Download Confidence Interval Estimation

Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 7
Confidence Interval Estimation
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-1
Chapter Goals
After completing this chapter, you should be able to:
 Distinguish between a point estimate and an interval
estimate
 Construct and interpret a confidence interval estimate for a
population mean using the t distribution
 Form and interpret a confidence interval estimate for a
population proportion using the Z distribution
 Determine the required sample size to estimate a mean or
proportion within a specified margin of error
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-2
Point and Interval Estimates
 A point estimate is a single number,
 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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-3
Point Estimates
We can estimate a
Population Parameter …
with a Sample
Statistic
(a Point Estimate)
Mean
μ
X
Proportion
p
ps
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-4
Confidence Interval Estimate
 An interval gives a range of values:
 Takes into consideration the variation in
sample statistics from sample to sample
 Based on observation from 1 sample
 Gives information about closeness to
unknown population parameters
 Stated in terms of level of confidence
 Can never be 100% confident
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-5
Confidence Level, (1-)




Suppose confidence level = 95%
Also written (1 - ) = .95
Where  is the risk of being wrong
A relative frequency interpretation:
 In the long run, 95% of all the confidence
intervals that can be constructed will contain the
unknown parameter
 A specific interval either will contain or will
not contain the true parameter
 No probability involved in a specific interval
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-6
Estimation Process
Random Sample
Population
(mean, μ, is
unknown)
Mean
X = 50
I am 95%
confident that
μ is between
40 & 60.
Sample
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-7
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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-8
Intervals and Level of Confidence
Sampling Distribution of the Mean
/2
1 
/2
x
μx  μ
x1
Confidence
x2
Confidence Intervals
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-9
Confidence Interval for μ
(σ Unknown)
 If the population standard deviation σ is
unknown, we can substitute the sample
standard deviation, s as an estimate
 This introduces extra uncertainty, since s is
different from sample to sample
 In these circumstances the t distribution is used
instead of the normal distribution
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-10
Student’s t Distribution
Note: t
Normal as n increases
Standard
Normal
(t with df > 30)
t (df = 13)
t-distributions are bellshaped and symmetric, but
have ‘fatter’ tails than the
normal
t (df = 5)
0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
t
Chap 7-11
Confidence Interval for μ
(σ Unknown)
(continued)
 Assumptions
 Population standard deviation is unknown
 Population is not highly skewed
 Population is normally distributed or the sample
size is large (>30)
 Use Student’s t Distribution
 Confidence Interval Estimate:
X  t n-1
(where t is the critical value of the t distribution with n-1 d.f. and an
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
S
n
Chap 7-12
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 t /2 , n1  t.025,24  2.0639
The confidence interval is
X  t /2, n-1
S
8
 50  (2.0639)
n
25
46.698 …..  ….. 53.302
46.698    53.302
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-13
Example
d.f. = n – 1 = 24, so t /2 , n1  t.025,24  2.0639
To get a t value use the TINV function. The value of
alpha (1-confidence)/2 and n-1 degrees of freedom
are the inputs needed. For 95% confidence use
.025 and for a sample size of 25 use 24 df
Result 2.0639
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-14
Confidence Intervals
Confidence
Intervals
Population
Mean
σ Known
Population
Proportion
σ Unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-15
Confidence Intervals for the
Population Proportion, p
(continued)
 Recall that the distribution of the sample
proportion is approximately normal if the
sample size is large, with standard deviation
p(1  p)
σp 
n
 We will estimate this with sample data:
ps(1  ps)
Sps 
n
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-16
Confidence Interval Endpoints
 Upper and lower confidence limits for the
population proportion are calculated with the
formula
ps(1  ps)
ps  Z
n
 To get a Z value use the NORMSINV function with
alpha/2 for 95% confidence use .025
 Result 1.96
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-17
Example
 A random sample of 100 people shows
that 25 are left-handed.
 Form a 95% confidence interval for the
true proportion of left-handers
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-18
Example
(continued)
 A random sample of 100 people shows
that 25 are left-handed. Form a 95%
confidence interval for the true proportion
of left-handers.
1. ps  25/100  .25
2. Sps  ps(1  ps )/n  .25(.75)/1 00  .0433
3.
.25  1.96 (.0433)
0.1651  p  0.3349
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-19
Interpretation
 We are 95% confident that the true
percentage of left-handers in the population
is between
16.51% and 33.49%.
 Although this range 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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-20
Determining Sample Size
Determining
Sample Size
For the
Mean
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
For the
Proportion
Chap 7-21
Determining Sample Size
Determining
Sample Size
For the
Mean
σ
XZ
n
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Sampling error
(margin of error)
σ
eZ
n
Chap 7-22
Determining Sample Size
(continued)
Determining
Sample Size
For the
Mean
σ
eZ
n
Z σ
n
2
e
2
Now solve
for n to get
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
2
Chap 7-23
If σ is unknown
 If σ is unknown it can be estimated
from experience or
 Select a pilot sample and estimate σ with
the sample standard deviation, s
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-24
Determining Sample Size
Determining
Sample Size
For the
Proportion
ps(1  ps)
ps  Z
n
p(1  p)
eZ
n
Sampling error
(margin of error)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-25
Determining Sample Size
(continued)
Determining
Sample Size
For the
Proportion
p(1  p)
eZ
n
Now solve
for n to get
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Z 2 p (1  p)
n
2
e
Chap 7-26
PHStat Interval Options
options
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-27
PHStat Sample Size Options
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-28
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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-29
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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-30
Applications in Auditing
 Advantages of statistical sampling in auditing
 Sample result is objective and defensible
 Sample size estimation is done in advance on an
objective basis
 Provides an estimate of the sampling error
 Can provide more accurate conclusions than a
census of the population
 Samples can be combined and evaluated by different
auditors
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-31
Confidence Interval for
Population Total Amount
 Point estimate:
Population total  NX
 Confidence interval estimate:
S
NX  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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-32
Confidence Interval for
Population Total: Example
An 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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-33
Confidence Interval for
Total Difference
 Point estimate:
Total Difference  ND
 Where the average difference, D, is:
n
D
D
i1
i
n
where Di  audited value - original value
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-34
Confidence Interval for
Total Difference
(continued)
 Confidence interval estimate:
SD
ND  N( t n1 )
n
where
Nn
N 1
n
SD 
2
(
D

D
)
 i
i1
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
n 1
Chap 7-35
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, 4e © 2004 Prentice-Hall, Inc.
Chap 7-36
Chapter Summary




Introduced the concept of confidence intervals
Discussed point estimates
Developed confidence interval estimates
Determined confidence interval estimates for the
mean (σ unknown)
 Created confidence interval estimates for the
proportion
 Determined required sample size for mean and
proportion estimation samples
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-37
Chapter Summary
(continued)
 Developed applications of confidence interval
estimation in auditing
 Confidence interval estimation for population total
 Confidence interval estimation for total difference
in the population
 Addressed confidence interval estimation and ethical
issues
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 7-38