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Statistics

Sampling and Sampling Distribution

- MeadWestvaco Corporation’s
products include textbook

paper, magazine paper, and

office products.

- MeadWestvaco’s internal consulting
group uses sampling to provide information that enables the company to obtain significant productivity benefits and remain competitive.

- Managers need reliable and accurate information about the timberlands and forests to evaluate the company’s ability to meet its future raw material needs.
- Data collected from sample plots throughout the forests are the basis for learning about the population of trees owned by the company.

- The Electronics Associates Sampling Problem
- Simple Random Sampling
- Point Estimation
- Introduction to Sampling Distributions
- Sampling Distribution of p
- Properties of Point Estimators
- Other Sampling Methods

Statistical Inference

- The purpose of statistical inference is to obtain
- information about a population from
- information contained in a sample.

- A population is the set of all the elements of
- interest.

- A sample is a subset of the population.

Statistical Inference

- The sample results provide only estimates of
- the values of the population characteristics.

- With proper sampling methods, the sample
- results can provide “good” estimates of the
- population characteristics.

- A parameter is a numerical characteristic of a
- population.

- Often the cost of collecting information from a sample is substantially less than from a population,
- Especially when personal interviews must be conducted to collect the information.

- Finite populations are often defined by lists such as:
- Organization membership roster
- Credit card account numbers
- Inventory product numbers

- A simple random sample of size n from a
- finite population of size N is a sample
- selected such that each possible sample of
- size n has the same probability of being
- selected.

- Replacing each sampled element before
- selecting subsequent elements is called
- sampling with replacement.

- Sampling without replacement is the
- procedure used most often.

- Random Numbers: the numbers in the table are random, these four-digit numbers are equally likely.

- Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely.

- A simple random sample from an infinite
- population is a sample selected such that the
- following conditions are satisfied.
- Each element selected comes from the same
- population.
- Each element is selected independently.

- In the case of infinite populations, it is
- impossible to obtain a list of all elements
- in the population.

- The random number selection procedure
- cannot be used for infinite populations.

- We refer to as the point estimator of the
- population mean .

Point Estimation

- In point estimation we use the data from the
- sample to compute a value of a sample statistic
- that serves as an estimate of a population
- parameter.

- is the point estimator of the population
- proportion p.

Point Estimation

- sis the point estimator of the population standard
- deviation .

- Example: to estimate the population mean, the population standard deviation and population proportion.

- When the expected value of a point estimator
- is equal to the population parameter, the point
estimator is said to be unbiased.

- The absolute value of the difference between an
unbiased point estimate and the corresponding

population parameter is called the

sampling error.

- Sampling error is the result of using a subset
- of the population (the sample), and not the
- entire population.

- Statistical methods can be used to make
- probability statements about the size of the
- sampling error.

for sample mean

for sample standard deviation

for sample proportion

Sampling Error

- The sampling errors are:

St. Andrew’s College

receives 900 applications

annually from

prospective students.

The application form

contains a variety of information

including the individual’s scholastic aptitude

test (SAT) score and whether or not the

individual desires on-campus housing.

The director of admissions

would like to know the

following information:

- the average SAT score
for the 900 applicants,

and

- the proportion of applicants that want to live on campus.

We will now look at three

alternatives for obtaining

The desired information.

- Conducting a census of
the entire 900 applicants

- Selecting a sample of 30
applicants, using a random number table

- Selecting a sample of 30 applicants, using Excel

Conducting a Census

- If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3.
- We will assume for the moment that conducting a census is practical in this example.

- Population Mean SAT Score

- Population Standard Deviation for SAT Score

- Population Proportion Wanting On-Campus Housing

Simple Random Sampling

- Now suppose that the necessary data on the
current year’s applicants were not yet entered

in the college’s database.

- Furthermore, the Director of Admissions must
- obtain estimates of the population parameters of
- interest for a meeting taking place in a few hours.

Simple Random Sampling

- Now suppose that the necessary data on the current year’s applicants were not yet entered in the college’s database.

- Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours.

Simple Random Sampling

- The applicants were numbered, from 1 to 900, as their applications arrived.

- Taking a Sample of 30 Applicants

- Because the finite population has 900 elements, we will need 3-digit random numbers to randomly select applicants numbered from 1 to 900.
- We will use the last three digits of the 5-digit random numbers in the third column of the textbook’s random number table , and continue into the fourth column as needed.

- Taking a Sample of 30 Applicants

- The numbers we draw will be the numbers of
- the applicants we will sample unless the
- random number is greater than 900 or
- the random number has already been used.
- We will continue to draw random numbers

- (We will go through all of column 3 and
- part of column 4 of the random number table,
- encountering in the process five numbers
- greater than 900 and one duplicate, 835.)

- Use of Random Numbers for Sampling

3-Digit

Random Number

Applicant

Included in Sample

744

No. 744

436

No. 436

865

No. 865

790

No. 790

835

No. 835

902

Number exceeds 900

190

No. 190

836

No. 836

. . . and so on

- Sample Data

Random

Number

SAT

Score

Live On-

Campus

No.

Applicant

1744 Conrad Harris1025 Yes

2436 Enrique Romero 950 Yes

3865 Fabian Avante1090 No

4790 Lucila Cruz1120 Yes

5835 Chan Chiang 930 No

. . . . .

. . . . .

30 498 Emily Morse 1010 No

- Taking a Sample of 30 Applicants

- Computers can be used to generate random
- numbers for selecting random samples.

- For example, Excel’s function
- = RANDBETWEEN(1,900)
- can be used to generate random numbers
- between 1 and 900.

- Then we choose the 30 applicants
- corresponding to the 30 smallest random
- numbers as our sample.

- as Point Estimator of

–

- pas Point Estimator of p

- s as Point Estimator of

Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.

= Sample mean

SAT score

= Sample pro-

portion wanting

campus housing

Summary of Point Estimates

Obtained from a Simple Random Sample

Population

Parameter

Parameter

Value

Point

Estimator

Point

Estimate

m = Population mean

SAT score

990

997

80

s = Sample std.

deviation for

SAT score

75.2

s = Population std.

deviation for

SAT score

.72

.68

p = Population pro-

portion wanting

campus housing

- Example: Relative Frequency Histogram of Sample
Mean Values from 500 Simple Random Samples of

30 each.

- Example: Relative Frequency Histogram of Sample
Proportion Values from 500 Simple Random

Samples of 30 each.

The value of is used to

make inferences about

the value of m.

The sample data

provide a value for

the sample mean .

Sampling Distribution of

- Process of Statistical Inference

A simple random sample

of n elements is selected

from the population.

Population

with mean

m = ?

E( ) =

Sampling Distribution of

The sampling distribution of is the probability

distribution of all possible values of the sample

mean .

Expected Value of

where:

= the population mean

Sampling Distribution of

- Standard Deviation of

InfinitePopulation

Finite Population

- A finite population is treated as being
- infinite if n/N< .05.

Sampling Distribution of

- is the finite correction factor.

- is referred to as the standard error of the mean.

Form of the Sampling Distribution

of

- If we use a large (n> 30) simple random sample,
- the central limit theorem enables us to conclude
- that the sampling distribution of can be
- approximated by a normal distribution.

- When the simple random sample is small (n < 30),
- the sampling distribution of can be considered
- normal only if we assume the population has a
- normal distribution.

- Illustration of The Central Limit Theorem

- A Comparison of The Sampling Distributions of Sample Mean for Simple Random Samples of n = 30 and n = 100.

Sampling Distribution offor SAT Scores

Sampling

Distribution

of

Sampling Distribution offor SAT Scores

What is the probability that a simple

random sample of 30 applicants will provide

an estimate of the population mean SAT score

that is within +/-10 of the actual population mean ?

In other words, what is the probability that

will be between 980 and 1000?

Sampling Distribution offor SAT Scores

Step 1: Calculate the z-value at the upper

endpoint of the interval.

z = (1000 - 990)/14.6= .68

Step 2: Find the area under the curve to the

left of the upper endpoint.

P(z< .68) = .7517

Sampling Distribution offor SAT Scores

Cumulative Probabilities for

the Standard Normal Distribution

Sampling Distribution offor SAT Scores

Sampling

Distribution

of

Area = .7517

990

1000

Sampling Distribution offor SAT Scores

P(z< -.68) = P(z> .68)

= 1 - P(z< .68)

= 1 - . 7517

= .2483

Step 3: Calculate the z-value at the lower

endpoint of the interval.

z = (980 - 990)/14.6= - .68

Step 4: Find the area under the curve to the

left of the lower endpoint.

Sampling Distribution offor SAT Scores

Sampling

Distribution

of

Area = .2483

980

990

Sampling Distribution offor SAT Scores

P(-.68 <z< .68) = P(z< .68) - P(z< -.68)

= .7517 - .2483

= .5034

Step 5: Calculate the area under the curve

between the lower and upper endpoints

of the interval.

The probability that the sample mean SAT score will be between 980 and 1000 is:

P(980 << 1000) = .5034

Sampling Distribution offor SAT Scores

Sampling

Distribution

of

Area = .5034

980

990

1000

Relationship Between the Sample Size and the Sampling Distribution of

- E( ) = m regardless of the sample size.
- in our example, E( ) remains at 990.

- Suppose we select a simple random sample
of 100 applicants instead of the 30 originally

considered.

Relationship Between the Sample Size and the Sampling Distribution of

- Whenever the sample size is increased, the
standard error of the mean is decreased.

With the increase in the sample size to n = 100,

the standard error of the mean is decreased to:

Relationship Between the Sample Size and the Sampling Distribution of

With n = 100,

With n = 30,

Relationship Between the Sample Size and the Sampling Distribution of

- We follow the same steps to solve for
- P(980 << 1000)
- when n = 100 as we showed earlier when n = 30.

- Recall that when n = 30,
- P(980 << 1000) = .5034.

Relationship Between the Sample Size and the Sampling Distribution of

- Because the sampling distribution with n = 100
- has a smaller standard error, the values of have
- less variability and tend to be closer to the
- population mean than the values of with n = 30.

- Now, with n = 100, P(980 << 1000) = .7888.

Relationship Between the Sample Size and the Sampling Distribution of

Sampling

Distribution

of

Area = .7888

980

990

1000

- Example: Relative Frequency Histogram of Sample Proportion Values from 500 Simple Random Samples of 30 each.

The sample data

provide a value for the

sample proportion .

The value of is used

to make inferences

about the value of p.

Sampling Distribution of p

- Making Inferences about a Population Proportion

A simple random sample

of n elements is selected

from the population.

Population

with proportion

p = ?

Sampling Distribution of

The sampling distribution of p is the probability

distribution of all possible values of the sample

proportion p .

Expected Value of p

where:

p = the population proportion

is referred to as the standard error of the proportion.

Sampling Distribution of

Standard Deviation of p

Infinite Population

Finite Population

- The sampling distribution of can be
- approximatedby a normal distribution whenever
- the sample size is large.

The sample size is considered large whenever these conditions are satisfied:

Form of the Sampling Distribution of

and

np> 5

n(1 – p) > 5

- For values of p near .50, sample sizes as
- small as 10 permit a normal approximation.

- With very small (approaching 0) or very
- large (approaching 1) values of p, much
- larger samples are needed.

Form of the Sampling Distribution of

Sampling Distribution of

- Example: St. Andrew’s College

Recall that 72% of the prospective students

applying to St. Andrew’s College desire

on-campus housing.

Sampling Distribution of

- Example: St. Andrew’s College

What is the probability that a simple random

sample of 30 applicants will provide an estimate

of the population proportion of applicant

desiring on-campus housing that is within plus or

minus .05 of the actual population proportion?

Sampling Distribution of

For our example, with n = 30 and p = .72,

the normal distribution is an acceptable

approximation because:

np= 30(.72) = 21.6 > 5

and

n(1 - p) = 30(.28) = 8.4 > 5

Sampling

Distribution

of

Sampling Distribution of

Sampling Distribution of

Step 1: Calculate the z-value at the upper

endpoint of the interval.

z = (.77 - .72) /.082 = .61

Step 2: Find the area under the curve to the

left of the upper endpoint.

P(z< .61) = .7291

Sampling Distribution of

Cumulative Probabilities for

the Standard Normal Distribution

Sampling

Distribution

of

Sampling Distribution of

Area = .7291

.72

.77

Sampling Distribution of

Step 3: Calculate the z-value at the lower

endpoint of the interval.

z = (.67 - .72) /.082 = - .61

Step 4: Find the area under the curve to the

left of the lower endpoint.

P(z< -.61) = P(z> .61)

= 1 - P(z< .61)

= 1 - . 7291

= .2709

Sampling

Distribution

of

Sampling Distribution of

Area = .2709

.67

.72

P(.67 << .77) = .4582

Sampling Distribution of

Step 5: Calculate the area under the curve between

the lower and upper endpoints of the interval.

P(-.61 <z< .61) = P(z< .61) - P(z< -.61)

= .7291 - .2709

= .4582

The probability that the sample proportion of

applicants wanting on-campus housing will be

within +/-.05 of the actual population proportion :

Sampling

Distribution

of

Sampling Distribution of

Area = .4582

.67

.72

.77

- Notations:
- θ = the population parameter of interest.
For example, population mean, population standard deviation, population proportion, and so on.

^

- Notations:
- θ = the sample statistic or point estimator of θ .
Represents the corresponding sample statistic such as the sample mean, sample standard deviation, and sample proportion.

- The notation θ is the Greek letter theta.
- the notation θ is pronounced “theta-hat.”

^

^

Properties of Point Estimators

- Before using a sample statistic as a point estimator, statisticians check to see whether the sample statistic has the following properties associated with good point estimators.

- Unbiased

- Efficiency

- Consistency

Properties of Point Estimators

- Unbised

If the expected value of the sample statistic is equal to the population parameter being estimated, the sample statistic is said to be an unbiased estimator of the population parameter.

- Unbised
The sample statistic θis unbiased estimator of the population parameter θ if

^

^

E(θ)=θ

where

^

E(θ)=the expected value of the sample statistic θ

- Examples of Unbiased and Biased Point Estimators

Properties of Point Estimators

- Efficiency

Given the choice of two unbiased estimators of the same population parameter, we would prefer to use the point estimator with the smaller standard deviation, since it tends to provide estimates closer to the population parameter.

The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other.

- Example: Sampling Distributions of Two
Unbiased Point Estimators.

Properties of Point Estimators

- Consistency

A point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as the sample size becomes larger.

- Stratified Random Sampling(分層隨機抽樣)

- Cluster Sampling(部落抽樣）

- Systematic Sampling（系統抽樣）

- Convenience Sampling（便利抽樣）

- Judgment Sampling(判斷抽樣）

Stratified Random Sampling

- The population is first divided into groups of
- elements called strata.

- Each element in the population belongs to one
- and only one stratum.

- Best results are obtained when the elements
- withineach stratum are as much alike as possible
- (i.e. a homogeneous group).

- Diagram for Stratified Random Sampling

Stratified Random Sampling

- A simple random sample is taken from each
- stratum.

- Formulas are available for combining the
- stratum sample results into one population
- parameter estimate.

Stratified Random Sampling

- Advantage: If strata are homogeneous, this
- method is as “precise” as simple random
- sampling but with a smaller total sample size.

- Example: The basis for forming the strata
- might be department, location, age, industry type,
- and so on.

Cluster Sampling

- The population is first divided into separate
- groupsof elements called clusters.

- Ideally, each cluster is a representative small-
- scale version of the population (i.e.
- heterogeneous group).

- A simple random sample of the clusters is then
- taken.

- All elements within each sampled (chosen)
- cluster form the sample.

- Diagram for ClusterSampling

Cluster Sampling

- Example: A primary application is area
- sampling, where clusters are city blocks or
- other well-defined areas.

- Advantage: The close proximity of elements
- can be cost effective (i.e. many sample
- observations can be obtained in a short time).

- Disadvantage: This method generally requires a
- larger total sample size than simple or stratified
- random sampling.

Systematic Sampling

- If a sample size of n is desired from a population
- containing N elements, we might sample one
- element for every n/N elements in the population.

- We randomly select one of the first n/N elements
- from the population list.

- We then select every n/Nth element that follows in
- the population list.

Systematic Sampling

- This method has the properties of a simple
- random sample, especially if the list of the
- population elements is a random ordering.

Advantage: The sample usually will be easier to

identify than it would be if simple random

sampling were used.

Example: Selecting every 100th listing in a

telephone book after the first randomly selected

listing

Convenience Sampling

It is a nonprobability sampling technique.

Items are included in the sample without known

probabilities of being selected.

The sample is identified primarily by convenience.

Example: A professor conducting research might

use student volunteers to constitute a sample.

Convenience Sampling

- Advantage: Sample selection and data collection
- are relatively easy.

- Disadvantage: It is impossible to determine how
- representative of the population the sample is.

Judgment Sampling

- The person most knowledgeable on the subject of
- the study selects elements of the population that
- he or she feels are most representative of the
- population.

- It is a nonprobability sampling technique.

- Example: A reporter might sample three or four
- senators, judging them as reflecting the general
- opinion of the senate.

Judgment Sampling

- Advantage: It is a relatively easy way of
- selecting a sample.

- Disadvantage: The quality of the sample results
- depends on the judgment of the person selecting
- the sample.