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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-1
Chapter 7
Confidence Interval Estimation
Statistics for ManagersUsing Microsoft® Excel
4th Edition
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-2
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 7-3
Point and Interval Estimates
A point estimate is a single number, a confidence interval provides additional
information about variability
Point Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Width of confidence interval
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-4
We can estimate a Population Parameter …
Point Estimates
with a SampleStatistic
(a Point Estimate)
Mean
Proportion psp
Xμ
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-5
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 7-6
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 7-7
Estimation Process
(mean, μ, is unknown)
Population
Random Sample
Mean X = 50
Sample
I am 95% confident that μ is between 40 & 60.
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-8
Confidence Intervals
Population Mean
σ Unknown t Distribution
ConfidenceIntervals
PopulationProportion
NormalDistribution Z
σ Known Normal Distribution
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-9
μμx
Intervals and Level of Confidence
Confidence Intervals
Sampling Distribution of the Mean
x
x1
x2
/2 /21
Confidence
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-10
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
Confidence Interval for μ(σ Unknown)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-11
Student’s t Distribution
t0
t (df = 5)
t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
Standard Normal
(t with df > 30)
Note: t Normal as n increases
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-12
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:
(where t is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail)
Confidence Interval for μ(σ Unknown)
n
StX 1-n
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-13
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
2.0639tt .025,241n,/2
25
8(2.0639)50
n
StX 1-n /2,
46.698 ….. ….. 53.302 46.698 53.302
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-14
Example
d.f. = n – 1 = 24, so 2.0639tt .025,241n,/2
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 7-15
Confidence Intervals
Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-16
Confidence Intervals for the Population Proportion, p
Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation
We will estimate this with sample data:
(continued)
n
)p(1pS
ssps
n
p)p(1σp
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-17
Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
To get a Z value use the NORMSINV function with alpha/2
for 95% confidence use .025
Result 1.96
n
)p1(pZp
sss
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-18
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 7-19
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.
1.
2.
3.
.0433 00.25(.75)/1)/np(1pS
.2525/100
ss
sp
sp
0.3349 p 0.1651
(.0433) 1.96 .25
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-20
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 7-21
Determining Sample Size
For the Mean
DeterminingSample Size
For theProportion
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-22
Determining Sample Size
For the Mean
DeterminingSample Size
n
σZX
n
σZe
Sampling error (margin of error)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-23
Determining Sample Size
For the Mean
DeterminingSample Size
n
σZe
(continued)
2
22
e
σZn Now solve
for n to get
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-24
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 7-25
Determining Sample Size
n
)p1(pZp
sss
n
)p1(pZe
DeterminingSample Size
For theProportion
Sampling error (margin of error)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-26
Determining Sample Size
DeterminingSample Size
For theProportion
2
2
e
)p1(pZn
Now solve
for n to getn
)p1(pZe
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-27
PHStat Interval Options
options
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-28
PHStat Sample Size Options
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-29
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 7-30
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 7-31
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 7-32
Confidence Interval for Population Total Amount
Point estimate:
Confidence interval estimate:
XN total Population
1N
nN
n
S)t(NXN 1n
(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 7-33
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 7-34
Point estimate:
Where the average difference, D, is:
DN Difference Total
Confidence Interval for Total Difference
value original - value auditedD where
n
DD
i
n
1ii
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-35
Confidence interval estimate:
where
1N
nN
n
S)t(NDN D
1n
Confidence Interval for Total Difference
(continued)
1n
)DD(S
n
1i
2i
D
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-36
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 7-37
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 7-38
Chapter Summary
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
(continued)