© 2003 Prentice-Hall, Inc. Chap 6-1

Business Statistics: A First Course

(3rd Edition)

Chapter 6Sampling Distributions

andConfidence Interval

Estimation

© 2003 Prentice-Hall, Inc. Chap 6-2

Chapter Topics

Sampling Distribution of the Mean

Estimation Process Point Estimates Interval Estimates Confidence Interval Estimation for the

Mean ( Known)

Determining Sample Size

© 2003 Prentice-Hall, Inc. Chap 6-3

Chapter Topics

Confidence Interval Estimation for the Mean ( Unknown)

Confidence Interval Estimation for the Proportion

Confidence Interval Estimation and Ethical Issues

(continued)

© 2003 Prentice-Hall, Inc. Chap 6-4

Why Study Sampling Distributions

Sample Statistics are Used to Estimate Population Parameters e.g. estimates the population mean

Problem: Different Samples Provide Different Estimates Large sample gives better estimate; large

sample costs more How good is the estimate?

Approach to Solution: Theoretical Basis is Sampling Distribution

50X

© 2003 Prentice-Hall, Inc. Chap 6-5

Sampling Distribution

Theoretical Probability Distribution of a Sample Statistic

Sample Statistic is a Random Variable Sample mean, sample proportion

Results from Taking All Possible Samples of the Same Size

© 2003 Prentice-Hall, Inc. Chap 6-6

Developing Sampling Distributions

Suppose There is a Population … Population Size N=4 Random Variable, X,

is Age of Individuals Values of X : 18, 20,

22, 24 Measured inYears A

B C

D

© 2003 Prentice-Hall, Inc. Chap 6-7

1

2

1

18 20 22 2421

4

2.236

N

ii

N

ii

X

N

X

N

.3

.2

.1

0 A B C D (18) (20) (22) (24)

Uniform Distribution

P(X)

X

Developing Sampling Distributions

(continued)

Summary Measures for the Population Distribution

© 2003 Prentice-Hall, Inc. Chap 6-8

1st 2nd Observation Obs 18 20 22 24

18 18,18 18,20 18,22 18,24

20 20,18 20,20 20,22 20,24

22 22,18 22,20 22,22 22,24

24 24,18 24,20 24,22 24,24

All Possible Samples of Size n=2

16 Samples Taken with Replacement

16 Sample Means1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Developing Sampling Distributions

(continued)

© 2003 Prentice-Hall, Inc. Chap 6-9

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Sampling Distribution of All Sample Means

18 19 20 21 22 23 240

.1

.2

.3

Sample Means

Distribution

16 Sample Means

Developing Sampling Distributions

(continued)

P X

X

© 2003 Prentice-Hall, Inc. Chap 6-10

1

2

1

2 2 2

18 19 19 2421

16

18 21 19 21 24 211.58

16

N

ii

X

N

i Xi

X

X

N

X

N

Summary Measures of Sampling Distribution

Developing Sampling Distributions

(continued)

© 2003 Prentice-Hall, Inc. Chap 6-11

Comparing the Population with its Sampling

Distribution

18 19 20 21 22 23 240

.1

.2

.3

Sample Means Distribution

n = 2

A B C D (18) (20) (22) (24)

0

.1

.2

.3

PopulationN = 4

21 2.236 21 1.58X X P X P X

X X

© 2003 Prentice-Hall, Inc. Chap 6-12

Properties of Summary Measures

i.e. is unbiased

Standard Error (Standard Deviation) of the Sampling Distribution is Less than the Standard Error of Other Unbiased Estimators

For Sampling with Replacement or without Replacement from Large or Infinite Populations:

As n increases, decreases

X

X

Xn

X

X

© 2003 Prentice-Hall, Inc. Chap 6-13

Unbiasedness

BiasedUnbiased

X X

f X

© 2003 Prentice-Hall, Inc. Chap 6-14

Less Variability

Sampling Distribution of Median Sampling

Distribution of Mean

X

f X

© 2003 Prentice-Hall, Inc. Chap 6-15

Effect of Large Sample

Larger sample size

Smaller sample size

X

f X

© 2003 Prentice-Hall, Inc. Chap 6-16

When the Population is Normal

Central Tendency

Variation

Population Distribution

Sampling Distributions

X

Xn

X50X

4

5X

n

16

2.5X

n

50

10

© 2003 Prentice-Hall, Inc. Chap 6-17

When the Population is Not Normal

Central Tendency

Variation

Population Distribution

Sampling Distributions

X

Xn

X50X

4

5X

n

30

1.8X

n

50

10

© 2003 Prentice-Hall, Inc. Chap 6-18

Central Limit Theorem

As Sample Size Gets Large Enough

Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X

© 2003 Prentice-Hall, Inc. Chap 6-19

How Large is Large Enough?

For Most Distributions, n>30 For Fairly Symmetric Distributions, n>15 For Normal Distribution, the Sampling

Distribution of the Mean is Always Normally Distributed

© 2003 Prentice-Hall, Inc. Chap 6-20

Example:

8 =2 25

7.8 8.2 ?

n

P X

Sampling Distribution

Standardized Normal

Distribution2

.425

X 1Z

8X 8.2 Z0Z

0.5

7.8 8 8.2 87.8 8.2

2 / 25 2 / 25

.5 .5 .3830

X

X

XP X P

P Z

7.8 0.5

.1915

X

© 2003 Prentice-Hall, Inc. Chap 6-21

Estimation Process

Mean, , is unknown

Population Random Sample I am 95%

confident that is between 40 &

60.

Mean X = 50

Sample

© 2003 Prentice-Hall, Inc. Chap 6-22

Point Estimates

Estimate Population

Parameters …

with SampleStatistics

Mean

Proportion

Variance

Difference

p

2

1 2

X

SP

2S

1 2X X

© 2003 Prentice-Hall, Inc. Chap 6-23

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

© 2003 Prentice-Hall, Inc. Chap 6-24

Confidence Interval Estimates

Mean

Unknown

ConfidenceIntervals

Proportion

Known

© 2003 Prentice-Hall, Inc. Chap 6-25

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

© 2003 Prentice-Hall, Inc. Chap 6-26

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

© 2003 Prentice-Hall, Inc. Chap 6-27

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 No probability involved in a specific interval

100 1 %

100 1 %

© 2003 Prentice-Hall, Inc. Chap 6-28

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

1 100%

100 %

/ 2 XZ / 2 XZ

© 2003 Prentice-Hall, Inc. Chap 6-29

Factors Affecting Interval Width

(Precision)

Data Variation Measured by

Sample Size

Level of Confidence

Intervals Extend from

© 1984-1994 T/Maker Co.

Xn

100 1 %

to X XX Z X Z

© 2003 Prentice-Hall, Inc. Chap 6-30

Determining Sample Size (Cost)

Too Big:

• Requires more resources

Too small:

• Won’t do the job

© 2003 Prentice-Hall, Inc. Chap 6-31

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

© 2003 Prentice-Hall, Inc. Chap 6-32

Determining Sample Size for Mean in PHStat

PHStat | Sample Size | Determination for the Mean …

Example in Excel Spreadsheet

Microsoft Excel Worksheet

© 2003 Prentice-Hall, Inc. Chap 6-33

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

© 2003 Prentice-Hall, Inc. Chap 6-34

Student’s t Distribution

Zt

0

t (df = 5)

t (df = 13)Bell-ShapedSymmetric

‘Fatter’ Tails

Standard Normal

© 2003 Prentice-Hall, Inc. Chap 6-35

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

© 2003 Prentice-Hall, Inc. Chap 6-36

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

© 2003 Prentice-Hall, Inc. Chap 6-37

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

© 2003 Prentice-Hall, Inc. Chap 6-38

PHStat | Confidence Interval | Estimate for the Mean, sigma unknown

Example in Excel Spreadsheet

Confidence Interval for( Unknown) in PHStat

Microsoft Excel Worksheet

© 2003 Prentice-Hall, Inc. Chap 6-39

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

© 2003 Prentice-Hall, Inc. Chap 6-40

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

© 2003 Prentice-Hall, Inc. Chap 6-41

Confidence Interval Estimate for Proportion in PHStat

PHStat | Confidence Interval | Estimate for the Proportion …

Example in Excel Spreadsheet

Microsoft Excel Worksheet

© 2003 Prentice-Hall, Inc. Chap 6-42

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

© 2003 Prentice-Hall, Inc. Chap 6-43

Determining Sample Size for Proportion in PHStat

PHStat | Sample Size | Determination for the Proportion …

Example in Excel Spreadsheet

Microsoft Excel Worksheet

© 2003 Prentice-Hall, Inc. Chap 6-44

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

© 2003 Prentice-Hall, Inc. Chap 6-45

Chapter Summary

Discussed Sampling Distribution of the Sample Mean

Illustrated Estimation Process Discussed Point Estimates Addressed Interval Estimates Discussed Confidence Interval

Estimation for the Mean ( Known) Addressed Determining Sample Size

© 2003 Prentice-Hall, Inc. Chap 6-46

Chapter Summary

Discussed Confidence Interval Estimation for the Mean ( Unknown)

Discussed Confidence Interval Estimation for the Proportion

Addressed Confidence Interval Estimation and Ethical Issues

(continued)