AP Statistics

Term	Short Definition
Population	Whole group being studied.
Sample	Smaller group selected from the population to study.
Simple Random Sample (SRS)	Every possible sample of that size has an equal chance of being selected.
Stratified Sampling	Randomly pick some individuals from every group (strata).
Cluster Sampling	Randomly pick entire groups (clusters); study everyone within them.
Systematic Sampling	Select a random start, then choose every $k^{\text{th}}$ individual (e.g., every $5^{\text{th}}$ , $10^{\text{th}}$ ).
Convenience Sampling	Pick individuals who are easiest to reach.
Voluntary Response	Individuals choose whether to join an open survey.
Nonresponse	Selected individuals cannot be contacted or refuse to participate.
Response Bias	An influence (such as question wording or interviewer behavior) leads to inaccurate answers.
Undercoverage	Some groups in the population are left out or are less likely to be chosen.
Bias	A sampling method consistently underestimates or overestimates the true population value.

Practice Problems and Applications

Employee Connection Survey (Simple Random Sample)
- Scenario: A corporation wants to survey a simple random sample of $100$ employees to evaluate workplace connection.
- Options:
  - A) Using the employee ID format, randomly generate $100$ employee ID numbers and survey those employees (ignoring duplicates and unassigned numbers).
  - B) Split employees by tenure ( $\ge 5$ years and $< 5$ years) and randomly select $50$ from each group.
  - C) Randomly select $2$ of $20$ floors and survey all employees on those floors.
  - D) Select every $5^{\text{th}}$ employee from an ID roster until $100$ are selected.
- Answer: A
- Reasoning: Option A gives every possible group of $100$ employees an equal chance of selection. Option B is stratified sampling, Option C is cluster sampling, and Option D is systematic sampling.
Veterinary Clinic Pet Insurance (Population vs. Sample)
- Scenario: A clinic surveys pet owners among its $2{,}500$ clients. A sample of $75$ dog owners and $50$ cat owners ( $125$ total) is chosen; $38$ state they would purchase insurance.
- Answer: B (The population is the $2{,}500$ pet owners who bring pets to the clinic, and the sample is the $125$ pet owners selected).
- Reasoning: The population is the entire group of interest ( $2{,}500$ ), while the sample is the subset selected to provide data ( $125$ ).
Candy Factory Quality Control (Cluster Sampling)
- Scenario: Boxes of candy are packaged in shipping cartons of $12$ boxes. The company selects a sample during an $8$ -hour shift.
- Answer: D (Randomly select $8$ shipping packages and inspect every box in those $8$ packages).
- Reasoning: A cluster sample selects entire pre-existing groups (shipping packages) at random and evaluates every unit within the selected groups.
Recycling Club Survey (Sampling Method)
- Scenario: Officers stand near recycling and trash bins, questioning people as they dispose of items.
- Answer: C (Convenience sampling).
- Reasoning: The researchers select participants who are readily available and easiest to access.
Baby Food Mall Survey (Response Bias)
- Scenario: A researcher approaches mothers of infants at a shopping center.
- Answer: D (The researcher gives sample baby food to the mother before asking survey questions).
- Reasoning: Offering free product samples can influence participants to give overly favorable or inaccurate responses.
High School Homecoming Alma Mater Survey
- a. Sampling Method and Bias: Convenience sampling. Alumni attending homecoming are likely more invested in school traditions than alumni who do not attend.
- b. Additional Source of Bias: Response bias. Framing the song as "a symbol of the school and its history" leads respondents toward opposing a change.
- c. Direction of Bias: The sample estimate will be lower than the true percentage of all alumni favoring a rewrite, because tradition-minded attendees are overrepresented and the wording discourages change.
Yoga Studio Flexibility Survey
- a. Type of Bias: Nonresponse bias. Out of $100$ selected clients, $62$ failed to respond.
- b. Direction of Bias: The estimate ( $6.5$ out of $7$ ) is likely higher than the true population mean, as clients confident in their flexibility are more motivated to complete the survey.
Tomato Plant Pest Inspection
- Method 1 (Simple Random Sample): Assign each of the $4{,}000$ plants an integer label from $1$ to $4{,}000$ . Use a random number generator to select $200$ unique integers, skipping repeats. Inspect the corresponding plants.
- Method 2 (Cluster Sample): Divide into a $10 \times 10$ grid ( $100$ plots), randomly choose $5$ plots, and inspect every plant within those $5$ plots.
  - Benefit: Reduces field travel time and operational costs because sampled plants are clustered together.
- Method 3 (Stratified Sample): Divide plants into strata by variety (roma, cherry, beefsteak) and take a random sample from each variety ( $100$ roma, $50$ cherry, $50$ beefsteak).
  - Benefit: Guarantees representation from every tomato variety, reducing variability across strata.