Simple Random Sample (SRS):

Every possible subset of size $n$ has an equal chance of being selected. Use random methods (random number generator, lottery) with no systematic pattern.

Example: Select 100 voters from a list of 10,000 registered voters by numbering them and using random digits.

Pros: Unbiased, conceptually simple Cons: Requires complete list; doesn't guarantee representation of subgroups

Stratified Sample:

Divide the population into strata (groups by characteristic), then randomly sample from each stratum.

Example: A university has 2,000 freshmen, 1,800 sophomores, 1,700 juniors, 1,500 seniors. To survey campus housing, randomly sample 40 from each class (proportional to size).

Pros: Ensures representation of each subgroup; more efficient for comparing groups Cons: Requires knowing strata beforehand; more complex

Cluster Sample:

Divide population into clusters (geographic or natural groups), randomly select a few clusters, then survey all or random sample within selected clusters.

Example: To survey 5,000 high school students across 50 schools, randomly select 5 schools, then survey all or random sample within those 5 schools.

Pros: Cost-efficient (less travel), practical for geographically dispersed populations Cons: May introduce bias if clusters are not representative; within-cluster similarity can distort results

When to use:

• SRS: Small population, complete list available
• Stratified: Known subgroups important to represent fairly
• Cluster: Large geographically dispersed population, cost constraints

Method type: This is systematic sampling (every 10th person) combined with convenience sampling (mall intercept). It appears to use a systematic rule but is actually non-probability because:

1. Not everyone in the population has an equal (known) chance of selection
2. Only people at the mall on Saturday are included

Potential biases:

1. Undercoverage: Missing people who don't visit malls (elderly, disabled, those who shop online, night-shift workers)

2. Temporal bias: Saturday shoppers differ from weekday shoppers (leisure vs. work-focused, different age/income mix)

3. Location bias: Mall shoppers differ from non-mall shoppers; may not represent the broader community

4. Voluntary response bias (if participation is optional): People willing to stop and answer differ from those in a hurry

Result: The sample is not representative. If surveying about shopping habits or product preferences, the results would overrepresent mall shoppers and be unreliable for the general population.

Better approach: Use true SRS from voter rolls or census data (if available) or conduct random-digit dialing for phone surveys.

Plan A: SRS of 1,000

Pros:

• Unbiased; every resident has equal chance
• Large enough (n=1,000) for reasonable precision
• No need to know neighborhood structure

Cons:

• Requires complete list of all 500,000 people (costly)

Plan B: Stratified sample (100 per neighborhood)

Pros:

• Ensures each neighborhood is represented
• Can compare income across neighborhoods
• More practical than getting a citywide list
• Better precision if neighborhoods are similar within but different between

Cons:

• Assumes neighborhoods are meaningful strata
• May still miss income groups (e.g., homeless, institutionalized)

Plan C: Convenience sample (university students)

Pros:

• Cheap and easy

Cons:

• Severely biased: University students are younger, better educated, and earn less (or earn future income) than city average
• Sample is not representative; results would underestimate true city average income
• This would be a terrible estimate

Best choice: Plan B (Stratified by neighborhood)

Reasoning:

• Better than A (more practical, avoids needing a complete citywide list)
• Much better than C (actually representative; reduces bias)
• Provides neighborhood-level insights
• Balances simplicity with statistical validity

Conclusion: C is obviously wrong (not representative). Between A and B, stratified sampling is more practical and achieves high precision with manageable data collection.