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10th Edition
Chapter 7 Sampling Distributions

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc..

Chap 7-1

Learning Objectives
In this chapter, you learn:
? The concept of the sampling distribution ? To compute probabilities related to the sample mean and the sample proportion ? The importance of the Central Limit Theorem ? To distinguish between different survey sampling methods ? To evaluate survey worthiness and survey errors

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-2

Sampling Distributions
Sampling Distributions

Sampling Distribution of the Mean

Sampling Distribution of the Proportion

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-3

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

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-4

Developing a Sampling Distribution
? Assume there is a population … ? Population size N=4
A B C D

? Random variable, X, is age of individuals ? Values of X: 18, 20, 22, 24 (years)

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-5

Developing a Sampling Distribution
(continued)

Summary Measures for the Population Distribution:

?X μ?
N

i

P(x)
.3 .2 .1 0
18 20 22 24

18 ? 20 ? 22 ? 24 ? ? 21 4

σ?

? (X ? μ)
i

2

N

? 2.236

x

A

B

C

D

Uniform Distribution
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 7-6

Developing a Sampling Distribution
(continued)

Now consider all possible samples of size n=2
1st Obs 18 20 22 2nd Observation 18 18,18 20,18 22,18 20 18,20 20,20 22,20 22 18,22 20,22 22,22 24 18,24 20,24 22,24

16 Sample Means
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
Chap 7-7

24

24,18

24,20

24,22

24,24

16 possible samples (sampling with replacement)
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Developing a Sampling Distribution
(continued)

Sampling Distribution of All Sample Means

16 Sample Means
1st 2nd Observation Obs 18 20 22 24

Sample Means Distribution _
P(X) .3 .2 .1 0
18 19 20 21 22 23 24

18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

_
X
Chap 7-8

(no longer uniform)

Developing a Sampling Distribution
(continued)

Summary Measures of this Sampling Distribution:

μX

?X ?
N

i

18 ? 19 ? 21? ? ? 24 ? ? 21 16

σX ? ?

( Xi ? μ X )2 ? N (18 - 21)2 ? (19 - 21)2 ? ? ? (24 - 21)2 ? 1.58 16
Chap 7-9

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Comparing the Population with its Sampling Distribution
μ ? 21
P(X) .3
.2 .1

Population N=4

σ ? 2.236

Sample Means Distribution n=2

μX ? 21
_

σ X ? 1.58

P(X) .3
.2 .1
18 20 22 24

0

X

0

18 19

20 21 22 23

24

_
X
Chap 7-10

A

B

C

D

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Sampling Distribution of the Mean

Sampling Distributions

Sampling Distribution of the Mean

Sampling Distribution of the Proportion

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-11

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)

σ σX ? n
? Note that the standard error of the mean decreases as the sample size increases
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 7-12

If the Population is Normal
? If a population is normal with mean μ and standard deviation σ, the sampling distribution of X is also normally distributed with

μX ? μ

and

σ σX ? n

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-13

Z-value for Sampling Distribution of the Mean
? Z-value for the sampling distribution of X :

Z?

( X ? μX ) σX

( X ? μ) ? σ n

where:

μ = population mean

X = sample mean

σ = population standard deviation
n = sample size

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-14

Sampling Distribution Properties

?

μx ? μ
(i.e.

Normal Population Distribution

μ

x is unbiased )

x

Normal Sampling Distribution (has the same mean)

μx
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

x
Chap 7-15

Sampling Distribution Properties
(continued)

As n increases,

σ x decreases
Smaller sample size

Larger sample size

μ
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

x
Chap 7-16

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:

μx ? μ
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

and

σ σx ? n
Chap 7-17

Central Limit Theorem
As the sample size gets large enough…
n↑

the sampling distribution becomes almost normal regardless of shape of population

x
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 7-18

If the Population is not Normal
(continued)

Population Distribution

Sampling distribution properties:
Central Tendency

μx ? μ
Variation

μ
Sampling Distribution (becomes normal as n increases) Smaller sample size

x
Larger sample size

σ σx ? n

μx
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

x
Chap 7-19

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

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-20

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?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-21

Example
(continued)

Solution:
? Even if the population is not normally distributed, the central limit theorem can be used (n > 30)

? … so the sampling distribution of approximately normal ? … with mean
μx ? …and standard deviation

x

is

= 8

σ 3 σx ? ? ? 0.5 n 36
Chap 7-22

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Example
(continued)

Solution (continued):
? ? ? 7.8 - 8 X -μ 8.2 - 8 ? P(7.8 ? X ? 8.2) ? P? ? ? ? 3 σ 3 ? ? 36 n 36 ? ? ? P(-0.4 ? Z ? 0.4) ? 0.3108
Population Distribution ??? ? ?? ? ?? ? ? Sampling Distribution
Sample

Standard Normal Distribution
Standardize

.1554 +.1554

?

μ?8

X

7.8

μX ? 8

8.2

x

-0.4

μz ? 0

0.4

Z
Chap 7-23

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Sampling Distribution of the Proportion

Sampling Distributions

Sampling Distribution of the Mean

Sampling Distribution of the Proportion

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-24

Population Proportions
π = the proportion of the population having some characteristic
? Sample proportion ( p ) provides an estimate of π:
p? X number of items in the sample having the characteri of interest stic ? n sample size

? 0≤ p≤1 ? p has a binomial distribution
(assuming sampling with replacement from a finite population or without replacement from an infinite population)

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-25

Sampling Distribution of p
? Approximated by a normal distribution if:
?
Sampling Distribution

P( ps)
.3 .2 .1 0 0

np ? 5
and

n(1? p) ? 5
where

.2

.4

.6

8

1

p

μp ? π

and

π(1? π ) σp ? n
Chap 7-26

(where π = population proportion)
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Z-Value for Proportions
Standardize p to a Z value with the formula:

p ?? Z? ? σp

p ?? ? (1? ? ) n

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-27

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) ?

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-28

Example
(continued)

?

if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ?
? (1? ? )
n 0.4(1? 0.4) ? ? 0.03464 200

Find σ p : σ p ?

0.45 ? 0.40 ? ? 0.40 ? 0.40 Convert to P(0.40 ? p ? 0.45) ? P? ?Z? ? standard 0.03464 ? ? 0.03464 normal: ? P(0 ? Z ? 1.44)

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-29

Example
(continued)

?

if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ?
P(0 ≤ Z ≤ 1.44) = 0.4251
Standardized Normal Distribution
0.4251
Standardize

Use standard normal table:
Sampling Distribution

0.40

0.45

p

0

1.44

Z
Chap 7-30

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Reasons for Drawing a Sample
? Less time consuming than a census ? Less costly to administer than a census

? Less cumbersome and more practical to administer than a census of the targeted population

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-31

Types of Samples Used
? Nonprobability Sample
? Items included are chosen without regard to their probability of occurrence

? Probability Sample
? Items in the sample are chosen on the basis of known probabilities

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-32

Types of Samples Used
(continued)

Samples

Non-Probability Samples

Probability Samples

Judgement Quota

Chunk Convenience

Simple Random

Stratified Cluster

Systematic

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-33

Probability Sampling
? Items in the sample are chosen based on known probabilities
Probability Samples

Simple Random

Systematic

Stratified

Cluster

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-34

Simple Random Samples
? Every individual or item from the frame has an equal chance of being selected

? Selection may be with replacement or without replacement
? Samples obtained from table of random numbers or computer random number generators

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-35

Systematic Samples
? Decide on sample size: n ? Divide frame of N individuals into groups of k individuals: k=N/n

? Randomly select one individual from the 1st group
? Select every kth individual thereafter
N = 64 n=8 k=8
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 7-36

First Group

Stratified Samples
? 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

Population Divided into 4 strata

Sample
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 7-37

Cluster Samples
? 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

Population divided into 16 clusters.
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Randomly selected clusters for sample
Chap 7-38

? 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)
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 7-39

Evaluating Survey Worthiness
? ? ? ? ? What 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

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-40

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
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc. Chap 7-41

Types of Survey Errors
(continued)

? Coverage error ? Non response error ? Sampling error ? Measurement error
Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Excluded from frame
Chap 7-42

Chapter Summary
? 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 distributions ? Described different types of samples and sampling techniques ? Examined survey worthiness and types of survey errors

Basic Business Statistics, 10e ? 2006 Prentice-Hall, Inc.

Chap 7-43

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