© 2004 Prentice-Hall, Inc. Chap 8-1

Basic Business Statistics (9th Edition)

Chapter 8Confidence Interval

Estimation

© 2004 Prentice-Hall, Inc. Chap 8-2

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)

© 2004 Prentice-Hall, Inc. Chap 8-3

Chapter Topics Confidence Interval Estimation and

Sample Size Determination for the Proportion

Confidence Interval Estimation for Population Total

Confidence Interval Estimation for Total Difference in the Population

Estimation and Sample Size Determination for Finite Population (CD-ROM Topic)

Confidence Interval Estimation and Ethical Issues

(continued)

© 2004 Prentice-Hall, Inc. Chap 8-4

Estimation Process

Mean, , is unknown

Population Random Sample I am 95%

confident that is between 40 &

60.

Mean X = 50

Sample

© 2004 Prentice-Hall, Inc. Chap 8-5

Point Estimates

Estimate Population

Parameters …

with SampleStatistics

Mean

Proportion

Variance

Difference

p

2

1 2

X

SP

2S

1 2X X

© 2004 Prentice-Hall, Inc. Chap 8-6

Interval Estimates

Provide 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

© 2004 Prentice-Hall, Inc. Chap 8-7

Confidence Interval Estimates

Mean

Unknown

ConfidenceIntervals

Proportion

Known

© 2004 Prentice-Hall, Inc. Chap 8-8

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

is called the sampling error or margin of error

/ 2 / 2X Z X Zn n

Standard ErrorCritical Value

/ 2e Zn

© 2004 Prentice-Hall, Inc. Chap 8-9

Elements of Confidence Interval Estimation

Level of Confidence Confidence that 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

© 2004 Prentice-Hall, Inc. Chap 8-10

Level of Confidence

Denoted by A Relative Frequency Interpretation

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

A Specific Interval Will Either Contain or Not Contain the Parameter

100 1 %

100 1 %

© 2004 Prentice-Hall, Inc. Chap 8-11

Interval and Level of Confidence

Confidence Intervals

Intervals extend from

to

of intervals constructed contain ;

do not.

_Sampling Distribution of the Mean

XX Z

/ 2/ 2

XX

1

XX Z

1 100%

100 %

/ 2 XZ / 2 XZ

© 2004 Prentice-Hall, Inc. Chap 8-12

Example

Confidence Interval Estimate for the Mean

Population Standard Deviation 195000Sample Mean 215000Sample Size 15Confidence Level 99%Standard Error of the Mean 50348.7835Z Value -2.57583451Interval Half Width 129690.1343Interval Lower Limit 85309.86569Interval Upper Limit 344690.1343

PHStat output

85309.9 344690.1 The 99% CI for the population mean:

A random sample of 15 stocks traded on the NASDAQ market showed an average of 21500 shares traded. From past experience, it is believed that the population standard deviation of shares traded is 195000 and the shares traded are very closed to a normal distribution. Construct a 99% confidence interval for the average shares traded on the NASDAQ market. Interpret your result.

© 2004 Prentice-Hall, Inc. Chap 8-13

Example: Interpretation(continued)

We are 99% confident that the population average number of shares traded on the NASDAQ is between 85309.9 and 344690.1.

If all possible samples of size 15 are taken and the corresponding 99% confidence intervals are constructed, 99% of the confidence intervals that are constructed will contain the true unknown population mean.

Using the confidence interval method on repeated sampling, the probability that we will have constructed a confidence interval that will contain the unknown population mean is 99%.

© 2004 Prentice-Hall, Inc. Chap 8-14

Obtaining Confidence Interval

in PHStat

PHStat | Confidence Interval | Estimates for the Mean, Sigma Known

© 2004 Prentice-Hall, Inc. Chap 8-15

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 %

© 2004 Prentice-Hall, Inc. Chap 8-16

Determining Sample Size (Cost)

Too Big:

• Requires more resources

Too small:

• Won’t do the job

© 2004 Prentice-Hall, Inc. Chap 8-17

Determining Sample Sizefor 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

© 2004 Prentice-Hall, Inc. Chap 8-18

Determining Sample Size for Mean in PHStat

PHStat | Sample Size | Determination for the Mean …

Example in Excel Spreadsheet

Microsoft Excel Worksheet

Sample Size Determination

DataPopulation Standard Deviation 45Sampling Error 5Confidence Level 90%

Z Value -1.644853Calculated Sample Size 219.1488528

Sample Size Needed 220Result

Intemediate Calculations

© 2004 Prentice-Hall, Inc. Chap 8-19

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

Margin of ErrorStandard Error

© 2004 Prentice-Hall, Inc. Chap 8-20

Student’s t Distribution

Zt

0

t (df = 5)

t (df = 13)Bell-ShapedSymmetric

‘Fatter’ Tails

Standardized Normal

© 2004 Prentice-Hall, Inc. Chap 8-21

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

© 2004 Prentice-Hall, Inc. Chap 8-22

Example

/ 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

A random sample of 25 has 50 and 8.

Set up a 95% confidence interval estimate for

n X S

We are 95% confident that the unknown true population

mean is somewhere between 46.69 and 53.30.

.

© 2004 Prentice-Hall, Inc. Chap 8-23

PHStat | Confidence Interval | Estimate for the Mean, Sigma Unknown

Example in Excel Spreadsheet

Confidence Interval for( Unknown) in PHStat

Microsoft Excel Worksheet

Confidence Interval Estimate for the Mean

DataSample Standard Deviation 8Sample Mean 50Sample Size 25Confidence Level 95%

Standard Error of the Mean 1.6Degrees of Freedom 24t Value 2.063898137Interval Half Width 3.302237019

Interval Lower Limit 46.70Interval Upper Limit 53.30

Intermediate Calculations

Confidence Interval

© 2004 Prentice-Hall, Inc. Chap 8-24

Confidence Interval Estimatefor 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

Margin of Error

Standard Error

© 2004 Prentice-Hall, Inc. Chap 8-25

ExampleA random sample of 400 voters showed that 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

We are 95% confident that the proportion of voters who prefer Candidate A is somewhere between 0.053 and 0.107.

© 2004 Prentice-Hall, Inc. Chap 8-26

Confidence Interval Estimate for Proportion in PHStat

PHStat | Confidence Interval | Estimate for the Proportion …

Example in Excel Spreadsheet

Microsoft Excel Worksheet

Confidence Interval Estimate for the Mean

DataSample Size 400Number of Successes 32Confidence Level 95%

Sample Proportion 0.08Z Value -1.95996108Standard Error of the Proportion 0.01356466Interval Half Width 0.026586206

Interval Lower Limit 0.053413794Interval Upper Limit 0.106586206

Intermediate Calculations

Confidence Interval

© 2004 Prentice-Hall, Inc. Chap 8-27

Determining Sample Sizefor Proportion

What sample size is needed to be within ±5% with 90% confidence if past studies show about 30% are defective?

Round Up

2 2

2 2

1 1.645 0.3 0.7

Error 0.05227.3 228

Z p pn

© 2004 Prentice-Hall, Inc. Chap 8-28

Determining Sample Size for Proportion in PHStat

PHStat | Sample Size | Determination for the Proportion …

Example in Excel Spreadsheet

Microsoft Excel Worksheet

Sample Size Determination

DataEstimate of True Proportion 0.3Sampling Error 0.05Confidence Level 90%

Z Value -1.644853Calculated Sample Size 227.265477

Sample Size Needed 228

Intermediate Calculations

Result

© 2004 Prentice-Hall, Inc. Chap 8-29

Confidence Interval for Population Total Amount

Point Estimate

Confidence Interval Estimate

NX

/ 2, 1 1n

N nSNX N t

Nn

© 2004 Prentice-Hall, Inc. Chap 8-30

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 the 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.

© 2004 Prentice-Hall, Inc. Chap 8-31

Example Solution

/ 2, 1

1000 50 $1076.39 $273.62

1

273.62 1000 501000 1076.39 1000 2.0096

1000 1501,076,390 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.

© 2004 Prentice-Hall, Inc. Chap 8-32

Example Solution in PHStat

PHStat | Confidence Intervals | Estimate for the Population Total

Excel Spreadsheet for the Voucher Example

Microsoft Excel Worksheet

© 2004 Prentice-Hall, Inc. Chap 8-33

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

© 2004 Prentice-Hall, Inc. Chap 8-34

Estimation for Finite Population

(CD-ROM Topic)

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

© 2004 Prentice-Hall, Inc. Chap 8-35

Sample Size (n ) Determination for Finite Population (CD-ROM

Topic) 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

0

0 1

n Nn

n N

© 2004 Prentice-Hall, Inc. Chap 8-36

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

© 2004 Prentice-Hall, Inc. Chap 8-37

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)

© 2004 Prentice-Hall, Inc. Chap 8-38

Chapter Summary Discussed Confidence Interval Estimation

for the Proportion Addressed Confidence Interval

Estimation for Population Total Discussed Confidence Interval Estimation

for Total Difference in the Population Addressed Estimation and Sample Size

Determination for Finite Population (CD-ROM Topic)

Addressed Confidence Interval Estimation and Ethical Issues

(continued)