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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 Managers Using Microsoft ® Excel 4 th Edition

Sampling Excel

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Page 1: Sampling Excel

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

Page 2: Sampling Excel

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

Page 3: Sampling Excel

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

Page 4: Sampling Excel

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

Page 5: Sampling Excel

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

Page 6: Sampling Excel

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

Page 7: Sampling Excel

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.

Page 8: Sampling Excel

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

Page 9: Sampling Excel

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

Page 10: Sampling Excel

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)

Page 11: Sampling Excel

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

Page 12: Sampling Excel

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)

Page 13: Sampling Excel

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

Page 14: Sampling Excel

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

Page 15: Sampling Excel

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-15

Confidence Intervals

Population Mean

σ Unknown

ConfidenceIntervals

PopulationProportion

σ Known

Page 16: Sampling Excel

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

Page 17: Sampling Excel

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

Page 18: Sampling Excel

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

Page 19: Sampling Excel

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)

Page 20: Sampling Excel

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.

Page 21: Sampling Excel

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 7-21

Determining Sample Size

For the Mean

DeterminingSample Size

For theProportion

Page 22: Sampling Excel

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)

Page 23: Sampling Excel

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

Page 24: Sampling Excel

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

Page 25: Sampling Excel

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)

Page 26: Sampling Excel

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)

Page 27: Sampling Excel

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

PHStat Interval Options

options

Page 28: Sampling Excel

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

PHStat Sample Size Options

Page 29: Sampling Excel

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

Page 30: Sampling Excel

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)

Page 31: Sampling Excel

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

Page 32: Sampling Excel

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)

Page 33: Sampling Excel

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.

Page 34: Sampling Excel

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

Page 35: Sampling Excel

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

Page 36: Sampling Excel

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

Page 37: Sampling Excel

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

Page 38: Sampling Excel

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)