MELJUN CORTES Research seminar 1_sampling_theory_and_methods

Sampling Theory and Methods

Lesson 1Basics of Sampling Theory

Sampling

• Very important part of the research process.

• Concerned with selecting a relatively small number of elements (sample) from a larger group of elements (population).

• Often used in a survey research when it is difficult or impossible to conduct a census where data are gathered from every element of the population.

SAMPLINGTERMINOLOGIES

Population

•A set of elements usually grouped (e.g. people, organizations) from which data pertinent to a problem may be collected.

•Defined target population – consists of elements identified as key informants essential for investigation based on the limitations of the research project.


Sample

•Synonymous to an element of the population

•A person or object from which data are collected and information is derived.


Sampling units

•Elements of the target population available for selection during the sampling process

Sampling Distribution

•Theoretical concept referring to the frequency distribution of a specific sample statistics.


Sampling Error

•Any type of bias that results from mistakes either in the determination of the sample size or in the utilization of the sampling technique or method.


Confidence Interval (Margin of Error)

•Statistical range of values within which the true value of the target population parameter is expected to lie.

•It determines how much higher or lower than the population mean you are willing to let your sample fall.


Confidence Level

•How confident do you want to be that the actual mean falls within your confidence interval?

•Most common confidence intervals 90% confident, 95% confident, and 99% confident.


Central Limit Theorem

•For almost all target populations, the sampling distributions of the means or the percentage value derived from a simple random sample will be approximately be normally distributed provided that the sample size is sufficiently large. (Hair, 2006)

•The probability is high when the mean of any sample taken from the target population closely approach as one increases the size of the sample.


Standard Deviation

•Variance expected in responses

•Use 0.5, a safe decision and ensures that the sample will be large enough

SIZEDETERMINATION

• Factors that determine sample size are:– Nature of data and data analysis– Kind and number of comparisons– Number of variables to be investigated– Desired degree of accuracy– Homogeneity of samples

• Determining Sample size for:– Unknown population– Known population

SIZEDETERMINATION

• Confidence level corresponds to a Z-score, a constant value needed for the equation.

• Z-scores for the most common confidence levels:– 80% - Z-score = 1.281– 90% - Z-score = 1.645– 95% - Z-score = 1.96– 97% - Z-score = 2.17

• For other values (http://www.utdallas.edu/dept/abp/zscoretable.pdf)

SIZEDETERMINATION

• Sample sizes as small as 30 are generally adequate to ensure that the sampling distribution of the mean will approximate the normal curve (Shott, 1990 as cited in Cristobal, 2013)

• When the total population is equal to or less than 100, this same number may serve as the sample size (universal sampling).

FORMULAUnknown Population Size

• Used in determining sample sizes from unknown population

Necessary Sample Size = (Z-score)2 * StdDev * (1 – StdDev) / (margin of

error)2

Sample size = 0.25 (desired certainty / acceptable error) 2

Example 1• Assuming you chose 95% confidence level, 0.5 standard

deviation, and a margin of error (confidence interval) of +/- 5%. How many respondents are needed?

recall: Necessary Sample Size = (Z-score)2 * StdDev * (1 – StdDev) / (margin of error)2

(1.96)2 * 0.5 (0.5) / (0.05)2

(3.8416 * 0.25) / 0.00250.9604 / 0.0025384.16 (385 respondents rounded up)

Example 2• If a survey of a group of teachers requires a

confidence level of 95% and 5 % acceptable error, then the researcher needs a sample size of 384 teachers that will have to be surveyed.

For 95% certainty, use 1.96

0.25 (1.96 / 0.05)2 = ?0.25 (1,536.64) = ?

385 teachers (rounded up) = ?

Example 3

• If a researcher wants to examine a group of students with 97% certainty, then.

For 97% certainty, use 2.17

0.25 (2.17 / 0.03)2 = ?0.25 (5,232.111) = ?

1,308 students = ?

Exercise

• Assume you wish to randomly sample teachers across the southeast USA. With a 90% confidence level, what sample size would be needed?

0.25 (1.645/0.10)2 = ?0.25 (270.6025) = ?67.65 = ?

68 teachers (rounded up)

Exercise

• You wish to estimate the proportion of all buyers that are young buyers with 95% confidence. How many observations are needed to estimate the population proportion within 0.06 error?– Assuming that nothing is known about the

population proportion.– Assuming that the population proportion has

recently been estimated to be 0.34.

Exercise

– Assuming that nothing is known about the population proportion.

(1.96)2 * 0.5 (0.5) / (0.06)2

(3.8416 * 0.25) /0.00360.9604 / 0.0036266.7778 (267 respondents rounded up)

Exercise– Assuming that the population proportion has recently

been estimated to be 0.34.

Since a prior estimate of the population proportion is known (0.34), use it as the standard deviation.

(1.96)2 * 0.34 (1 – 0.34) / (0.06)2

(3.8416) * 0.34(0.66) / 0.0036(3.8416 * 0.2244) /0.00360.8620/ 0.0036240 respondents rounded up

FORMULAKnown Population Size

• Use of Slovin’s Formula (used to calculate an appropriate sample size from a population).

n = N / (1 + Ne2)

Where:n = sample sizeN = populatione = estimate of error (acceptable error)

* The error tolerance, e, can be given (in a question) or if you want to figure it out on your own, just subtract the confidence level from 1.

Example 1

• Estimate the sample size from a population of 5000 freshmen students using 5% acceptable error.

n = N / (1+ Ne2)= 5000 / (1 + (5000)(0.05)2)= 5000 / (1 + 12.5)= 5000 / 13.5= 370 students

Exercise• A researcher plans to conduct a survey. If the

population on Smith City is 1,000,000 find the sample size if the margin of error is 15%

n = N / (1+ Ne2)= 1,000,000 / (1 + (1,000,000)(0.15)2)= 1,000,000 / (1 + (1,000,000)(0.0225))= 1,000,000 / (1 + 22,500)= 1,000,000 / 22,501= 45 people (rounded up)

SAMPLING THEORY AND METHODS

Lesson 2Sampling Methods

The Problem

• Suppose a school would like to determine the weekly food expenditure of its students, if there are 1,000 students and the guidance counselor decided to use only 100 students as a sample, who will be included in the sample?

Sampling Method

• Sampling Method or Sampling Technique is the process of determining or selecting those sample units which would provide the required estimates with associated margins of error, arising from examining only a part and not the whole.

• Two methods are used: probability sampling and non-probability sampling.

Sampling MethodSampling

Technique

Probability Sampling

Simple Random Sampling

Systematic Random Sampling

Stratified Random Sampling

Cluster Sampling

Multi-stage Sampling

Non-Probability Sampling

Convenience Sampling

Quota Sampling

Purposive Sampling

Probability Sampling

• Each eligible member of the population has a specific and known chance of being included in the sample.

• Also termed as scientific sampling.

• When using this technique, it is important to have a complete list of the members of the population.

Probability SamplingSimple Random Sampling

• Lottery method– Write names or codes on a piece of paper– Put in a container– Randomly select the desired number of samples

• Fishbowl technique

Probability SamplingSystematic Random Sampling

• Choose a starting point then select every kth element of the population using k = N / n.

• Ex. The town of Fairfax is divided up into 576 blocks which are numbered consecutively. A 10 percent sample blocks is to be taken for the study. If the random number chosen between 1 to k is 3, what are the blocks considered for the sample?

Probability SamplingSystematic Random Sampling

• Solution:– Determine k; k = 576 / 10 percent of 576

= 576 / 57.6 = 10this means you have to include every 10th

member of the population after choosing a random start.

– Random number = 3 (can be determined via the lottery method)

– Blocks considered for the ample are: 03, 13, 23, 33, 43, …, 573

Probability SamplingStratified Random Sampling

• Population is subdivided into at least two different subpopulations (or strata) that share the same characteristics (such as gender, age). Samples are derived from these strata using the other sampling techniques.


• Ex. Assuming a barangay of 10,000 families belonging to different income brackets, a survey to find out how many are in favor of the RH bill is to be conducted. To ensure that all income groups are represented, respondents will be divided into High-Income, Average-Income, and Low-Income.


Strata Number of familiesHigh – Income 2,000

Average – Income 5,000Low – Income 3,000

N = 10,000


• Using a 5% margin of error, how many families should be included in the sample?

• Using either proportional or equal allocation, how many from each group should be taken as samples?


• Using a 5% margin of error, how many families should be included in the sample?

n = 10,000 / (1 + (10,000)(0.05)2 )n = 10,000 / (1 + (10,000)(0.0025)n = 10,000 / (1 + 25)n = 10,000 / 26n = 385 families (rounded up)


• Using either proportional or equal allocation, how many from each group should be taken as samples?

Probability SamplingStratified Random SamplingStrata Number of

families Percent Proportional Equal

High – Income 2,000 2,000 / 10,000

= 0.20 or 20% 0.2(385) = 77 385/3 = 128

Average – Income 5,000 5,000 / 10,000

= 0.50 or 50% 0.5(385) = 193 129

Low – Income 3,000 3,000 / 10,000

= 0.30 or 30% 0.3(385) = 115 128

N = 10,000 100% 385 385

Probability SamplingCluster Sampling

• Normally used in large-scale studies in which the population is geographically spread out where sampling procedures may be difficult and time consuming.

• Process: – Divide the population area into sections (or clusters)– Then randomly select a few of those sections– Randomly select a sample from those sections

Probability SamplingCluster Sampling

• Ex. Suppose we want to determine the average daily expenses of families in the City of Santa Rosa. We can draw a random sample of 5 barangays from the total 15 using random sampling and then get a certain number of families from each of the 5 barangays.

Probability SamplingMulti-Stage Sampling

• Combination of several sampling techniques discussed

Non-Probability Sampling

• Members of the sample size are drawn based on the judgment of the researcher.

• It is a process of selection in which not all members of the entire population are given a chance of being selected as samples.

• Sometimes referred to as subjective sampling or non-scientific sampling.

Non-Probability SamplingConvenience Sampling

• Simply use results that are readily available.

• Also called accidental or incidental sampling.

Non-Probability SamplingQuota Sampling

• Similar to stratified sampling in which the researcher first identifies the strata and their proportions as they are represented in the population.

• Convenience or judgment sampling is used to select the required number of subjects from each stratum.

Non-Probability SamplingPurposive Sampling

• Handpicking of subjects; also called judgmental sampling

• Selects members or elements based on the particular purpose of the experiment or study.

• Useful for situations where you need to reach a targeted sample quickly and where sampling for proportionality is not the primary concern.

The Problem

• Suppose a school would like to determine the weekly food expenditure of its students, if there are 1,000 students and the guidance counselor decided to use only 100 students as a sample, who will be included in the sample?

• Find a partner and help the school decide on which sampling technique should they use in selecting students for the sample. Present three (3) choices.

Exercise

• Identify which sampling technique was used in the following scenarios:1. When she wrote Women and Emoticons, author KC

Pascual based conclusions on 4,500 responses from 100,000 questionnaires distributed to women.

2. The Guidance Counselor of a high school surveys all students from each of the 20 randomly selected classes.

3. A sociologist at selects 15 men and 15 women from each of 4 Math classes.

Exercise

• Identify which sampling technique was used in the following scenarios (cont’d):4. IBM selects 200th compact disk from the assembly

line and conducts a thorough test of quality5. The court secretary writes the name of each

Municipal Judge on a separate card, shuffles the cards, and then draws 3 names.

6. Pro-RH bill lobbyists polls 300 men and 300 women about their views concerning the use of contraceptives.

Exercise

• Identify which sampling technique was used in the following scenarios (cont’d):7. The marketing manager of ebay.ph tests a new

sales strategy by randomly selecting 150 consumers with less than P100,000 in gross income and 150 consumers with gross income of at least P100,000.

8. A market researcher for Champion Detergent interviews all passengers on each of 10 randomly selected PUVs.

Exercise

• Identify which sampling technique was used in the following scenarios (cont’d):9. A medical researcher from Unilever Phils.

interviews all leukemia patients in each of 20 randomly selected hospitals.

10.In conducting research for the evening news, a reporter for ABS-CBN interviews 15 people as they leave a shopping mall.

Exercise

• Solve the following:– A researcher would like to investigate the

perception of students on mathematics. He divided the population into sub-populations as shown in the table. Use stratified random sampling if the sample to be drawn consists of 500 students.

ExerciseStrata Number of StudentsFirst year 1600Second year 1500Third year 1400Fourth year 1000

MELJUN CORTES Research seminar 1_sampling_theory_and_methods

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