chapter eleven sampling foundations. copyright © houghton mifflin company. all rights reserved.11 |...

Chapter Eleven

Sampling Foundations

Copyright © Houghton Mifflin Company. All rights reserved. 11 | 2

Chapter Objectives• Define and distinguish between sampling and census

studies• Discuss when to use a probability versus a

nonprobability sampling method and implement the different methods

• Explain sampling error and sampling distribution• Construct confidence intervals for population means

and proportions• List the factors to consider in determining sample

size, and compute the required sample size to achieve a specific degree of precision at a desired confidence level


Gallup Poll on Sampling: China• 12,500 counties, cities, and urban districts were divided

into 50 strata based on their geographic location, degree of economic development, and proportion of non-agricultural population

• One primary sampling unit (PSU), consisting of either a county or a city, was selected from each stratum based on probability proportional to population size

• Within each PSU, the populations of all neighborhoods and villages were compiled. From this listing, four neighborhoods or villages were selected proportional to size.

• From each of these four neighborhoods or villages, five households were selected at random


Gallup Poll on Sampling: China (Cont’d)

• One respondent was selected from each of the selected households, ensuring proper representation in the sample of all age groups by both genders

• The respondent to be interviewed is then selected according to a prescribed systematic procedure

• If the designated respondent was not at home, or could not be reached, a second or, if needed, a third adult family member was selected systematically from among the household members remaining on the list

• If contact with the designated respondent could not be made after a total of three separate visits to the household, an interview with a respondent in a substitute household in the same locality was permitted

• Two substitute households were kept in reserve for each five assigned households in the interviewing area


Gallup Poll on Sampling: China (Cont’d)

• By following this methodology and correcting for any rural/urban sampling issues the Gallup China polls are statistically accurate to within + or – 2%


National Poll –Sample Size

• Harris Poll – A weekly study that monitors the reactions of

the American public to a variety of economic, political, and social issues

• Sample Size – Based on a nationally representative

telephone survey of 1,000 adults age 18 or over


AC Nielsen Scantrack Index

• Offers valuable scanner-based sales and brand share data on a regular basis to manufacturers of a wide variety of consumer products such as food, drugs, and cosmetics

• Sample Size– Sales and brand share estimates are gathered

weekly from a representative sample of more than 4,800 stores representing over 800 retailers in 50 major markets


Sampling vs. Census Studies

• A census study draws inferences from the entire body of units of interest (the population)

• A sample study draws inferences from a sample drawn from the population


Advantages of Sampling

• Low Cost• Reduced time


Sampling and Nonsampling Errors

• Sampling error: The difference between a statistic value that is generated through a sampling procedure and the parameter value, which can be determined only through a census study

• Nonsampling error: Any error in a research study other than sampling error (which arises purely because a sample, rather than the entire population, is studied)


Minimizing Sampling Errors

• Increase the sample size• Use a statistically efficient sampling plan• Make the sample as representative of the

population as possible


Types of Nonsampling Errors

• Nonsampling Error – Any error other than sampling error

• Sampling Frame Error – Sampling frame not being representative of ideal

population

• Nonresponse Error – Final sample not representative of planned sample

• Data Error – Distortions in collected data and mistakes in data

coding, analysis, or interpretation


Potential Causes of Sampling Frame Errors

• Incomplete sampling frame over-represents some population segments and underrepresents others

• Sampling frame contains irrelevant units


Minimizing Sampling Frame Errors

• Start with a complete sampling frame• Modify the sampling frame to make it

representative of the ideal population using plus-one dialing in telephone surveys


Potential Causes of Nonresponse Errors

• Mail surveys/Internet surveys– Certain types of sample units being more likely

to respond than others

• Telephone and personal interview surveys– Person not-at‑home problem and respondent

refusal problem


Minimizing Nonresponse Errors

• Mail surveys: increase response rates through the use of incentives, follow-up mailings, etc.– Caution: increase in response rate per se may

not reduce non-response error

• Telephone and personal interview surveys: make call-backs and spread out the time blocks during which interviews are conducted


Potential Causes of Data Errors

• Respondents’ reluctance/inability to give accurate answers

• Ill-trained interviewers• Unscrupulous interviewers• Poorly designed questionnaire• Mistakes in coding data• Erroneous analysis• Incorrect/ inappropriate interpretation of

results


Exhibit 11.1 Types and Potential Causes of Nonsampling Errors


When Census Studies Are Appropriate

• The feasibility condition – Whenever a population is relatively small or

can be accessed easily

• The necessity condition– When the population units are extremely

varied and each population unit is likely to be very different from all the other units


Probability and Nonprobability Sampling

• Probability sampling is an objective procedure in which the probability of selection is known in advance for each population unit

• Nonprobability sampling is a subjective procedure in which the probability of selection for each population unit is unknown beforehand


Exhibit 11.3 Classification of Sampling Methods


Probability Sampling Methods

• Simple Random Sampling• Stratified Random Sampling• Cluster Sampling


Gallup Poll: USA

• Identify and describe the population that a given poll is attempting to represent

• Choose or design a method that will enable Gallup to sample the target population randomly

• Random Digit Dialing (RDD): a procedure that creates a list of all possible household phone numbers in America and then selects a sub-set of numbers from that list for Gallup to call


Simple Random Sampling

• Every possible sample of a certain size within a population has a known and equal probability of being chosen as the study sample


Stratified Random Sampling

• Two Types of Stratified Random Sampling– Proportionate Stratified Random Sampling

– Disproportionate Stratified Random Sampling


Proportionate Stratified Random Sampling

• Sample consists of units selected from each population stratum in proportion to the total number of units in the stratum


Kirkwood University- Proportionate Stratified Random Sampling

• Administrators of Kirkwood University wanted to determine the attitudes of their students toward various aspects of the university

• They selected a proportionate stratified random sample of 500 students for conducting the attitude survey


Table 11.2 Proportionate Allocation of Total Sample of Kirkwood University Students


Disproportionate Stratified Random Sampling

• Sample consists of units selected from each population stratum according to how varied the units are within the stratum


Exhibit 11.4 Disproportionate Stratified Random Sampling Used by A.C. Nielsen Company

Copyright © ACNielsen Company. Reprinted by permission.


Cluster Sampling

• Clusters of population units are selected at random and then all or some units in the chosen clusters are studied


Systematic Sampling Steps

• An organized procedure, selecting a sample from a list containing all the population units

Steps:

1) Determine the sampling interval,

number of units in the population

k = ------------------------------------------

number of units desired in the sample


Systematic Sampling Steps (Cont’d)

2) Choose randomly one unit between the first and kth units in the population list

3) The randomly chosen unit and every kth unit thereafter are designated as part of the sample


Practical Considerations: Probability Sampling Methods

• Probability sampling techniques are generally used by large commercial marketing research firms that maintain national samples or panels that can be readily accessed for conducting periodic research surveys


Nonprobability Sampling Methods

• Convenience Sampling• Judgment Sampling• Quota sampling


Convenience Sampling

• Researcher's convenience forms the basis for selecting a sample of units– The administrators of a college have announced a

sharp increase in tuition fees for the next year.

– A TV reporter covering this news item is shown standing on campus talking to several students, one at a time, about their reactions to the proposed tuition fee increase.

– TV Reporter says: “While some of the students feel that the 10 percent fee hike is justified, most of them consider it to be unfair.”


Judgment Sampling

• A procedure in which a researcher exerts some effort in selecting a sample that he or she believes is most appropriate for a study

• Example– The administrators of a college have announced a

sharp increase in tuition fees for the next year

– A judgment sample of student officers may be more representative than a convenience sample of students

– The researcher should be knowledgeable about the ideal population for a study


Quota Sampling

• Involves sampling a quota of units to be selected from each population cell based on the judgment of the researchers and/or decision makers

Steps

1) Divide the population into segments (referred to as cells) based on certain control characteristics

2) Determine the quota of units for each cell (quotas are determined by the researchers and/or decision makers)

3) Instruct the interviewers to fill the quotas assigned to the cells


Parameter & Statistic

• Parameter – The actual, or true, population mean value or

population proportion for any variable • income, product ownership

• Statistic– An estimate of a parameter from sample data


Sampling Error

• Sampling Error = Parameter Value - Statistic Value

• Difference between a statistic value that is generated through a sampling procedure and the parameter value, which can be determined only through a census study


Sampling Distribution

• Representation of the sample statistic values obtained from every conceivable sample of a certain size chosen from a population by using a specified sampling procedure along with the relative frequency of occurrence of those statistic values


µX SX

Sampling Distribution


5001045094008350730062505200415031002501

Annual expenditure for eating out ($)

Family Number

Table 11.4 Expenditures for Eating Out for a Hypothetical Population


4759,10

3755,10;6,9;7,8

2751,10;2,9;3,8;4,7;5,6

1751,6;2,5;3,4

751,2

Sample Mean Values ($)

Samples of Two Families

Table 11.5 Partial List of Possible Samples and Sample Means


Exhibit 11.5 Sampling Distribution (Bar Chart) for Simple Random Samples of Two Units


Exhibit 11.6 Sampling Distribution Shown as a Histogram


Central Limit Theorem

• When the sample size is sufficiently large, the sampling distribution associated with the sampling procedure display the properties of a normal distribution.


Confidence Estimation for Interval Data

n = number of units in the sample

X = sample mean value

Sx = s / n

S = standard deviation


• Given n = 100, x = 1,278 units, and s = 399 units

• To Construct 95 percent confidence interval s 399

sx = --- = ----- = 39.9 units n 100

• The 95 percent confidence interval is

x ± 1.96 sx = 1,278 ± (1.96)(39.9) = 1,278 ± 78.204 = 1,278 ± 78, approximately

Confidence Estimation for Interval Data (Cont’d)


Confidence Estimation for Interval Data (Cont’d)

• Interpretation– From the sample data, we can be 95 percent

confident that the average annual sales of men's suits, across all men's clothing stores in the population, are between 1,200 and 1,356 units


= true population proportion (i.e., the parameter value)Confidence Intervals for Population proportion:

p - 1.96sp p + 1.96sp

p = proportion obtained from a single sample (i.e., the statistic value)

sp = estimate of the standard error of the sample proportion

p = number of sample units having a certain feature total number of sample units (i.e., n)

sp = p (1 - p) n

Finding Confidence Intervals for Population Proportions


Given n = 100 and p = .64. To construct a 95 percent confidence interval for the population proportion

sp = p (1 - p) n (.64)(.36) = .048

100 The 95 percent confidence interval is p ± 1.96 sp = .64 ± (1.96)(.048)

= .64 ± .09408 = .64 ± .09, approximately.

Finding Confidence Intervals for Population Proportions (Cont’d)


Finding Confidence Intervals for Population Proportions (Cont’d)

• Interpretation– This confidence interval can also be

expressed in percentage terms: 64% ± 9%

– In other words, we can be 95 percent confident that between 55 and 73 percent of all grocery stores in the city carry potted plants


Factors Influencing Sample Size

• Desired precision level• Desired confidence level• Degree of variability • Resources available


Methods for Determining Sample Size

• The desired precision level• The desired confidence level• An estimate of the degree of variability in the

population, expressed in the form of a standard deviation


zq2 s2

N = ------ H2

zqs H = ----

n

Sample Size Estimation

• H-> Desired precision level • q-> Desired confidence level • S-> Sample Standard deviation • N-> Population mean


Sample Size Estimation (Cont’d)

• A marketing manager of a frozen-foods firm wants to estimate within ±$10 the average annual amount that families in a certain city spend on frozen foods per year and have 99 percent confidence in the estimate

• He estimates that the standard deviation of annual family expenditures on frozen foods is about $100

• How many families must be chosen for this study?


Sample Size Estimation (Cont’d)

H = $10, s = $100, and zq = 2.575 (corresponding to a confidence level of 99 percent)

n = (2.575)2(100)2 = 663 families,approximately

(10)2


Determining Sample Size

• A sporting goods marketer wants to estimate the proportion of tennis players among high school students in the United States

• The marketer wants the estimate to be accurate within ±.02 and wants to have 95 percent confidence in the interval estimate

• A pilot telephone survey of 50 high school students showed that 20 of them played tennis. Estimate the required sample size for the final study from the given data

• What should the sample size be if the desired precision and confidence levels are to be guaranteed?


Determining Sample Size (Cont’d)

H = .02 and zq = 1.96. p = 20/50 =0.4

s = (20/50)(1 - 20/50) = (.4)(.6) = .24

z2q s2 (l.96)2(.24 )2

n = ------------ = ------------------ H2 (.02)2

= 2,305 students, approximately

The maximum sample size is .25z2

q

nmax = ------------ = 2,401 students H2

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