1. Statistics for Managers Using Microsoft® Excel 7th Edition
Chapter 8
Confidence Interval Estimation
Statistics for Managers Using Microsoft Excel, 7e © 2014 Pearson Prentice-Hall, Inc Philip A. Vaccaro , PhD

2. Learning Objectives In this chapter, you will learn:
To construct and interpret confidence interval estimates for the population mean and the population proportion ( μ and π ) .
How to determine the sample size necessary to develop a confidence interval for the population mean or population proportion
How to use confidence interval estimates in auditing (optional ) .

3. Confidence Interval Estimate
I am 95% confident that the true average
income of U.S. appliance factory workers
is between $51, and $56,000.00
per year.
I am 90% confident that the true proportion
of potential voters supporting candidate “B”
is between 35% and 38%.
EXAMPLES

4. Chapter Outline Confidence Intervals for the Population Mean, μ
when Population Standard Deviation σ is known
when Population Standard Deviation σ is unknown
Confidence Intervals for the Population Proportion, π
Determining the Required Sample Size, n

5. developing the confidence
Point Estimates
The starting point for
developing the confidence
interval for μ or π
The value of a
single sample
statistic: mean
or proportion
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
A range of
numbers
constructed
around the
point estimate
Lower Confidence
Limit
Upper Confidence
Limit
Point Estimate
Width of
confidence interval

6. 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 one sample
Gives information about closeness to unknown population parameters
Stated in terms of level of confidence
Example: 95% confidence, 99% confidence
Can never be 100% confident * *
Recall that the
sample mean
will vary from
sample to
sample

7. Confidence Interval Estimates
The general formula for all confidence intervals is:
σ / √ n or s / √ n
Sample Mean or
Sample Proportion
The “z” or “t”
Critical Value
Point Estimate ± (Critical Value) (Standard Error)

8. Confidence Level Confidence Level
A confidence interval estimate
of 100% would be so wide as to be meaningless
for practical decision making
Confidence Level
Confidence in which the interval will contain the unknown population parameter ( μ or π )
A percentage ( less than 100% )

9. Confidence Level Suppose confidence level = 95%
Also written (1 - ) = .95
A relative frequency interpretation:
In the long run, 95% of all the confidence intervals that could be constructed will contain the unknown true parameter ( μ , π )
A specific interval either will contain or will not contain the true parameter

10. The Level of Significance (α)
Because we only select one sample, and μ or π
are unknown, we never really know whether the
confidence interval includes the true population
mean or proportion, or not.
The level of significance, or “α” risk is the chance
we take that the true population parameter is not
contained in the confidence interval.
Therefore, a 95% confidence interval would have
an “α” of 5%

11. The Level of Significance (α)
If α = .05, then
each tail has
.025 area
The critical values of “z” that
define the “α” areas are
and
95% Confidence Interval z z
a = .025
a = .025
.0250
Point Estimate
.9750 Z
.06
- 1.9
.0250 Z
.06
+ 1.9
.9750
“ α “ is the proportion in the tails of the sampling distribution
that is outside the established confidence interval.

12. distribution corresponds
The ‘z’ Table
.0250 of the area under the
standardized normal
distribution corresponds
to z .

13. distribution corresponds
The ‘z’ Table
.9750 of the area under the
standardized normal
distribution corresponds
to z .

14. Confidence Interval for μ ( when σ is Known )
Assumptions
Population standard deviation σ is known
Population is normally distributed
If population is not normal, use large sample ( n > 30 )
Confidence interval estimate:
(where Z is the standardized normal distribution critical value for a probability of α/2 in each tail)

15. Finding the Critical Value, Z
Consider a 95% confidence interval:
The value of
‘z’ is needed
for constructing
the confidence
interval z z
Z= -1.96
Z= 1.96
Z units:
Lower
Confidence
Limit
Upper
Confidence
Limit
X units:
Point Estimate

16. Finding the Critical Value, Z
Consider a 99% confidence interval:
The value of
‘z’ is needed
for constructing
the confidence
interval
1 - α = .99 Z
.08
+ 2.5
.9951 Z
.08
- 2.5
.0049
.005
.005 z z
Z units: Z= Z=
X units:
Lower
Confidence
Limit
Upper
Confidence
Limit
Point Estimate

17. Finding the Critical Value, Z
Confidence Level
Confidence
Coefficient
‘z’
Value
‘a’ / 2 Value
80%
.80
1.28
.1000
90%
.90
1.645
.0500
95%
.95
1.96
.0250
98%
.98
2.33
.0100
99%
.99
2.58
.0050
99.8%
.998
3.08
.0010
99.9%
.999
3.27
.0005
Commonly
used
confidence
levels
are
90%,
95%,
and
99%

18. Intervals and Level of Confidence
Means vary from
sample to sample
Accordingly,
confidence
intervals of
the same
percentage
will have
different lower
and upper limits
The true
population mean
may not be in
them at all !
Sampling Distribution of the Mean x
Intervals extend from to x1
( 1- ) x 100% of intervals constructed contain μ;
(  ) x 100% do not. x2
Confidence Intervals

19. 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.

20. Confidence Interval for μ ( σ Known ) Example
Detailed Computations
/ (.35 / )
/ (.10553) / = =
We are 95% confident that the true mean resistance is between and 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

21. 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
“S” is variable from sample
to sample, and even more
so, in very small samples

22. Confidence Interval for μ ( σ Unknown )
Actually, we rarely know
the true population
standard deviation.
We must instead
develop a confidence
interval using the
sample standard
deviation
Assumptions
Population standard deviation is unknown
Population is normally distributed
If population is not normal, use large sample ( n > 30 )
Use Student’s ‘t’ Distribution
The Confidence Interval Estimate:
(where t is the critical value of the t distribution with ‘n-1’ degrees of freedom and an area of α/2 in each tail)
Sample
Standard
Deviation
Degrees of
Freedom

23. Student ‘ t ‘ Distribution
it is bell-shaped
has fatter tails
has flatter center ( peak or “kurtosis” )
the sampling distributions of small
samples follow the ‘t ’ distribution
the smaller the sample size taken,
the more variable the ‘t ’ distribution

24. Student’s ‘ t ‘ Distribution
d.f. = n - 1
Sample size
The ‘ t ’ value depends on degrees of freedom ( d.f. )
The number of observations in the sample that are free
to vary after the sample mean has been calculated.
The ‘ t ’ distribution changes as the number of degrees
of freedom changes.
small samples have greater variability and the ‘ t ’ reflects
that variability when we are finding areas under the sam-
pling distribution curve.

25. Student’s t Distribution
Peaks
rise as
d..f.
increase
df = 1
df = 2
df = 5
df =10
df = ∞
William S. Gosset
of the
Guiness Brewery
who published
under the pen
name “student”
(1913)
Tails
flatten as
d.f.
increase

26. Degrees of Freedom
Concept: The number of observations that are free to vary after the sample mean has been calculated
Example: Suppose the mean of 3 numbers is 8.0
Let X1 = 7
Let X2 = 8
What is X3?
If the mean of these three values is 8.0,
then X3 must be 9
(i.e., X3 is not free to vary)
2 values can be any numbers, that is, free to vary, but the third is not free to vary for a given mean

27. Student’s t Distribution
Note: ‘t’ approaches ‘Z’ as ‘n’ increases
As n and d.f. increase, ‘s’ becomes a better
estimate of the population standard deviation,
and the sampling distribution eventually
becomes the standardized normal distribution
when n = 120 !
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) t

28. The body of the table contains t values, not probabilities !
Student’s t Table
It’s set up for the
upper tail area only !
Upper Tail Area
Let: n = df = n - 1 =  = /2 =.05
For
90%
Confidence
Interval df
.25
.10
.05 1
1.000
3.078
6.314 2
0.817
1.886
2.920
/2 = .05 3
0.765
1.638
2.353
90% Confidence
The body of the table contains t values, not probabilities ! t
2.920
( INFERRED )

29. 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
Detailed
Calculations
50 + / - (2.0639)( 8/5)
50 + / - (2.0639)(1.6)
50 + / = =
( , )
d.f.
α = .025 24
2.0639

30. Confidence Intervals for the Population Proportion, π
An confidence interval estimate for the population proportion ( π ) can be calculated. p p p
‘p’ is the sample proportion
Recall, that if np => 5 and if n ( 1 - p ) => 5, that is, if the sample
size is large enough, we can approximate the proportion’s
binomial distribution with a normally distributed
sampling distribution.

31. Confidence Intervals for the Population Proportion, π
Again, 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:
Estimated σp
( ‘p’ is the sample proportion )

32. Confidence Intervals for the Population Proportion, π
Upper and lower confidence limits for the population proportion are calculated with the formula:
where
Z is the standardized normal value for the level of confidence desired
p is the sample proportion
n is the sample size
Alternately,
p +/- Z ( estimated σp )

33. 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.
Detailed Computations
.25 +/ √ / 100
.25 +/ √
.25 +/ (.0433)
.25 +/
= .3349
= .1651
alternately, =< π =< .3349

34. Confidence Intervals for the Population Proportion: Example
We are 95% confident that the true percentage of IRA openers in the population is between 16.51% and 33.49%.
Although the interval from to 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.

35. Determining Sample Size
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 or tolerated or allowable 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
We can
determine
sample size
for the
mean
and the
proportion

36. Determining Sample Size
To determine the required sample size for the mean, you must know:
The desired level of confidence (1 - ), which determines the critical ‘Z’ value
The acceptable sampling error (margin of error), ‘e’
The standard deviation, ‘σ’
Now solve for n to get

37. Determining Sample Size
If  = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?
interpolated Z value
So the required sample size is n = 220
We always round up to the meet or exceed the requirements for the confidence interval
and the margin of error.

38. Determining Sample Size
( ) ( 2025 )
n = 25
n = = 25
Detailed Computations

39. There is no ‘z’ value for ‘.9500’ We need to interpolate:
The ‘z’ Table
There is no ‘z’ value for ‘.9500’
in the table.
We need to interpolate: Z
.04
1.6
.9495 Z
.05
1.6
.9505
Z becomes 1.645

40. Finding the Critical Value, Z
Consider a 90% confidence interval:
The value of
‘z’ is needed
for constructing
the confidence
interval
1 - α = .90 Z
.05
+ 1.6
.9505 Z
.04
+ 1.6
.9495
.05
.05 z z Z= Z=
Z units:
Lower
Confidence
Limit
Upper
Confidence
Limit
X units:
Point Estimate

41. Determining Sample Size
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
Also consider Rg ( data range ) divided by ‘6’

42. Determining Sample Size for the Proportion
To determine the required sample size for the proportion, you must know 3 factors:
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)
Now solve for n to get

43. Determining Sample Size for the Proportion
Ironically, π is actually the population parameter that we are
trying to estimate !
1. Consider using an “educated” guess for π .
2. Consider using π = 0.50 which will generate the
maximum sample size needed.
3. Be aware the the maximum sample size results
in the highest sampling costs.
4. That said, there is then increased precision in
estimating π, because the confidence interval
is narrowed !

44. 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)

45. Determining Sample Size
Solution:
For 95% confidence, use Z = 1.96
e = .03
p = .12, so use this to estimate π Z
.06
+ 1.9
.9750 Z
.06
- 1.9
.0250
So use n = 451
Detailed Computations
( ) ( .12 ) ( .88 ) / = / = ≈ 451

46. Determining Sample Size
Solution:
For 95% confidence, use Z = 1.96
e = .03
p = .50, if using this to estimate π (max ‘n’) Z
.06
+ 1.9
.9750 Z
.06
- 1.9
.0250
(.50)( )
1,067
Detailed Computations
( ) ( .50 ) ( .50 ) / = (3.8416) (.25) / = / = 1, ≈ 1,067
So use n = 1,067

47. 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

48. 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

49. Population Total Amount
Point estimate:
Confidence interval estimate:
This is sampling without
replacement, so use the
finite population
correction in the
confidence interval
formula

50. 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.

51. Population Total Amount
The 95% confidence interval for the population total balance is $82, to $92,362.48

52. Confidence Interval for Total Difference
Point estimate:
Where the mean difference is:

53. Confidence Interval for Total Difference
Confidence interval estimate:
where:

54. One Sided Confidence Intervals
Application: find the upper bound for the proportion of items that do not conform with internal controls
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

55. 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

56. 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

57. 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