Transcript point estimate - Sonoma State University

Statistics for Managers
Using Microsoft® Excel
5th Edition
Chapter 8
Confidence Interval Estimation
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-1
Learning Objectives
In this chapter, you will learn:
 To construct and interpret confidence interval
estimates for the mean and the proportion
 How to determine the sample size necessary
to develop a confidence interval for the mean
or proportion
 How to use confidence interval estimates in
auditing
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-2
Chapter Outline
Confidence Intervals for the Population
Mean, μ
 when Population Standard Deviation σ is
Known
 when Population Standard Deviation σ is
Unknown
2. Confidence Intervals for the Population
Proportion, π
3. Determining the Required Sample Size
1.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-3
Point Estimates
 A point estimate is a single number. For the
population mean (and population standard
deviation), a point estimate is the sample mean
(and sample standard deviation).
 A confidence interval provides additional
information about variability
Lower Confidence
Limit
Point Estimate
Upper Confidence
Limit
Width of
confidence interval
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-4
Confidence Interval Estimates
 A confidence interval gives a range estimate of
values:
 Takes into consideration variation in sample
statistics from sample to sample
 Based on all the observations from 1 sample
 Gives information about closeness to unknown
population parameters
 Stated in terms of level of confidence
 Ex. 95% confidence, 99% confidence
 Can never be 100% confident
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-5
Confidence Level
 Confidence Level
 Confidence in which the interval will contain the
unknown population parameter
 A percentage (less than 100%)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-6
Confidence Level
 Suppose the confidence level = 95%
 Also written as (1 - ) = .95
 A relative frequency interpretation:
 In the long run, 95% of all the confidence
intervals that can be constructed will contain
the unknown true parameter
 A specific interval either will contain or will
not contain the true parameter
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-7
Confidence Interval Estimates
 The general formula for all
confidence intervals is:
Point Estimate ± (Critical Value) (Standard Error)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-8
Confidence Intervals
Confidence
Intervals
Population
Mean
Population
Proportion
Normal
Distribution Z
σ Known
Normal
Distribution
σ Unknown
t Distribution
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-9
Confidence Interval for μ
(σ Known) seldom used
Assumptions
 Population standard deviation σ is known
 Population is normally distributed
 If population is not normal, use large sample
Confidence interval estimate:
σ
XZ
n
(where Z is the standardized normal distribution critical value
for a probability of α/2 in each tail)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-10
Finding the Critical Value, Z
Consider a 95% confidence interval:
1   .95
α
 .025
2
Z units:
X units:
α
 .025
2
Z= -1.96
Lower
Confidence
Limit
0
Point Estimate
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Z= 1.96
Upper
Confidence
Limit
Chap 8-11
Intervals and Level of
Confidence
Sampling Distribution
of the Mean
/2
Intervals
extend from
σ
X Z
n
1 
x
μx  μ
x1
x2
to
σ
X Z
n
/2
Confidence Intervals
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
(1-)x100%
of intervals
constructed
contain μ;
()x100% do
not.
Chap 8-12
Confidence Interval for μ
(σ Known) Example
 A sample of 11 circuits from a large normal
population has a mean resistance of 2.20
ohms. We know from past testing that the
population standard deviation is .35 ohms.
 Determine a 95% confidence interval for the
true mean resistance of the population.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-13
Confidence Interval for μ
(σ Known) Example
σ
X Z
n
 2.20  1.96 (.35/ 11)
 2.20  .2068
(1.9932 , 2.4068)
 We are 95% confident that the true mean resistance is between
1.9932 and 2.4068 ohms
 Although the true mean may or may not be in this interval, 95%
of intervals formed in this manner will contain the true mean
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-14
Confidence Interval for μ
(σ Unknown)
 If the population standard deviation σ is
unknown, we can substitute the sample
standard deviation, S
 This introduces extra uncertainty, since S is
variable from sample to sample
 So we use the t distribution instead of the
normal distribution
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-15
Confidence Interval for μ
(σ 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:
S
X  t n -1
n
(where t is the critical value of the t distribution with n-1
d.f. and an area of α/2 in each tail)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-16
Student’s t Distribution
Note: t
Z as n increases
Standard
Normal
(t with df = ∞)
t (df = 13)
t-distributions are bell-shaped
and symmetric, but have
‘fatter’ tails than the normal
t (df = 5)
0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
t
Chap 8-17
Confidence Interval for μ
(σ Unknown) Example
A random sample of n = 25 has X = 50 and S = 8. Form a 95%
confidence interval for μ
 d.f. = n – 1 = 24, so
 The confidence interval is
X  t/2, n -1
S
8
 50  (2.0639)
n
25
(46.698 , 53.302)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-18
Confidence Intervals for the
Population Proportion, π
 An interval estimate for the population
proportion ( π ) can be calculated by adding
an allowance for uncertainty to the sample
proportion ( p )
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-19
Confidence Intervals for the
Population Proportion, π
Recall that the distribution of the sample proportion is
approximately normal if the sample size is large, with
standard deviation
σp 
 (1   )
n
We will estimate this with sample data:
p(1 p)
n
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-20
Confidence Intervals for the
Population Proportion, π
Upper and lower confidence limits for the population proportion
are calculated with the formula
p(1  p)
pZ
n
where
 Z is the standardized normal value for the level of
confidence desired
 p is the sample proportion
 n is the sample size
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-21
Confidence Intervals for the
Population Proportion, Example
A random sample of 100 people shows that 25 have opened
IRA’s this year. Form a 95% confidence interval for the true
proportion of the population who have opened IRA’s.
p  Z p(1  p)/n
 25/100  1.96 .25(.75)/1 00
 .25  1.96 (.0433)
(0.1651 , 0.3349)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-22
Confidence Intervals for the
Population Proportion, Example
 We are 95% confident that the true
percentage of those who have opened IRA’s
in the population is between 16.51% and
33.49%. Although the interval from .1651 to
.3349 may or may not contain the true
proportion, 95% of intervals formed from
samples of size 100 in this manner will
contain the true proportion.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-23
Determining Sample Size
Determining
Sample Size
For the
Mean
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
For the
Proportion
Chap 8-24
Determining Sample Size for
Intervals of the mean
 The required sample size can be found to
reach a desired margin of error (e) with a
specified level of confidence (1 - )
 The margin of error is also called sampling
error
 the amount of imprecision in the estimate of
the population parameter
 the amount added and subtracted to the point
estimate to form the confidence interval
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-25
Determining Sample Size
 known
 Given the desired level of confidence (1 - ), which
determines the critical Z value
 The acceptable sampling error e
 The standard deviation, σ
 And the interval estimate formula
X Z
σ
eZ
n

n

X  e then
Z σ
n
2
e
2
Now solve
for n to get
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
2
Chap 8-26
Determining Sample Size
If  = 45, what sample size is needed to estimate
the mean within ± 5 with 90% confidence?
Z 2 σ 2 (1.645) 2 (45) 2
n

 219.19
2
2
e
5
So the required sample size is n = 220
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-27
Determining Sample Size
 unknown
 If unknown, σ can be estimated when using
the required sample size formula
 Use a value for σ that is expected to be at least
as large as the true σ
 Select a pilot sample and estimate σ with the
sample standard deviation, S
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-28
Determining Sample Size
Proportion
To determine the required sample size for the proportion, you
must know:
 The desired level of confidence (1 - ), which
determines the critical Z value
 The acceptable sampling error (margin of error), e
 The true proportion of “successes”, π
 π can be estimated with a pilot sample, if necessary (or
conservatively use π = .50)
eZ
 (1   )
n
Now solve
for n to get
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Z  (1   )
n
e2
2
Chap 8-29
Determining Sample Size
How large a sample would be necessary to
estimate the true proportion defective in a large
population within ±3%, with 95% confidence?
 (Assume a pilot sample yields p = .12)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-30
Determining Sample Size
Solution:
For 95% confidence, use Z = 1.96
e = .03
p = .12, so use this to estimate π
Z 2  (1   ) (1.96) 2 (.12)(1  .12)
n

 450.74
2
2
e
(.03)
So use n = 451
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-31
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
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-32
Applications in Auditing
 Can provide more accurate conclusions on 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
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-33
Population Total Amount
 Point estimate:
Population total  N X
 Confidence interval estimate:
S
N X  N (t n 1 )
n
N n
N 1
(This is sampling without replacement, so use the finite population correction
in the confidence interval formula)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-34
Population Total Amount
 A firm has a population of 1000 accounts and wishes
to estimate the total population value.
 A sample of 80 accounts is selected with average
balance of $87.6 and standard deviation of $22.3.
 Find the 95% confidence interval estimate of the
total balance.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-35
Population Total Amount
N  1000, n  80, X  87.6, S  22.3
S
N X  N (t n 1 )
n
Nn
N 1
22.3 1000  80
 (1000)(87. 6)  (1000)(1.9 905)
80 1000  1
 87,600  4,762.48
The 95% confidence interval for the population total balance
is $82,837.52 to $92,362.48
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-36
Confidence Interval for
Total Difference
 Point estimate:
Total Difference  ND
 Where the mean difference, D , is:
n
D
D
i 1
i
n
where Di  audited value - original value
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-37
Confidence Interval for
Total Difference
Confidence interval estimate:
SD
N D  N (t n 1 )
n
where
N n
N 1
n
SD 
2
(
D

D
)
 i
i 1
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
n 1
Chap 8-38
One Sided Confidence
Intervals
Application: find the upper bound for the proportion of items that
do not conform with internal controls
p(1  p) N  n
Upper bound  p  Z
n
N 1
where
 Z is the standardized normal value for the level of
confidence desired
 p is the sample proportion of items that do not conform
 n is the sample size
 N is the population size
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-39
PHStat Interval Options
options
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-40
PHStat Sample Size Options
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-41
Using PHStat
(for μ, σ unknown)
A random sample of n = 25 has X = 50 and
S = 8. Form a 95% confidence interval for μ
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-42
Using PHStat
(sample size for proportion)
How large a sample would be necessary to estimate the true
proportion defective in a large population within 3%, with 95%
confidence?
(Assume a pilot sample yields ps = .12)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-43
Ethical Issues
 A confidence interval (reflecting sampling
error) should always be reported along with a
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
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-44
Chapter Summary
In this chapter, we have
 Introduced the concept of confidence intervals
 Discussed point estimates
 Developed confidence interval estimates
 Created confidence interval estimates for the mean
(σ known)
 Determined confidence interval estimates for the
mean (σ unknown)
 Created confidence interval estimates for the
proportion
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-45
Chapter Summary
In this chapter, we have
 Determined required sample size for mean and
proportion settings
 Developed applications of confidence interval
estimation in auditing
 Confidence interval estimation for population total
 Confidence interval estimation for total difference in
the population
 One sided confidence intervals
 Addressed confidence interval estimation and ethical
issues
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 8-46