Statistics

Statistics

Sampling and Sampling Distribution

1/101

STATISTICS in PRACTICE 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.

2/101

STATISTICS in PRACTICE 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.

3/101

Contents

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

4/101

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

Statistical Inference

A population is the set of all the elements of interest. A sample is a subset of the population.

5/101

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

A parameter is a numerical characteristic of a population.

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

Statistical Inference

6/101

The Electronics Associates Sampling Problem

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.

7/101

Simple Random Sampling:Finite Population

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

8/101


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.

9/101


Sampling without replacement is the procedure used most often.

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

10/101

Simple Random Sampling Random Numbers: the numbers in the table are

random, these four-digit numbers are equally likely.

11/101

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.

Simple Random Sampling:Infinite Population

12/101


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.

13/101


The random number selection procedure

cannot be used for infinite populations.

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

14/101

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.

Point Estimation

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

x

15/101

s is the point estimator of the population standard deviation .

Point Estimation

is the point estimator of the population proportion p.

p

16/101

Point Estimation Example: to estimate the population mean,

the population standard deviation and population proportion.

17/101

Sampling Error

The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error.

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

18/101

Sampling Error

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

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

19/101

Example: St. Andrew’s St. Andrew’s College

receives 900 applications annually from prospective students. The application form contains a variety of informationincluding the individual’s scholastic aptitudetest (SAT) score and whether or not the individual desires on-campus housing.

21/101

Example: St. Andrew’s The director of admissionswould like to know thefollowing information: the average SAT score for the 900 applicants, and the proportion of applicants that want to

live on campus.22/101

Example: St. Andrew’sWe 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

23/101

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.

24/101

990900ix

2( ) 80900ix

Conducting a Census

648 .72900p

Population Mean SAT Score

Population Standard Deviation for SAT Score

Population Proportion Wanting On-Campus Housing

25/101

Simple Random Sampling

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

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

26/101


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

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

entered in the college’s database.

27/101


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

28/101

Taking a Sample of 30 Applicants

Simple Random Sampling:Using a Random Number Table

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.29/101



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

until we have selected 30 applicants for our sample.

30/101


(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.)

31/101

Use of Random Numbers for Sampling


744436865790835902190836

. . . and so on

3-DigitRandom Number

ApplicantIncluded in Sample

No. 436No. 865No. 790No. 835

Number exceeds 900No. 190No. 836

No. 744

32/101

Sample Data


1 744 Conrad Harris 1025 Yes2 436 Enrique Romero 950 Yes3 865 Fabian Avante 1090 No

4 790 Lucila Cruz 1120 Yes5 835 Chan Chiang 930 No. . . . .

30 498 Emily Morse 1010 No

No.RandomNumber Applicant

SAT Score

Live On-Campus

. . . . .

33/101


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

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

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

Simple Random Sampling:Using a Computer

34/101

29,910 99730 30ix

x

2( ) 163,996 75.229 29ix x

s

Point Estimation

s as Point Estimator of

as Point Estimator of i i

i

wxx

w

– p as Point Estimator of p

.6820/30 p

35/101

Point Estimation

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

36/101

PopulationParameter

PointEstimator

PointEstimate

ParameterValue

= Population mean SAT score

990 997

= Population std. deviation for SAT score

80 s = Sample std. deviation for SAT score

75.2

p = Population proportion wanting campus housing

.72 .68

Summary of Point EstimatesObtained from a Simple Random Sample

= Sample mean SAT score

x

= Sample proportion wanting campus housing

p

37/101

Sampling Distribution Example: Relative Frequency Histogram of Sample Mean Values from 500 Simple Random Samples of 30 each.

38/101

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

39/101

Process of Statistical Inference

The value of is used tomake inferences aboutthe value of .

x The sample data provide a value forthe sample mean .x

A simple random sampleof n elements is selectedfrom the population.

Population with mean = ?

Sampling Distribution of i i

i

wxx

w

The sampling distribution of is the probabilitydistribution of all possible values of the sample mean .

Sampling Distribution of

where: = the population mean

E( ) = x

Expected Value of i i

i

wxx

w

i i

i

wxx

w

i i

i

wxx

w

i i

i

wxx

w

41/101

Finite Population Infinite Population

x n

N nN

( )1

x n

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

Standard Deviation ofx

Sampling Distribution of x

42/101

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

x

• is the finite correction factor.

)1/()( NnN

Sampling Distribution of x

43/101

Form of the Sampling Distribution of i i

i

wxx

w

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.

x

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.

x

44/101

Central Limit Theorem Illustration of The Central Limit Theorem

45/101

Relationship Between the Sample Size and the Sampling Distribution of Sample Mean

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

46/101

Sampling Distribution of for SAT Scores

i i

i

wxx

w

80 14.630x n

( ) 990E x x

SamplingDistributionof x

47/71

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?


i i

i

wxx

w

i i

i

wxx

w

48/101

Step 1: Calculate the z-value at the upper endpoint of the interval.

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

P(z < .68) = .7517

Step 2: Find the area under the curve to the left of the upper endpoint.


i i

i

wxx

w

49/101

Cumulative Probabilities for the Standard Normal Distribution

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09. . . . . . . . . . ..5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389. . . . . . . . . . .


i i

i

wxx

w

50/101

x990


14.6x

1000

Area = .7517


i i

i

wxx

w

51/101

Step 3: Calculate the z-value at the lower endpoint of the interval.

Step 4: Find the area under the curve to the left of the lower endpoint.

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

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

= .2483= 1 - . 7517= 1 - P(z < .68)


i i

i

wxx

w

52/101

x980 990

Area = .2483


14.6x


i i

i

wxx

w

53/101

Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval.

P(-.68 < z < .68) = P(z < .68) - P(z < -.68)= .7517 - .2483= .5034

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

P(980 < < 1000) = .5034x


i i

i

wxx

w

54/101

x1000980 990

Area = .5034


14.6x


i i

i

wxx

w

55/101

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 i i

i

wxx

w

E( ) = m regardless of the sample size.

in our example, E( ) remains at 990.

i i

i

wxx

w

i i

i

wxx

w

56/101


i

wxx

w

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:

8.010080

x n

σσ

xσ

57/101


i

wxx

w

( ) 990E x x

14.6x With n = 30,

8x With n = 100,

58/101

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

i

wxx

w

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

i i

i

wxx

w

i i

i

wxx

w

Relationship Between the Sample Size and the Sampling Distribution of

59/101

Relationship Between the Sample Size and the Sampling Distribution of

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

i

wxx

w

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.

i i

i

wxx

w

i i

i

wxx

w

i i

i

wxx

w

60/101

x1000980 990

Area = .7888


8x


i

wxx

w

61/101

Sampling Distribution Example: Relative Frequency Histogram of

Sample Proportion Values from 500 Simple Random Samples of 30 each.

62/101

A simple random sampleof n elements is selected

from the population.

Population with proportion

p = ?

Making Inferences about a Population Proportion

The sample data provide a value for thesample proportion .p

The value of is usedto make inferences

about the value of p.

p

Sampling Distribution of p

63/101

E p p( )

where:p = the population proportion

The sampling distribution of p is the probabilitydistribution of all possible values of the sampleproportion p .

Expected Value of p


64/101

pp p

nN nN

( )11

pp p

n ( )1

is referred to as the standard error of the proportion.

p


Finite Population Infinite Population

Standard Deviation of p

p

The sampling distribution of can be approximated by a normal distribution whenever the sample size is large.

p

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

np > 5 n(1 – p) > 5and

Form of the Sampling Distribution of p

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.

pForm of the Sampling Distribution of

67/101

Recall that 72% of the prospective students applying to St. Andrew’s College desire on-campus housing.

Example: St. Andrew’s College


68/101

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 applicantdesiring on-campus housing that is within plus orminus .05 of the actual population proportion?

p

69/101

For our example, with n = 30 and p = .72, the normal distribution is an acceptable approximation because:

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

np = 30(.72) = 21.6 > 5


p

SamplingDistribution

of p


082.30

)72.1(72.

p

72.)( pE

Step 1: Calculate the z-value at the upper endpoint of the interval.

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

P(z < .61) = .7291

Step 2: Find the area under the curve to the left of the upper endpoint.


72/101

Cumulative Probabilities for the Standard Normal Distribution

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09. . . . . . . . . . ..5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389. . . . . . . . . . .


73/101

.77.72

Area = .7291

p


of p.082p


74/101

Step 3: Calculate the z-value at the lower endpoint of the interval.

Step 4: Find the area under the curve to the left of the lower endpoint.

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

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

= .2709= 1 - . 7291= 1 - P(z < .61)


75/101

.67 .72

Area = .2709

p


of p.082p


76/101

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

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 :


77/101

.77.67 .72

Area = .4582

p


of p.082p


78/101

Point Estimators Notations: θ = the population parameter of interest. For example, population mean, population

standard deviation, population proportion, and so on.

^

79/101

Point Estimators 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.”

^

^

80/101

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.

Consistency

Efficiency

Unbiased

81/101


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

82/101

Properties of Point Estimators Unbised The sample statistic θ is unbiased estimator

of the population parameter θ if E(θ)= θ

^

where

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

^

83/101

Properties of Point Estimators Examples of Unbiased and Biased Point

Estimators

84/101


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.

Efficiency

85/101

Properties of Point Estimators Example: Sampling Distributions of Two Unbiased Point Estimators.

86/101


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.

Consistency

87/101

Other Sampling Methods 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）

88/101

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

Stratified Random Sampling

Each element in the population belongs to one and only one stratum. Best results are obtained when the elements within each stratum are as much alike as possible (i.e. a homogeneous group).

89/101

Stratified Random Sampling Diagram for Stratified Random Sampling

90/101

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.

91/101

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.

92/101

Cluster Sampling The population is first divided into separate groups of 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.

93/101

Cluster Sampling Diagram for Cluster Sampling

94/101

Cluster Sampling

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.

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

95/101

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.

96/101

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 97/101

Convenience Sampling It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.

Example: A professor conducting research might use student volunteers to constitute a sample.

The sample is identified primarily by convenience.

98/101

Advantage: Sample selection and data collection are relatively easy. Disadvantage: It is impossible to determine how representative of the population the sample is.

Convenience Sampling

99/101

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.

100/101

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.

101/101

Statistics

Documents

Transcript of Statistics