audhesh paswan, ph.d. determining the sampling plan audhesh paswan, ph.d. university of north texas

Audhesh Paswan, Ph.D.

Determining the Sampling Plan


University of North Texas


Basic Concepts in Samples Population - the entire group under study as

specified by the objectives of the research project.

Sample - a subset of the population that should represent that entire group.

Sample unit - the basic level of investigation (e.g., household, individual).

Census - an accounting of the complete population.


Basic Concepts in Sampling

Sampling error - error in a survey that occurs because a sample is used (caused by two factors).• The method of sample selection• The size of the sample

Sample frame - a list of the population of interest. Sample frame error - the degree to which a

sample frame fails to account for all of the population.


Examples of sample frame errors

Phone book Yellow pages Any incomplete population lists


Why sample?

A sample is almost always more desirable than a census

Population size and expense. Cannot analyze the huge amount of

data generated by a census.


Basic Sampling Methods

Probability samples - members of the population have a known chance (I.e., probability) of being selected into the sample.

Nonprobability samples - the chances (I.e., probabilities) of selecting members from the population of interest into the sample are unknown.


Probability Methods - Simple Random Sampling

Probability is known and equal for all members of the population.P(selection)=(sample size)/(population

size) The "Blind Draw" Method The Table of Random Numbers

Method


Advantages and Disadvantages

Advantages• Derives unbiased estimates• Valid representation of the population

Disadvantages• Must pre-designate each population

member.• May be difficult to obtain a complete listing.• May be too cumbersome


Systematic Sampling One of most prevalent types used Advantage: "economic efficiency" (i.e., quick

and easy). It employs a random starting point Every kth element in the population is designated

for inclusion in the sample (after a random start). Create a sample that is almost identical in

quality to simple random sampling.


Systematic Sample

Example 1: Sample the population of phone customers in Denton by taking every 10th number in the phone book. (Be sure to start randomly on one of the first 10 numbers.)

Example 2: Sample every 5th customer

Skip interval = (population list size)/(sample size)


How to Take a Systematic Sample

Step 1: Identify a listing of the population that contains an acceptable level of sample frame error.

Step 2: Compute the skip interval.

Step 3: Using random number(s), determine a starting position.

Step 4: Apply the skip interval.

Step 5: Treat the list as "circular."


Cluster Sampling

Population is divided into subgroups (cluster)

Each cluster represents the entire population.

Must identify clusters that are identical to the population and to each other.

The parent population is divided into mutually exclusive and exhaustive subsets.


Cluster Sampling

Subgroups should be heterogeneous within and

homogeneous between. (i.e., Subsets should

each look representative of the total population.)

Advantages: less cost to obtain a sample; good

for personal interviews (proximity)

Limitations: difficult to find subsets that truly

meet the criteria mentioned above; lower

statistical efficiency (higher error)


Area Sampling as a Form of Cluster Sampling

Population subdivided into areas (e.g., cities or

neighborhoods)

One-step approach - one area is selected

randomly; perform a census of the cluster

Two-step approach

• Step 1: Select a random sample of clusters.

• Step 2: Randomly select individuals within the

clusters.


How to Take an Area Sampling Using Subdivisions

Step 1: Determine the geographic area to be surveyed, and identify its subdivisions. Each subdivision should be highly similar to all others.

Step 2: Decide on the use of one-step or two-step cluster sampling.

Step 3: Using random numbers, select the subdivisions to be sampled (Assuming two-step).

Step 4: Using some probability method of sample selection, select the members of each chosen subdivision to be included in the sample.


Stratified Sampling Separates the population into different subgroups

and then samples all of the subgroups. Does not assume the population has a "normal"

distribution. Addresses "skewed" distribution problems. Weighted mean

Meanpopulation = (meanA)(proportionA)+ (meanB)(proportionB)


Stratified Sample

A probability sample distinguished by a two-step procedure:1 Divide the population into mutually exclusive and

collectively exhaustive subsets.2 Take a simple random sample of elements from

each subset (independently). The subsets are called “strata”. Each population member can be assigned to

one and only one stratum.


Stratified Sampling

Advantages:• Produces a more concentrated distribution of

estimates (leads to more precise statistics and smaller sampling error); fewer possible sample means that deviate widely from the true population mean.

• Can reduce variation within each stratum, which reduces the error of the estimate

• Guarantees representation of certain subgroups of interest.

Limitation: cost of sampling several strata


Bases for Stratification

Strata should be divided by a known characteristic

that is expected to be related to the characteristic of

interest.

Example: If we are interested in magazine

readership, we can stratify on the basis of education

level. This should result in less variation within each

stratum.

The strata should be homogeneous within and

heterogeneous between groups.


How to Take a Stratified Sample

Step 1: Be assured that the population's distribution for some key

factor is not bell-shaped and that separate populations exist.

Step 2: Use this factor or some surrogate variable to divide the

population into strata consistent with the separate sub-populations

identified.

Step 3: Select a probability sample from each stratum

Step 4: Examine each stratum for managerially relevant differences.

Step 5: If strata sample sizes are not proportionate to the stratum

sizes in the population, use the weighted mean formula to

estimate the population value(s).


Nonprobability Samples

Convenience samples - drawn at the convenience of the interviewer.

Judgement samples - requires and "educated guess" as to who should represent the population.

Referral samples - a.k.a. "snowball samples" Quota samples - a specified quota for various

types of individuals to be interviewed is established.


Stratified versus Quota Sample

Similarities:• Population is divided into segments (strata).• Elements are selected from each segment.

Key Difference:• Stratified sampling uses probability methods.• Quota samples are based on a researcher’s

judgment.• Therefore, stratified sampling allows the

establishment of the sampling distribution, confidence intervals and statistical tests.


Developing a Sample Plan

Step 1: Define the relevant population

Step 2: Obtain sample frame

Step 3: Design the sample plan

Step 4: Access the population

Step 5: Draw the sample

Step 6: Validate the sample

Step 7: Resample, if necessary.


Determining Sample Size


Sample Accuracy

How close the sample’s profile is to the true population’s profile

Sample size is not related to representativeness,

Sample size is related to accuracy


Methods of Determining Sample Size

Compromise between what is theoretically perfect and what is practically feasible.

Remember, the larger the sample size, the more costly the research.

Why sample one more person than necessary?



Arbitrary• Rule of Thumb (ex. A sample should be at

least 5% of the population to be accurate• Not efficient or economical

Conventional• Follows that there is some “convention” or

number believed to be the right size• Easy to apply, but can end up with too small or

too large of a sample



Cost Basis• based on budgetary constraints

Statistical Analysis• certain statistical techniques require certain

number of respondents Confidence Interval

• theoretically the most correct method


Notion of Variability

Great variabilit

y

Little variability

Mean


Notion of Variability

Standard Deviation• approximates the average distance away

from the mean for all respondents to a specific question

• indicates amount of variability in sample• ex. compare a standard deviation of 500

and 1000, which exhibits more variability?


Measures of Variability

Standard Deviation: indicates the degree of variation or diversity in the values in such as way as to be translatable into a normal curve distribution

Variance = (x-x)2/ (n-1) With a normal curve, the midpoint (apex) of

the curve is also the mean and exactly 50% of the distribution lies on either side of the mean.

i


Normal Curve and Standard Deviation

Number ofstandard

deviationsfrom the

mean

Percent ofarea underthe curve

Percent ofarea to theright or left

+/- 1.00 st dev 68% 16%

+/- 1.64 st dev 90% 5%

+/- 1.96 st dev 95% 2.5%

+/- 2.58 st dev 99% 0.5%


Notion of Sampling Distribution

The sampling distribution refers to what would be found if the researcher could take many, many independent samples

The means for all of the samples should align themselves in a normal bell-shaped curve

Therefore, it is a high probability that any given sample result will be close to but not exactly to the population mean.


Midpoint

(mean)

Normal, bell-shaped curve


Notion of Confidence Interval

A confidence interval defines endpoints based on knowledge of the area under a bell-shaped curve.

Normal curve• 1.96 times the standard deviation theoretically

defines 95% of the population• 2.58 times the standard deviation theoretically

defines 99% of the population


Notion of Confidence Interval

Example• Mean = 12,000 miles• Standard Deviation = 3000 miles

We are confident that 95% of the respondents’ answers fall between 6,120 and 17,880 miles 12,000 + (1.96 * 3000) = 17,880 12,000 - (1.96 * 3000) = 6.120


Notion of Standard Error of a Mean Standard error is an indication of how far away

from the true population value a typical sample result is expected to fall.

Formula• S X = s / (square root of n)

• S p = Square root of {(p*q)/ n}– where S p is the standard error of the percentage

– p = % found in the sample and q = (100-p)

– S X is the standard error of the mean

– s = standard deviation of the sample

– n = sample size


Computing Sample Size Using The Confidence Interval Approach

To compute sample size, three factors need to be considered: • amount of variability believed to be in the

population• desired accuracy• level of confidence required in your

estimates of the population values


Determining Sample Size Using a Mean

Formula: n = (pqz2)/e2

Formula: n = (s2z2)/e2

Where• n = sample size• z = level of confidence (indicated by the number of

standard errors associated with it)• s = variability indicated by an estimated standard

deviation• p = estimated variability in the population• q = (100-p)• e = acceptable error in the sample estimate of the

population


Determining Sample Size Using a Mean: An Example

95% level of confidence (1.96) Standard deviation of 100 (from

previous studies) Desired precision is 10 (+ or -) Therefore n = 384

• (1002 * 1.962) / 102


Practical Considerations in Sample Size Determination

How to estimate variability in the population• prior research• experience• intuition

How to determine amount of precision desired• small samples are less accurate• how much error can you live with?


Practical Considerations in Sample Size Determination

How to calculate the level of confidence desired• risk• normally use either 95% or 99%


Determining Sample Size

Higher n (sample size) needed when:• the standard error of the estimate is high

(population has more variability in the sampling distribution of the test statistic)

• higher precision (low degree of error) is needed (i.e., it is important to have a very precise estimate)

• higher level of confidence is required Constraints: cost and access


Notes About Sample Size

Population size does not determine sample size.

What most directly affects sample size is the variability of the characteristic in the population.• Example: if all population elements have

the same value of a characteristic, then we only need a sample of one!

audhesh paswan, ph.d. determining the sampling plan audhesh paswan, ph.d. university of north texas

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