Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population...

26
QMIS 220, by Dr. M. Zainal Chapter 7 Student Lecture Notes 7-1 Chapter 7 Sampling Distributions Business Statistics: Department of Economics Dr. Mohammad Zainal ECON 509 Chapter Goals After completing this chapter, you should be able to: Define the concept of sampling error Determine the mean and standard deviation for the sampling distribution of the sample mean, x Determine the mean and standard deviation for the sampling distribution of the sample proportion, p Describe the Central Limit Theorem and its importance Apply sampling distributions for both x and p _ _ _ _ ECON 509, by Dr. M. Zainal Chap 7-2

Transcript of Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population...

Page 1: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-1

Chapter 7

Sampling Distributions

Business Statistics:

Department of Economics

Dr. Mohammad Zainal ECON 509

Chapter Goals

After completing this chapter, you should be

able to:

Define the concept of sampling error

Determine the mean and standard deviation for the

sampling distribution of the sample mean, x

Determine the mean and standard deviation for the

sampling distribution of the sample proportion, p

Describe the Central Limit Theorem and its importance

Apply sampling distributions for both x and p

_

_ _

_

ECON 509, by Dr. M. Zainal Chap 7-2

Page 2: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-2

Inferential statistics

Drawing conclusions and/or making decisions

concerning a population based only on

sample data

Consists of methods that use sample results

to help make decisions or predictions about a

population.

Elections

Review: Inferential Statistics

ECON 509, by Dr. M. Zainal Chap 7-3

Sample statistics Population parameters

(known) Inference (unknown, but can

be estimated from

sample evidence)

SamplePopulation

Review: Inferential Statistics

ECON 509, by Dr. M. Zainal Chap 7-4

Page 3: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-3

Review: Inferential Statistics

Estimation

e.g., Estimate the population mean

weight using the sample mean

weight

Hypothesis Testing

e.g., Use sample evidence to test

the claim that the population mean

weight is 120 pounds

Drawing conclusions and/or making decisions concerning a population based on sample results.

ECON 509, by Dr. M. Zainal Chap 7-5

Review: Key Definitions

A population is the entire collection of things

under consideration

A parameter is a summary measure computed to

describe a characteristic of the population

A sample is a portion of the population

selected for analysis

A statistic is a summary measure computed to

describe a characteristic of the sample

ECON 509, by Dr. M. Zainal Chap 7-6

Page 4: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-4

Review: Population vs. Sample

a b c d

ef gh i jk l m n

o p q rs t u v w

x y z

Population Sample

b c

g i n

o r u

y

ECON 509, by Dr. M. Zainal Chap 7-7

Review: Why Sample?

Less time consuming than a census

Less costly to administer than a census

It is possible to obtain statistical results of a

sufficiently high precision based on samples.

ECON 509, by Dr. M. Zainal Chap 7-8

Page 5: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-5

Review: Sampling Techniques

Convenience

Sampling Techniques

Nonstatistical Sampling

Judgment

Statistical Sampling

Simple

Random Systematic

Stratified Cluster

ECON 509, by Dr. M. Zainal Chap 7-9

Review: Statistical Sampling

Items of the sample are chosen based on

known or calculable probabilities

Statistical Sampling

(Probability Sampling)

Systematic Stratified Cluster Simple Random

ECON 509, by Dr. M. Zainal Chap 7-10

Page 6: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-6

Simple Random Sampling

Every possible sample of a given size has an

equal chance of being selected

Selection may be with replacement or without

replacement

The sample can be obtained using a table of

random numbers or computer random number

generator

ECON 509, by Dr. M. Zainal Chap 7-11

Stratified Random Sampling

Divide population into subgroups (called strata)

according to some common characteristic

Select a simple random sample from each

subgroup

Combine samples from subgroups into one

Population

Divided

into 4

strata

Sample ECON 509, by Dr. M. Zainal Chap 7-12

Page 7: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-7

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

Systematic Random Sampling

N = 64

n = 8

k = 8

First Group

ECON 509, by Dr. M. Zainal Chap 7-13

Cluster Sampling

Divide population into several “clusters,” each

representative of the population

Select a simple random sample of clusters

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. Randomly selected

clusters for sample

ECON 509, by Dr. M. Zainal Chap 7-14

Page 8: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-8

Examples of poor samplings

The technique of sampling has been widely

used, both properly and improperly, in the area of

politics.

During the 1936 presidential race where the

Literary Digest predicted Alf Landon to win the

election over Franklin D. Roosevelt.

ECON 509, by Dr. M. Zainal Chap 7-15

Sampling Error

So far, we have stressed the benefits of drawing

a sample from a population.

However, in statistics, as in life, there's no such

thing as a free lunch.

By sampling, we expose ourselves to errors that

can lead to inaccurate conclusions about the

population.

The type of error that a statistician is most

concerned about is called sampling error.

ECON 509, by Dr. M. Zainal Chap 7-16

Page 9: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-9

Sampling Error

Sample Statistics are used to estimate

Population Parameters

ex: X is an estimate of the population mean, μ

Problems:

Different samples provide different estimates of

the population parameter

Sample results have potential variability, thus

sampling error exits

ECON 509, by Dr. M. Zainal Chap 7-17

Sampling Error

As the entire population is rarely measured, the

sampling error cannot be directly calculated.

With inferential statistics, we'll be able to assign

probabilities to certain amounts of sampling error

later.

It occurs when we select a sample that is not a

perfect match to the entire population.

Sampling errors are a small price to pay to avoid

measuring an entire population.

ECON 509, by Dr. M. Zainal Chap 7-18

Page 10: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-10

Sampling Error

One way to reduce the sampling error of a

statistical study is to increase the size of the

sample.

In general, the larger the sample size, the

smaller the sampling error.

If you increase the sample size until it reaches

the size of the population, then the sampling

error will be reduced to 0.

But in doing so, we lose the benefits of sampling.

ECON 509, by Dr. M. Zainal Chap 7-19

Calculating Sampling Error

Sampling Error:

The difference between a value (a statistic)

computed from a sample and the corresponding

value (a parameter) computed from a population

Example: (for the mean)

where:

μ - xError Sampling

mean population μ

mean samplex

ECON 509, by Dr. M. Zainal Chap 7-20

Page 11: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-11

Review

Population mean: Sample Mean:

N

xμ i

where:

μ = Population mean

x = sample mean

xi = Values in the population or sample

N = Population size

n = sample size

n

xx

i

ECON 509, by Dr. M. Zainal Chap 7-21

Example

If the population mean is μ = 98.6 degrees

and a sample of n = 5 temperatures yields a

sample mean of = 99.2 degrees, then the

sampling error is

degrees0.698.699.2μx

x

ECON 509, by Dr. M. Zainal Chap 7-22

Page 12: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-12

Sampling Errors

Different samples will yield different sampling

errors

The sampling error may be positive or negative

( may be greater than or less than μ)

The expected sampling error decreases as the

sample size increases

x

ECON 509, by Dr. M. Zainal Chap 7-23

Sampling Distribution

A sampling distribution is a

distribution of the possible values of

a statistic for a given size sample

selected from a population

ECON 509, by Dr. M. Zainal Chap 7-24

Page 13: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-13

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

ECON 509, by Dr. M. Zainal Chap 7-25

.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

214

24222018

N

xμ i

2.236N

μ)(xσ

2

i

ECON 509, by Dr. M. Zainal Chap 7-26

Page 14: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-14

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

16 possible samples

(sampling with

replacement)

Now consider all possible samples of size n=2

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

(continued)

16 Sample

Means

ECON 509, by Dr. M. Zainal Chap 7-27

Developing a Sampling Distribution

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 24 0

.1

.2

.3

P(x)

x

Sample Means

Distribution 16 Sample Means

_

(continued)

(no longer uniform) ECON 509, by Dr. M. Zainal Chap 7-28

Developing a Sampling Distribution

Page 15: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-15

Summary Measures of this Sampling Distribution:

(continued)

2116

24211918

N

xμ i

x

1.5816

21)-(2421)-(1921)-(18

N

)μx(σ

222

2

xi

x

ECON 509, by Dr. M. Zainal Chap 7-29

Developing a Sampling Distribution

Comparing the Population with its Sampling 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 _

1.58σ 21μ xx 2.236σ 21μ

Sample Means Distribution n = 2

ECON 509, by Dr. M. Zainal Chap 7-30

Page 16: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-16

For any population,

the average value of all possible sample means computed from

all possible random samples of a given size from the population

is equal to the population mean:

The standard deviation of the possible sample means

computed from all random samples of size n is equal to the

population standard deviation divided by the square root of the

sample size:

Properties of a Sampling Distribution

μμx

n

σσx

Theorem 1

Theorem 2

ECON 509, by Dr. M. Zainal Chap 7-31

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

x

μμx n

σσx

Theorem 3

ECON 509, by Dr. M. Zainal Chap 7-32

Page 17: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-17

z-value for Sampling Distribution of x

Z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

σ

n

σ

μ)x(z

x

ECON 509, by Dr. M. Zainal Chap 7-33

Finite Population Correction

Apply the Finite Population Correction if:

the sample is large relative to the population

(n is greater than 5% of N)

and…

Sampling is without replacement

Then

1N

nN

n

σ

μ)x(z

ECON 509, by Dr. M. Zainal Chap 7-34

Page 18: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-18

Normal Population

Distribution

Normal Sampling

Distribution

(has the same mean)

Sampling Distribution Properties

The sample mean is an unbiased estimator

x

x

μμx

μ

ECON 509, by Dr. M. Zainal Chap 7-35

The sample mean is a consistent estimator

(the value of x becomes closer to μ as n increases):

Sampling Distribution Properties

Larger

sample size

Small

sample size

x

(continued)

nσ/σx

μ

x

x

Population

As n increases,

decreases

ECON 509, by Dr. M. Zainal Chap 7-36

Page 19: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-19

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

…and the sampling distribution will have

and μμx n

σσx

Theorem 4

ECON 509, by Dr. M. Zainal Chap 7-37

n↑

Central Limit Theorem

As the

sample

size gets

large

enough…

the sampling

distribution

becomes

almost normal

regardless of

shape of

population

xECON 509, by Dr. M. Zainal Chap 7-38

Page 20: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-20

Population Distribution

Sampling Distribution

(becomes normal as n increases)

Central Tendency

Variation

(Sampling with replacement)

x

x

Larger

sample

size

Smaller

sample size

If the Population is not Normal (continued)

Sampling distribution

properties:

μμx

n

σσx

μ

ECON 509, by Dr. M. Zainal Chap 7-39

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 is

sufficient

For normal population distributions, the

sampling distribution of the mean is always

normally distributed

ECON 509, by Dr. M. Zainal Chap 7-40

Page 21: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-21

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?

ECON 509, by Dr. M. Zainal Chap 7-41

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 = μ = 8

…and standard deviation

(continued)

x

0.536

3

n

σσx

ECON 509, by Dr. M. Zainal Chap 7-42

Page 22: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-22

Example

Solution (continued) -- find z-scores:

(continued)

0.31080.4)zP(-0.4

363

8-8.2

μ- μ

363

8-7.8P 8.2) μ P(7.8

x

x

z 7.8 8.2 -0.4 0.4

Sampling

Distribution

Standard Normal

Distribution .1554

+.1554

Population

Distribution

? ?

? ?

? ? ? ? ?

? ? ? Sample Standardize

8μ 8μx 0μz x x

ECON 509, by Dr. M. Zainal Chap 7-43

Population Proportions, π

π = the proportion of the population having some characteristic

Sample proportion ( p ) provides an estimate of π :

If two outcomes, p has a binomial distribution

size sample

sampletheinsuccessesofnumber

n

xp

ECON 509, by Dr. M. Zainal Chap 7-44

Page 23: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-23

Sampling Distribution of p

Approximated by a

normal distribution if:

where

and

(where π = population proportion)

Sampling Distribution P( p )

.3

.2

.1

0 0 . 2 .4 .6 8 1

p

πμp n

π)π(1σp

5π)n(1

5nπ

ECON 509, by Dr. M. Zainal Chap 7-45

z-Value for Proportions

If sampling is without replacement

and n is greater than 5% of the

population size, then must use

the finite population correction

factor:

1N

nN

n

π)π(1σp

n

π)π(1

πp

σ

πpz

p

Standardize p to a z value with the formula:

ECON 509, by Dr. M. Zainal Chap 7-46

Page 24: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-24

Example

If the true proportion of voters who support

Proposition A is π = .4, what is the probability

that a sample of size 200 yields a sample

proportion between .40 and .45?

i.e.: if π = .4 and n = 200, what is

P(.40 ≤ p ≤ .45) ?

ECON 509, by Dr. M. Zainal Chap 7-47

Example

if π = .4 and n = 200, what is

P(.40 ≤ p ≤ .45) ?

(continued)

.03464200

.4).4(1

n

π)π(1σp

1.44)zP(0

.03464

.40.45z

.03464

.40.40P.45)pP(.40

Find :

Convert to

standard

normal:

ECON 509, by Dr. M. Zainal Chap 7-48

Page 25: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-25

Example

z .45 1.44

.4251

Standardize

Sampling Distribution Standardized

Normal Distribution

if π = .4 and n = 200, what is

P(.40 ≤ p ≤ .45) ?

(continued)

Use standard normal table: P(0 ≤ z ≤ 1.44) = .4251

.40 0 p

ECON 509, by Dr. M. Zainal Chap 7-49

Chapter Summary

Discussed sampling error

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

Discussed sampling from finite populations

ECON 509, by Dr. M. Zainal Chap 7-50

Page 26: Department of Economics Business StatisticsChapter 7 Student Lecture Notes 7-18 Normal Population Distribution Normal Sampling Distribution (has the same mean) Sampling Distribution

QMIS 220, by Dr. M. Zainal

Chapter 7 Student Lecture Notes 7-26

Copyright

The materials of this presentation were mostly

taken from the PowerPoint files accompanied

Business Statistics: A Decision-Making Approach,

7e © 2008 Prentice-Hall, Inc.