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Sampling and Sampling

Distributions

Biswo Poudel

KUSOM

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Why Take Samples?

To save money, time

To maximize information gleaned out of

limited resource

Often it might be the only option. If access to

the population is impossible, it could be the

only option. (how would you survey the

owners of old Omega watches in Nepal?)

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Census vs sampling

Question:When is taking census a better

option than taking a sample?

Answer:When omission of a group of

population is not tolerable for the researcher.

Example: all airplanes are tested thoroughly

because their performance is individually so

important.

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Frame

The target population from which the sample istaken.

This population list, map, directory or other

source used to represent the population is calledframe.

Can also be school list, trade association lists, listssold by list brokers.

Frames may have overregistration(includingmore than target population) orunderregistration.

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Random Vs Nonrandom Sampling

Random Sampling: Every unit of the population has thesameprobability of being selected into the sample. Forexample: lottery outcomes. This is also calledprobability sampling.

Nonrandom Sampling: Not every unit of the populationhas the same probability of being selected into thesample. This is also called nonprobability sampling.Assigning the probability of occurrence in nonrandom

sampling is impossible. Nonrandom sampling data are not amenable to

analysis by most of the statistical techniques.

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Random Sample Techniques

There are four basic random sample

techniques.

1. Simple Random Sample Technique : this is

the most elementary technique. Number each

unit of the frame from 1 to N. Select n items

out of that into sample by using some random

number generator.

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2. Stratified Random Sampling

In this, population is subdivided intononoverlapping subpopulations called strata.

The researcher then extracts random sample

from each subpopulation. It has potential for reducing sample error.

How to choose strata? (a) must be internallyhomogenous, externally must contrast with eachother. (b) do stratification by demographicvariables such as gender, socioeconomic class,geographic region, religion and ethnicity.

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Stratified random sampling(SRS) could be

either proportionate or disproportionate.

Proportionate SRS occurs when the

percentage of the sample taken from each

stratum is proportionate to the percentage

that each stratum is within the whole

population. If the Sampling is notproportionate, then it is disproportionate SRS.

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Example of proportionate SRS: Suppose we

are sampling population of Kathmandu.

Kathmandu has 30% Newars. Suppose you

have divided your population into strata

involving ethnicity. If you are taking a sample

of 100 people, then you want to make sure 30

Newars are in the sample.

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3. Systematic Sampling

Used because of its convenience and relative

ease of administration.

Every kth item is selected to produce a sample

of size n from a population of size N.

Value of k, sometimes called sampling cycle, is

given by .

For this to be useful, the source of population

elements is random.

n

Nk

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4. Cluster (Area) Sampling

Divide population into nonoverlapping areas

(Clusters) that are internally heterogenous.

Each cluster is, in theory, a microcosm of the

population.

For example, Chitwan could be a cluster, when

thinking of taking a sample of Nepal. Other

cities, districts, metropolitan areas can also

qualify as a cluster.

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After choosing clusters, the researcher either selects allelements from the cluster or randomly selectsindividual elements into the sample from the clusters.

Two stage sampling: when clusters are too big, andanother cluster is picked up from within a big cluster.

Advantage:cost, convenience. Since all data are pickedfrom one cluster, the movement cost is reduced.

Disadvantage: If the elements are similar, then thecluster sampling may be inefficient compared to simplerandom sampling. If all elements of a cluster are same,then it is not better than sampling one individual.

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Nonrandom Sampling techniques

Also called nonprobability techniques sincechance is not used to select elements from thesamples.

Four nonrandom sampling techniques arepresented here.

1. Convenience Sampling: elements for the

sample are selected for the convenience ofthe researcher. Researcher chooses samplesthat are readily available.

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2 Judgment sampling

Elements selected for the sample are chosen by

the judgment of the researcher.

Researchers often believe they can obtain right

sample by using their sound judgment.

Sampling errors are hard to determine because

the samples are put together nonrandomly.

Problems: judgement error might be in onedirection (introducing bias), unlikely to include

extreme elements

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3 Quota Sampling

In essence, similar to Stratified RandomSampling(SRS).

Certain population subclasses are used as

strata. Use nonrandom sampling technique to gather

data from each strata.

For example: one may go to a Newarcommunity (say in Sundhara , Lalitpur) andinterview people there until the quota is filled.

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Advantage: cost, easy

Disadvantage: it is essentiallya nonrandom

sampling.

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4. Snowball Sampling

Survey subjects are selected based onreferrals from other survey respondents.

First pick a person who fits the profile of

subject wanted for the study. Then ask thisperson to refer others who have similarprofile.

Advantage: survey objects are identifiedcheaply and efficiently.

Disadvantage: this is nonrandom.

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Sampling Errors and Nonsampling

Errors

Sampling errors: error that occurs when thesample is not representative of thepopulation.

Nonsampling errors: all other errors such asmissing data, recording errors, in putprocessing errors, analysis errors, responseerrors, measurement instrument causederrors, defective questionairre error, poorconcept errors etc etc.

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Sample Mean and Sample Proportion

Whenever a research produces measurable data such as

weight, distance, time and income, the sample mean is often

the statistics of choice. If the research results in countable

items such as how many people in a sample choose Coca Cola,

the sample proportion is often the statistics of choice.

Sample Proportion ( )

sampletheinitemsofnumbern

sticscharacterithehavethatsampleainitemsofx

wheren

xp

#

p

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Distribution of Sample Mean

Central Limit Theorem: If samples of size nare drawn randomly from a population thathas a mean of and standard deviation ,

then the sample means are approximatelynormally distributed for sufficiently largesample sizes (greater than 30) regardless ofthe shape of the population distribution. If the

population is normally distributed, the samplemeans are normally distributed for any sizesample.

2

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Example:

Suppose during any hour in a large

department store, the average number of

shoppers is 448, with a standard deviation of

21 shoppers. What is the probability that arandom sample of 49 different shopping hours

will yield a sample mean between 441 and

446 shoppers? Answer: problem is to determine )446441( xP

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Notice that

This leads to the probability of the value beingbetween 441 and 446 to be 0.4901-

0.2486=0.2415; i.e. 24.15%.

2486.0;4901.0

67.0

49

21

448446

33.2

49

21

448441

valuesz

z

z

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Correction for finite sample

If the sample is taken from a finite population

of size N, then the z-value for sample size n

has to be calculated using the following

formula:

1

N

nN

n

xz

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Example

A production companys 350 hourly employees

average 37.6 years of age, with a standarddeviation of 8.3 years. If a random sample of 45

hourly employees is taken, what is the probabilitythat the sample will have an average age of lessthan 40 years?

Associate probability: 0.4808. Probability ofgetting average less than 40 years is: 0.9808

07.2

1350

45350

45

3.8

6.3740

1

N

nN

n

xz

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If the correction had not been used..

The answer would have been 0.9738 (with

associated z-value being 1.94).

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Sampling Distribution of proportion

Normal distribution approximates the shape

of the distribution of sample proportions if

n.p>5 and n.q>5 (where q=1-p).

Z-value for proportion

n

pq

ppz

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Example:

Suppose 60% of the electrical contractors in a

region use a particular brand of wire. What is

the probability of taking a random sample of

size 120 from these electrical contractors andfinding that 0.5 or less use that brand of wire?

Here

24.2

120

4.06.0

6.05.0

;120

;50.0

;60.0

n

pq

ppz

n

p

p

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Z-table associated with -2.24 is 0.4875. Hence

the probability of z getting less than this value

is less than 0.0125.

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Example: 2

If 10% of a population of parts is defective,

what is the probability of randomly selecting

80 parts and finding that 12 or more parts are

defective?

Here 12/80=.15; p= which is

associated with 0.0681.

49.1

80

9.01.0

1.015.0