Sampling Methods and Survey Design in Statistics

Sampling and Survey Design in Statistics

Exercise 2: Understanding Sampling Concepts in Customer Satisfaction Surveys

This exercise explores the fundamental concepts of sampling in the context of a customer satisfaction survey conducted by a department store. It covers key statistical terms and the representativeness of samples.

• Variable/Parameter: The variable being researched is customer satisfaction among store credit card purchasers. In statistics, a variable is a characteristic or attribute that can be measured or categorized.

• Population: The population is all 51,000 people who made credit card purchases at the store in the past month. The population is the entire group about which information is desired.

• Sample Frame: The sample frame is the list of 51,000 credit card purchasers' addresses from which the sample is drawn. The sample frame should ideally cover the whole population.

• Sample: The sample consists of 142 people who returned the survey out of the 1,500 surveys mailed. The sample is the subset of the population that actually provides data.

• Representativeness: The sample may not be fully representative of the population because only those who chose to respond are included. This can introduce nonresponse bias, as respondents may differ systematically from non-respondents.

Example: If only highly satisfied or dissatisfied customers respond, the results may not reflect the average customer experience.

Exercise 3: Comparing Sampling Methods

This exercise compares systematic sampling and simple random sampling (SRS) in the context of surveying college students about Medicare eligibility age.

• Systematic Sampling: Involves selecting every 20th person from an ordered list. While systematic sampling can be efficient, it may introduce bias if there is a pattern in the list that correlates with the variable of interest.

• Simple Random Sample (SRS): Each individual in the population has an equal chance of being selected. SRS is generally preferred for avoiding bias, as it does not depend on the order of the list.

• Bias Avoidance: SRS is most likely to avoid bias because it ensures randomness and equal probability for all members of the population.

Example: If the list is alphabetized, systematic sampling could over- or under-represent certain last names, while SRS would not.

Comparison Table: Systematic Sampling vs. Simple Random Sampling

Sampling Method	Description	Bias Risk
Systematic Sampling	Select every 20th person from a list	Possible if list order is related to variable
Simple Random Sample (SRS)	Randomly select individuals; equal chance for all	Low; best for avoiding bias

Exercise 4: Types of Sampling Plans

This exercise matches sampling plan descriptions to their statistical names, illustrating key sampling methods used in survey research.

• Simple Random Sample (SRS): Randomly select a fixed number of individuals from each group or the entire population. Example: Randomly select 30 seniors from each high school.

• Cluster Sample: Randomly select entire groups (clusters), then survey all individuals within those groups. Example: Randomly select 3 high schools and survey every senior in those schools.

• Stratified Sample: Divide the population into strata (groups) and randomly sample from each stratum. Example: Compile a list of all seniors and randomly select 500 from the list.

Sampling Plan Classification Table

Description of Sampling Plan	Name of the Plan
Randomly select 30 seniors from each high school	Simple Random Sample (SRS)
Randomly select 3 high schools and survey every senior in those schools	Cluster Sample
Compile a list of all seniors and randomly select 500 from that list	Stratified Sample

Key Terms and Formulas

• Population: The entire group of interest in a study.

• Sample: A subset of the population selected for analysis.

• Sample Frame: The list or database from which a sample is drawn.

• Sampling Bias: Systematic error due to non-random sampling.

• Simple Random Sample (SRS): Each member of the population has an equal probability of selection.

• Systematic Sampling: Selection of every $k$th member from a list.

• Cluster Sampling: Selection of entire groups (clusters) at random.

• Stratified Sampling: Division of population into strata, then random sampling within each stratum.

Formula for Probability of Selection in SRS

For a population of size $N$ and sample size $n$:

Probability that any individual is selected: $P = \frac{n}{N}$

Formula for Systematic Sampling Interval

Sampling interval $k$ for population size $N$ and sample size $n$:

$k = \frac{N}{n}$

Formula for Stratified Sampling

If strata sizes are $N_1, N_2, ..., N_h$ and sample sizes from each stratum are $n_1, n_2, ..., n_h$:

$P_i = \frac{n_h}{N_h}$ for each stratum $h$

Additional info: In practice, the choice of sampling method depends on the research question, population structure, and available resources. Proper sampling is crucial for valid statistical inference.