© 2002 Prentice-Hall, Inc. Chap 6-1

Statistics for Managers using Microsoft Excel

3rd Edition

Chapter 6Confidence Interval

Estimation

© 2002 Prentice-Hall, Inc. Chap 6-2

Chapter Topics Estimation process Point estimates Interval estimates Confidence interval estimation for the

mean ( known) Determining sample size Confidence interval estimation for the

mean ( unknown) Confidence interval estimation for the

proportion

© 2002 Prentice-Hall, Inc. Chap 6-3

Chapter Topics

Applications of confidence interval estimation in auditing Confidence interval estimation for population

total Confidence interval estimation for total

difference in the population Estimation and sample size

determination for finite population Confidence interval estimation and

ethical issues

(continued)

© 2002 Prentice-Hall, Inc. Chap 6-4

Estimation Process

Mean, , is unknown

Population Random Sample

Mean X = 50

Sample

I am 95% confident that is between 40 & 60.

© 2002 Prentice-Hall, Inc. Chap 6-5

Point Estimates

Estimate Population

Parameters …

with SampleStatistics

Mean

Proportion

Variance

Difference

p

2

1 2

X

SP

2S

1 2X X

© 2002 Prentice-Hall, Inc. Chap 6-6

Interval Estimates

Provides range of values Take into consideration variation in

sample statistics from sample to sample

Based on observation from 1 sample Give information about closeness to

unknown population parameters Stated in terms of level of confidence

Never 100% sure

© 2002 Prentice-Hall, Inc. Chap 6-7

Confidence Interval Estimates

Mean

Unknown

ConfidenceIntervals

Proportion

Known

© 2002 Prentice-Hall, Inc. Chap 6-8

Confidence Interval for( Known)

Assumptions Population standard deviation is

known Population is normally distributed If population is not normal, use large

sample Confidence interval estimate

/ 2 / 2X Z X Zn n

© 2002 Prentice-Hall, Inc. Chap 6-9

Elements of Confidence Interval

Estimation

Level of confidence Confidence in which the interval will

contain the unknown population parameter

Precision (range) Closeness to the unknown parameter

Cost Cost required to obtain a sample of size

n

© 2002 Prentice-Hall, Inc. Chap 6-10

Level of Confidence

Denoted by A relative frequency interpretation

In the long run, of all the confidence intervals that can be constructed will contain the unknown parameter

A specific interval will either contain or not contain the parameter No probability involved in a specific interval

100 1 %

100 1 %

© 2002 Prentice-Hall, Inc. Chap 6-11

Interval and Level of Confidence

Confidence Intervals

Intervals extend from

to

of intervals constructed contain ;

do not.

_Sampling Distribution of the Mean

XX Z

X/ 2

/ 2

XX

1

XX Z

100 1 %

100 %

/ 2 XZ / 2 XZ

© 2002 Prentice-Hall, Inc. Chap 6-12

Factors Affecting Interval Width (Precision)

Data variation Measured by

Sample size

Level of confidence

Intervals Extend from

© 1984-1994 T/Maker Co.

X - Z to X + Z xx

Xn

100 1 %

© 2002 Prentice-Hall, Inc. Chap 6-13

Determining Sample Size (Cost)

Too Big:

• Requires too much resources

Too small:

• Won’t do the job

© 2002 Prentice-Hall, Inc. Chap 6-14

Determining Sample Size for Mean

What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45.

Round Up

2 22 2

2 2

1.645 45219.2 220

Error 5

Zn

© 2002 Prentice-Hall, Inc. Chap 6-15

Determining Sample Size for Mean in PHStat

PHStat | sample size | determination for the mean …

Example in excel spreadsheet

Microsoft Excel Worksheet

© 2002 Prentice-Hall, Inc. Chap 6-16

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

Confidence Interval for( Unknown)

/ 2, 1 / 2, 1n n

S SX t X t

n n

© 2002 Prentice-Hall, Inc. Chap 6-17

Student’s t Distribution

Zt

0

t (df = 5)

t (df = 13)Bell-ShapedSymmetric

‘Fatter’ Tails

Standard Normal

© 2002 Prentice-Hall, Inc. Chap 6-18

Degrees of Freedom (df )

Number of observations that are free to vary after sample mean has been calculated

Example Mean of 3 numbers is 2

degrees of freedom = n -1 = 3 -1= 2

1

2

3

1 (or any number)

2 (or any number)

3 (cannot vary)

X

X

X

© 2002 Prentice-Hall, Inc. Chap 6-19

Student’s t Table

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0 2.920t Values

Let: n = 3 df = n - 1 = 2 = .10 /2 =.05

/ 2 = .05

© 2002 Prentice-Hall, Inc. Chap 6-20

Example

A random sample of 25 has 50 and 8.

Set up a 95% confidence interval estimate for

n X S

/ 2, 1 / 2, 1

8 850 2.0639 50 2.0639

25 2546.69 53.30

n n

S SX t X t

n n

© 2002 Prentice-Hall, Inc. Chap 6-21

PHStat | confidence interval | estimate for the mean, sigma unknown

Example in excel spreadsheet

Confidence Interval for( Unknown) in PHStat

Microsoft Excel Worksheet

© 2002 Prentice-Hall, Inc. Chap 6-22

Confidence Interval Estimate for Proportion

Assumptions Two categorical outcomes Population follows binomial distribution Normal approximation can be used if

and Confidence interval estimate

5np 1 5n p

/ 2 / 2

1 1S S S SS S

p p p pp Z p p Z

n n

© 2002 Prentice-Hall, Inc. Chap 6-23

ExampleA random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p.

/ /

1 1

.08 1 .08 .08 1 .08.08 1.96 .08 1.96

400 400.053 .107

s s s ss s

p p p pp Z p p Z

n n

p

p

© 2002 Prentice-Hall, Inc. Chap 6-24

Confidence Interval Estimate for Proportion in PHStat

PHStat | confidence interval | estimate for the proportion …

Example in excel spreadsheet

Microsoft Excel Worksheet

© 2002 Prentice-Hall, Inc. Chap 6-25

Determining Sample Size for Proportion

Out of a population of 1,000, we randomly selected 100 of which 30 were defective. What sample size is needed to be within ± 5% with 90% confidence?

Round Up

2 2

2 2

1 1.645 0.3 0.7

Error 0.05227.3 228

Z p pn

© 2002 Prentice-Hall, Inc. Chap 6-26

Determining Sample Size for Proportion in PHStat

PHStat | sample size | determination for the proportion …

Example in excel spreadsheet

Microsoft Excel Worksheet

© 2002 Prentice-Hall, Inc. Chap 6-27

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

© 2002 Prentice-Hall, Inc. Chap 6-28

Applications in Auditing Can provide more accurate conclusions on the

population than the actual investigation of 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

© 2002 Prentice-Hall, Inc. Chap 6-29

Confidence Interval for Population Total Amount

Point estimate

Confidence interval estimate

NX

/ 2, 1 1n

N nSNX N t

Nn

© 2002 Prentice-Hall, Inc. Chap 6-30

Confidence Interval for Population Total: Example

An auditor is faced with a population of 1000 vouchers and wishes to estimate the total value of the population of vouchers. A sample of 50 vouchers is selected with average voucher amount of $1076.39, standard deviation of $273.62. Set up the 95% confidence interval estimate of the total amount for the population of vouchers.

© 2002 Prentice-Hall, Inc. Chap 6-31

Example Solution

/ 2, 1

1000 50 $1076.39 $273.62

1

273.62 1000 501000 1076.39 1000 2.0096

1000 11001,076,390 75,830.85

n

N n X S

N nSNX N t

Nn

The 95% confidence interval for the population total amount of the vouchers is between 1,000,559.15, and 1,152,220.85

© 2002 Prentice-Hall, Inc. Chap 6-32

Example Solution in PHStat

PHStat | confidence intervals | estimate for the population total

Excel spreadsheet for the voucher example

Microsoft Excel Worksheet

© 2002 Prentice-Hall, Inc. Chap 6-33

Confidence Interval for Total Difference in the Population

Point estimate Where is the sample

average difference

Confidence interval estimate

Where

ND 1

n

ii

DD

n

/ 2, 1 1

Dn

N nSND N t

Nn

2

1

1

n

ii

D

D DS

n

© 2002 Prentice-Hall, Inc. Chap 6-34

Estimation for Finite Population

Samples are selected without replacement Confidence interval for the mean

( unknown)

Confidence interval for proportion

/ 2, 1 1n

N nSX t

Nn

/ 2

1

1S S

S

p p N np Z

n N

© 2002 Prentice-Hall, Inc. Chap 6-35

Sample Size Determination for Finite Population

Samples are selected without replacement When estimating the mean

When estimating the proportion

2 2/ 2

0 2

Zn

e

2/ 2

0 2

1Z p pn

e

© 2002 Prentice-Hall, Inc. Chap 6-36

Ethical Issues

Confidence interval (reflects sampling error) should always be reported along with the 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

© 2002 Prentice-Hall, Inc. Chap 6-37

Chapter Summary Illustrated estimation process Discussed point estimates Addressed interval estimates Discussed confidence interval estimation for

the mean ( known) Addressed determining sample size Discussed confidence interval estimation for

the mean ( unknown) Discussed confidence interval estimation for

the proportion

© 2002 Prentice-Hall, Inc. Chap 6-38

Chapter Summary

Addressed applications of confidence interval estimation in auditing Confidence interval estimation for population

total Confidence interval estimation for total

difference in the population Addressed estimation and sample size

determination for finite population Addressed confidence interval

estimation and ethical issues

(continued)