© 2003 Prentice-Hall, Inc. Chap 7-1

Basic Business Statistics (9th Edition)

Chapter 7Sampling Distributions

© 2003 Prentice-Hall, Inc. Chap 7-2

Chapter Topics

Sampling Distribution of the Mean

The Central Limit Theorem

Sampling Distribution of the Proportion

Sampling from Finite Population

© 2003 Prentice-Hall, Inc. Chap 7-3

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 7-4

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 7-5

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 7-6

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 7-7

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 7-8

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

X

Sample Means

Distribution

16 Sample Means

_

Developing Sampling Distributions

(continued)

P X

© 2003 Prentice-Hall, Inc. Chap 7-9

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 7-10

Comparing the Population with Its Sampling

Distribution

18 19 20 21 22 23 240

.1

.2

.3

X

Sample Means Distribution

n = 2

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

0

.1

.2

.3

PopulationN = 4

X_

21 2.236 21 1.58X X P X P X

© 2003 Prentice-Hall, Inc. Chap 7-11

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 7-12

Unbiasedness ( )

BiasedUnbiased

X X

X

f X

© 2003 Prentice-Hall, Inc. Chap 7-13

Less Variability

Sampling Distribution of Median Sampling

Distribution of Mean

X

f X

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

X

© 2003 Prentice-Hall, Inc. Chap 7-14

Effect of Large Sample

Larger sample size

Smaller sample size

X

f X

For sampling with replacement:

As increases, decreasesXn

© 2003 Prentice-Hall, Inc. Chap 7-15

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 7-16

When the Population isNot 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 7-17

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 7-18

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 Regardless of the Sample Size This is a property of sampling from a normal

population distribution and is NOT a result of the central limit theorem

© 2003 Prentice-Hall, Inc. Chap 7-19

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 7-20

Population Proportions Categorical Variable

E.g., Gender, Voted for Bush, College Degree

Proportion of Population Having a Characteristic

Sample Proportion Provides an Estimate

If Two Outcomes, X Has a Binomial

Distribution Possess or do not possess characteristic

number of successes

sample sizeS

Xp

n

p

p

© 2003 Prentice-Hall, Inc. Chap 7-21

Sampling Distribution ofSample Proportion

Approximated by Normal Distribution

Mean:

Standard error: p = population

proportion

Sampling Distributionf(ps)

.3

.2

.1 0

0 . 2 .4 .6 8 1ps

5np 1 5n p

Spp

1Sp

p p

n

© 2003 Prentice-Hall, Inc. Chap 7-22

Standardizing Sampling Distribution of Proportion

1S

S

S p S

p

p p pZ

p p

n

Sampling Distribution

Standardized Normal

Distribution

Sp 1Z

Sp Sp Z0Z

© 2003 Prentice-Hall, Inc. Chap 7-23

Example: 200 .4 .43 ?Sn p P p

.43 .4.43 .87 .8078

.4 1 .4

200

S

S

S pS

p

pP p P P Z

Sampling Distribution

Standardized Normal

DistributionSp

1Z

Sp

Sp Z0.43 .87

© 2003 Prentice-Hall, Inc. Chap 7-24

Sampling from Finite Sample

Modify Standard Error if Sample Size (n) is Large Relative to Population Size (N ) Use Finite Population Correction Factor (fpc)

Standard Error with FPC

1X

N n

Nn

1

1SP

p p N n

n N

.05 or / .05n N n N

© 2003 Prentice-Hall, Inc. Chap 7-25

Chapter Summary

Discussed Sampling Distribution of the Sample Mean

Described the Central Limit Theorem Discussed Sampling Distribution of the

Sample Proportion Described Sampling from Finite

Populations