Estimation

Level of confidence


Precision (range)


Confidence in which the interval will
contain the unknown population parameter
Closeness to the unknown parameter
Cost

Cost required to obtain a sample of size n
Level of Confidence


Denoted by 100 1    %
A relative frequency interpretation
 In the long run, 100 1    % 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
Interval and Level of Confidence
Sampling Distribution of the
_ Mean
  Z / 2 X
Intervals
extend from
 /2
X
1
  Z / 2 X
 /2
X  
X  Z X
X
100 1    %
of intervals
constructed
contain ;
to
X  Z X
100 % do
Confidence Intervals
not.
Factors Affecting
Interval Width (Precision)

Data variation


Sample size


Measured by 
X 
Intervals Extend from
X - Z
x
to X + Z 
x

n
Level of confidence

100 1    %
© 1984-1994 T/Maker Co.
Determining Sample Size (Cost)
Too Big:
Too small:
• Requires
too much
resources
• Won’t do
the job
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.
1.645  45
Z
n

2
2
Error
5
2
2
2
2
  219.2  220
Round Up
Determining Sample
Size for Mean in PHStat


PHStat | sample size | determination for the
mean …
Example in excel spreadsheet
Confidence Interval for
( Unknown) ]


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

X  t / 2,n 1
S
S
   X  t / 2,n 1
n
n
Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’
Tails
t (df = 13)
t (df = 5)
Z
0
t
Degrees of Freedom (df )


Number of observations that are free to
vary after sample mean has been calculated
Example
degrees of freedom

Mean of 3 numbers is 2
X 1  1 (or any number)
X 2  2 (or any number)
X 3  3 (cannot vary)
= n -1
= 3 -1
=2
Student’s t Table
Upper Tail Area
df
.25
.10
.05
Let: n = 3
df = n - 1 = 2
 = .10
/2 =.05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
 / 2 = .05
3 0.765 1.638 2.353
t Values
0 2.920
t
Example
A random sample of n  25 has X  50 and S  8.
Set up a 95% confidence interval estimate for 
S
S
X  t / 2,n 1
   X  t / 2,n 1
n
n
8
8
50  2.0639
   50  2.0639
25
25
46.69    53.30
Confidence Interval for 
( Unknown) in PHStat



PHStat | confidence interval | estimate for the
mean, sigma unknown
Example in excel spreadsheet
Confidence Interval
Estimate for Proportion

Assumptions





Two categorical outcomes
Population follows binomial distribution
Normal approximation can be used if
np  5 and n 1  p   5
Confidence interval estimate
pS  Z / 2
pS 1  pS 
 p  pS  Z / 2
n
pS 1  pS 
n
Example
A random sample of 400 Voters showed 32
preferred Candidate A. Set up a 95% confidence
interval estimate for p.
ps  Z / 
ps 1  ps 
 p  ps  Z / 
n
ps 1  ps 
n
.08 1  .08 
.08 1  .08 
.08  1.96
 p  .08  1.96
400
400
.053  p  .107
Confidence Interval Estimate for
Proportion in PHStat


PHStat | confidence interval | estimate for the
proportion …
Example in excel spreadsheet
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?
Z p 1  p  1.645  0.3 0.7 
n

2
2
Error
0.05
 227.3  228
2
2
Round Up
Determining Sample Size for Proportion in
PHStat


PHStat | sample size | determination for the
proportion …
Example in excel spreadsheet
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
Applications in Auditing

Can provide more accurate conclusions on the
population than the actual investigation of the
population


Samples can be combined and evaluated by
different auditors



Examination of the population can be time
consuming and subject to more nonsampling error
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
Confidence Interval for Population Total
Amount


Point estimate

NX
Confidence interval estimate

NX  N  t / 2,n1 
S
n
 N  n
 N  1
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.
Example Solution
N  1000
n  50
NX  N  t / 2,n 1 
S
n
X  $1076.39
S  $273.62
 N  n
 N  1
273.62 1000  50
 1000 1076.39   1000  2.0096 
1000  1
100
 1, 076,390  75,830.85
The 95% confidence interval for the population total
amount of the vouchers is between 1,000,559.15, and
1,152,220.85
Example Solution in PHStat


PHStat | confidence intervals | estimate for
the population total
Excel spreadsheet for the voucher example
Confidence Interval for Total Difference in
the Population

Point estimate


n
Where D 
difference
ND
D
i 1
i
is the sample average
n
Confidence interval estimate

SD
ND  N  t / 2,n1 
n
n

Where
SD 
 N  n
 N  1
 D  D
i 1
i
n 1
2
Estimation for Finite Population

Samples are selected without replacement

Confidence interval for the mean (  unknown)


X  t / 2,n1
S
n
 N  n
 N  1
Confidence interval for proportion

pS  Z / 2
pS 1  pS 
n
 N  n
 N  1
Sample Size Determination
for Finite Population

Samples are selected without replacement

When estimating the mean
Z / 2
n0 
2
e
2


2
When estimating the proportion
Z / 2 p 1  p 
n0 
2
e
2

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
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
Chapter Summary

Addressed applications of confidence interval
estimation in auditing




(continued)
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