This document chapter discusses confidence intervals, which provide a range of values that is likely to include an unknown population parameter. It explains how to construct confidence intervals for a population mean using the t-distribution when the population standard deviation is unknown. An example is provided to demonstrate calculating a 95% confidence interval for a population mean based on a sample size of 25. The chapter goals are to explain point and interval estimates and how to determine sample sizes needed to estimate parameters within a specified margin of error.

1. 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

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 7-2

3. Point and Interval Estimates
 A point estimate is a single number,
 a confidence interval provides additional
information about variability
Lower Upper
Confidence Confidence
Point Estimate Limit
Limit
Width of
confidence interval
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-3

4. Point Estimates
We can estimate a with a Sample
Population Parameter … Statistic
(a Point Estimate)
Mean μ X
Proportion p ps
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-4

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 7-5

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 7-6

7. Estimation Process
Random Sample I am 95%
confident that
μ is between
Population Mean 40 & 60.
(mean, μ, is X = 50
unknown)
Sample
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-7

8. Confidence Intervals
Confidence
Intervals
Population Population
Mean Proportion
Normal
Distribution Z
σ Known σ Unknown
Normal t Distribution
Distribution
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-8

9. Intervals and Level of Confidence
Sampling Distribution of the Mean
/2 1 /2
x
μx μ
Confidence x1
x2
Confidence Intervals
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-9

10. 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

11. Student’s t Distribution
Note: t Normal as n increases
Standard
Normal
(t with df > 30)
t (df = 13)
t-distributions are bell-
shaped and symmetric, but
have ‘fatter’ tails than the t (df = 5)
normal
0 t
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-11

12. 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: S
X t n-1
n
(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. Chap 7-12

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 t /2 , n 1 t.025,24 2.0639
The confidence interval is
S 8
X t /2, n-1 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

14. 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

15. Confidence Intervals
Confidence
Intervals
Population Population
Mean Proportion
σ Known σ Unknown
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-15

16. 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 )
S ps
n
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-16

17. 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

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 7-18

19. 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
Sps ps(1 ps )/n .25(.75)/1
00 .0433
2.
3. .25 1.96 (.0433)
0.1651 p 0.3349
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-19

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 7-20

21. Determining Sample Size
Determining
Sample Size
For the For the
Mean Proportion
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-21

22. Determining Sample Size
Determining
Sample Size
For the
Mean Sampling error
(margin of error)
σ σ
X Z e Z
n n
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-22

23. Determining Sample Size
(continued)
Determining
Sample Size
For the
Mean
σ Now solve Z σ
2 2
e Z for n to get n 2
n e
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-23

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 7-24

25. Determining Sample Size
Determining
Sample Size
For the
Proportion
ps(1 ps ) p(1 p )
ps Z e Z
n n
Sampling error
(margin of error)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-25

26. Determining Sample Size
(continued)
Determining
Sample Size
For the
Proportion
p(1 p ) Now solve Z 2 p (1 p)
e Z for n to get n 2
n e
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-26

27. PHStat Interval Options
options
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-27

28. PHStat Sample Size Options
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-28

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 7-29

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 7-30

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 7-31

32. Confidence Interval for
Population Total Amount
 Point estimate:
Population total NX
 Confidence interval estimate:
S N n
NX N( t n 1 )
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

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 7-33

34. Confidence Interval for
Total Difference
 Point estimate:
Total Difference ND
 Where the average difference, D, is:
n
Di
i 1
D
n
where Di audited value - original value
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-34

35. Confidence Interval for
Total Difference
(continued)
 Confidence interval estimate:
SD N n
ND N( t n 1 )
n N 1
where n
(Di D)2
i 1
SD
n 1
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-35

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 7-36

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 7-37

38. 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