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Chap 7- 1 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7- 1 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

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Chap 7- 1 Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall - PPT Presentation

Chap 7 1 Chapter 7 Sampling and Sampling Distributions Basic Business Statistics for 12 th Edition Chap 7 2 Copyright 2012 Pearson Education Inc publishing as Prentice Hall Chap 7 ID: 689548

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Slide1

Chap 7-1

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-1

Chapter 7Sampling and Sampling Distributions

Basic Business Statistics for

12

th

EditionSlide2

Chap 7-2

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-2

Learning Objectives

In this chapter, you learn:

To distinguish between different sampling methods

The concept of the sampling distribution

To compute probabilities related to the sample mean and the sample proportion

The importance of the Central Limit TheoremSlide3

Chap 7-3

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-3

Why Sample?

Selecting a sample is less time-consuming & less costly than selecting every item in the population (census).

An analysis of a sample is less cumbersome and more practical than an analysis of the entire population.

D

C

OVASlide4

Chap 7-4

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-4

A Sampling Process Begins With A Sampling FrameThe sampling frame is a listing of items that make up the population

Frames are data sources such as population lists, directories, or maps

Inaccurate or biased results can result if a frame excludes certain portions of the population

Using different frames to generate data can lead to dissimilar conclusions

D

C

OVASlide5

Chap 7-5

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-5

Types of Samples

Samples

Non-Probability Samples

Judgment

Probability Samples

Simple

Random

Systematic

Stratified

Cluster

Convenience

D

C

OVASlide6

Chap 7-6

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-6

Types of Samples:Nonprobability Sample

In a nonprobability sample, items included are chosen without regard to their probability of occurrence.

In

convenience sampling

, items are selected based only on the fact that they are easy, inexpensive, or convenient to sample.

In a

judgment sample,

you get the opinions of pre-selected experts in the subject matter.

D

C

OVASlide7

Chap 7-7

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-7

Types of Samples:Probability Sample

In a

probability sample

, items in the sample are chosen on the basis of known probabilities.

Probability Samples

Simple

Random

Systematic

Stratified

Cluster

D

C

OVASlide8

Chap 7-8

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-8

Probability Sample:Simple Random Sample

Every individual or item from the frame has an equal chance of being selected.

Selection may be with replacement (selected individual is returned to frame for possible reselection) or without replacement (selected individual isn’t returned to the frame).

Samples obtained from table of random numbers or computer random number generators.

D

C

OVASlide9

Chap 7-9

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-9

Selecting a Simple Random Sample Using A Random Number Table

Sampling Frame For Population With 850 Items

Item Name Item #

Bev R. 001

Ulan X. 002

. .

. .

. .

. .

Joann P. 849

Paul F. 850

Portion Of A Random Number Table

49280 88924 35779 00283 81163 07275

11100 02340 12860 74697 96644 89439

09893 23997 20048 49420 88872 08401

The First 5 Items in a simple random sample

Item # 492

Item # 808

Item # 892 -- does not exist so ignore

Item # 435

Item # 779

Item # 002

D

C

OVASlide10

Chap 7-10

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-10

Decide on sample size: n

Divide frame of

N

individuals into groups of

k

individuals:

k

=

N

/

n

Randomly select one individual from the 1

st

group

Select every k

th

individual thereafter

Probability Sample:

Systematic Sample

N = 40

n = 4

k = 10

First Group

D

C

OVASlide11

Chap 7-11

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-11

Probability Sample:Stratified Sample

Divide population into two or more subgroups (called

strata

) according to some common characteristic

A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes

Samples from subgroups are combined into one

This is a common technique when sampling population of voters, stratifying across racial or socio-economic lines.

Population

Divided

into 4

strata

D

C

OVASlide12

Chap 7-12

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-12

Probability SampleCluster Sample

Population is divided into several “clusters,” each representative of the population

A simple random sample of clusters is selected

All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique

A common application of cluster sampling involves election exit polls, where certain election districts are selected and sampled.

Population divided into 16 clusters.

Randomly selected clusters for sample

D

C

OVASlide13

Chap 7-13

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-13

Probability Sample:Comparing Sampling Methods

Simple random sample and Systematic sample

Simple to use

May not be a good representation of the population’s underlying characteristics

Stratified sample

Ensures representation of individuals across the entire population

Cluster sample

More cost effective

Less efficient (need larger sample to acquire the same level of precision)

D

C

OVASlide14

Chap 7-14

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-14

Evaluating Survey WorthinessWhat is the purpose of the survey?

Is the survey based on a probability sample?

Coverage error – appropriate frame?

Nonresponse error – follow up

Measurement error – good questions elicit good responses

Sampling error – always exists

D

C

OVASlide15

Chap 7-15

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-15

Types of Survey Errors

Coverage error or selection bias

Exists if some groups are excluded from the frame and have no chance of being selected

Nonresponse error or bias

People who do not respond may be different from those who do respond

Sampling error

Variation from sample to sample will always exist

Measurement error

Due to weaknesses in question design, respondent error, and interviewer’s effects on the respondent (“Hawthorne effect”)

D

C

OVASlide16

Chap 7-16

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-16

Types of Survey Errors

Coverage error

Non response error

Sampling error

Measurement error

Excluded from frame

Follow up on nonresponses

Random differences from sample to sample

Bad or leading question

(continued)

D

C

OVASlide17

Chap 7-17

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-17

Sampling Distributions

A sampling distribution is a distribution of all of the possible values of a sample statistic for a given size sample selected from a population.

For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are interested in the distribution of

the mean GPA from all possible samples of 50 students.

DCOV

ASlide18

Chap 7-18

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-18

Developing a Sampling Distribution

Assume there is a population …

Population size

N=4

Random variable, X,

is

age

of individuals

Values of X:

18, 20,

22, 24

(years)

A

B

C

D

DCOV

ASlide19

Chap 7-19

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-19

.3

.2

.1

0

18 20 22 24

A B C D

Uniform Distribution

P(x)

x

(continued)

Summary Measures for the Population Distribution:

Developing a

Sampling Distribution

DCOV

ASlide20

Chap 7-20

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-20

16 possible samples (sampling with replacement)

Now consider all possible samples of size n=2

(continued)

Developing a

Sampling Distribution

16 Sample Means

1

st

Obs

2

nd

Observation

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

DCOV

ASlide21

Chap 7-21

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-21

Sampling Distribution of All Sample Means

18 19 20 21 22 23 24

0

.1

.2

.3

P(X)

X

Sample Means Distribution

16 Sample Means

_

Developing a

Sampling Distribution

(continued)

(no longer uniform)

_

DCOV

ASlide22

Chap 7-22

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-22

Summary Measures of this Sampling Distribution:

Developing a

Sampling Distribution

(continued)

DCOV

A

Note: Here we divide by 16 because there are 16

different samples of size 2.Slide23

Chap 7-23

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-23

Comparing the Population Distributionto the Sample Means Distribution

18 19 20 21 22 23 24

0

.1

.2

.3

P(X)

X

18

20

22

24

A

B

C

D

0

.1

.2

.3

Population

N = 4

P(X)

X

_

Sample Means Distribution

n = 2

_

DCOV

ASlide24

Chap 7-24

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-24

Sampling Distribution of The Mean:Standard Error of the Mean

Different samples of the same size from the same population will yield different sample means

A measure of the variability in the mean from sample to sample is given by the

Standard Error of the Mean:

(This assumes that sampling is with replacement or

sampling is without replacement from an infinite population)

Note that the standard error of the mean decreases as the sample size increases

DCOV

ASlide25

Chap 7-25

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-25

Sampling Distribution of The Mean:If the Population is Normal

If a population is

normal

with mean

μ

and standard deviation

σ

, the sampling distribution of is

also normally distributed

with

and

DCOV

ASlide26

Chap 7-26

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-26

Z-value for Sampling Distributionof the Mean

Z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

DCOV

ASlide27

Chap 7-27

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-27

Normal Population Distribution

Normal Sampling Distribution

(has the same mean)

Sampling Distribution Properties

(i.e. is unbiased

)

DCOV

ASlide28

Chap 7-28

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-28

Sampling Distribution Properties

As n increases,

decreases

Larger sample size

Smaller sample size

(continued)

DCOV

ASlide29

Chap 7-29

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-29

Determining An Interval Including A Fixed Proportion of the Sample Means

Find a symmetrically distributed interval around

µ that will include 95% of the sample means when µ = 368,

σ

= 15, and n = 25.

Since the interval contains 95% of the sample means 5% of the sample means will be outside the interval.

Since the interval is symmetric 2.5% will be above the upper limit and 2.5% will be below the lower limit.

From the standardized normal table, the Z score with 2.5% (0.0250) below it is -1.96 and the Z score with 2.5% (0.0250) above it is 1.96.

DCOV

ASlide30

Chap 7-30

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-30

Determining An Interval Including A Fixed Proportion of the Sample Means

Calculating the lower limit of the interval

Calculating the upper limit of the interval

95% of all sample means of sample size 25 are between 362.12 and 373.88

(continued)

DCOV

ASlide31

Chap 7-31

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-31

Sampling Distribution of The Mean:If the Population is not

Normal

We can apply the

Central Limit Theorem

:

Even if the population is

not normal

,

…sample means from the population

will be

approximately normal

as long as the sample size is large enough.

Properties of the sampling distribution:

and

DCOV

ASlide32

Chap 7-32

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-32

Population Distribution

Sampling Distribution

(becomes normal as n increases)

Central Tendency

Variation

Larger sample size

Smaller sample size

Sample Mean Sampling Distribution:

If the Population is

not

Normal

(continued)

Sampling distribution properties:

DCOV

ASlide33

Chap 7-33

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-33

How Large is Large Enough?

For most distributions,

n ≥ 30

will give a sampling distribution that is nearly normal

For fairly symmetric distributions, n ≥ 15

For normal population distributions, the sampling distribution of the mean is always normally distributed

DCOV

ASlide34

Chap 7-34

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-34

Example

Suppose a population has mean

μ

= 8

and standard deviation

σ

= 3

. Suppose a random sample of size

n = 36

is selected.

What is the probability that the

sample mean

is between 7.8 and 8.2?

DCOV

ASlide35

Chap 7-35

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-35

Example

Solution:

Even if the population is not normally distributed, the central limit theorem can be used (n ≥ 30)

… so the sampling distribution of is approximately normal

… with mean

= 8

…and standard deviation

(continued)

DCOV

ASlide36

Chap 7-36

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-36

Example

Solution (continued):

(continued)

Z

7.8 8.2

-0.4 0.4

Sampling Distribution

Standard Normal Distribution

Population Distribution

?

?

?

?

?

?

?

?

?

?

?

?

Sample

Standardize

X

DCOV

ASlide37

Chap 7-37

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-37

Population Proportions

π

= the proportion of the population having

a characteristic of interest

Sample proportion

(p)

provides an estimate

of

π

:

0

≤ p ≤ 1

p is approximately distributed as a normal distribution when n is large

(assuming sampling with replacement from a finite population or without replacement from an infinite population)

DCOV

ASlide38

Chap 7-38

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-38

Sampling Distribution of p

Approximated by a

normal distribution if:

where

and

(where

π

= population proportion)

Sampling Distribution

P(

p

s

)

.3

.2

.1

0

0 . 2 .4 .6 8 1

p

DCOV

ASlide39

Chap 7-39

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-39

Z-Value for Proportions

Standardize p to a Z value with the formula:

DCOV

ASlide40

Chap 7-40

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-40

Example

If the true proportion of voters who support Proposition A is

π

= 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45?

i.e.:

if

π

= 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

DCOV

ASlide41

Chap 7-41

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-41

Example

if

π

= 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

Find :

Convert to standardized normal:

DCOV

ASlide42

Chap 7-42

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-42

Example

Z

0.45

1.44

0.4251

Standardize

Sampling Distribution

Standardized

Normal Distribution

if

π

= 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

Utilize the cumulative normal table:

P(0

≤ Z ≤ 1.44) = 0.9251 – 0.5000 = 0.4251

0.40

0

p

DCOV

ASlide43

Chap 7-43

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Chap 7-43

Chapter Summary

Discussed probability and nonprobability samples

Described four common probability samples

Examined survey worthiness and types of survey errors

Introduced sampling distributions

Described the sampling distribution of the mean

For normal populations

Using the Central Limit Theorem

Described the sampling distribution of a proportion

Calculated probabilities using sampling distributionsSlide44

Chap 7-44

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Online TopicSampling From Finite Populations

Basic Business Statistics for

12

th

EditionSlide45

Chap 7-45

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Learning Objectives

In this section, you learn: To know when finite population corrections are neededTo know how to utilize finite population correction factors in calculating standard errorsSlide46

Chap 7-46

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Finite Population Correction FactorsUsed to calculate the standard error of both the sample mean and the sample proportion

Needed when the sample size, n, is more than 5% of the population size N (i.e. n / N > 0.05)The Finite Population Correction Factor Is:

DCOV

ASlide47

Chap 7-47

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Using The fpc In Calculating Standard Errors

DCOVA

Standard Error of the Mean for Finite Populations

Standard Error of the Proportion for Finite PopulationsSlide48

Chap 7-48

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Using The fpc Reduces The Standard ErrorThe fpc is always less than 1So when it is used it reduces the standard error

Resulting in more precise estimates of population parameters

DCOV

ASlide49

Chap 7-49

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Using fpc With The Mean - Example

DCOVA

Suppose a random sample of size 100 is drawn from a

population of size 1,000 with a standard deviation of 40.

Here n=100, N=1,000 and 100/1,000 = 0.10 > 0.05.

So using the fpc for the standard error of the mean we get:Slide50

Chap 7-50

Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

Section Summary

Identified when a finite population correction should be used.Identified how to utilize a finite population correction factor in calculating the standard error of both a sample mean and a sample proportion