Study Notes on Sampling and Sampling Distribution

Definition of Census: The recording of information from an entire population is referred to as a census.
- Example: Collecting grades of all Grade 11 learners.
- Philippine Statistics Authority (PSA) conducts a decennial population census.
Challenges of Censuses:
- Censuses can be impractical for large populations and expensive.
- Analyzing a complete census can lead to difficulties, thus summarizing via sample data is often preferred.
Definition of Sampling:
- Sampling: The process of selecting a section of the population to gather information.
Sample Survey:
- A method of systematically gathering information about a segment of the population, like individuals or business firms, to infer attributes about the population.
- Sample: The fraction of the population being studied.

Cost:
- Sampling often provides reliable information at a lower cost than conducting a full census.
- Large populations make census impractical due to financial constraints.
Timeliness:
- Sampling typically yields more timely information as fewer data points are needed, which is critical when information is urgently required.
Accuracy:
- Sample data can be more accurate due to better control of data errors in smaller groups.
Detailed Information:
- Sample surveys often yield more detailed information than censuses, especially for flow data (e.g., agricultural production).
Destructive Testing:
- In situations like battery life testing, sampling avoids complete destruction of the product.

PSA Census of Population and Housing takes place every ten years, with financial costs being substantial compared to sample surveys.
Sample surveys can effectively gather traits about agricultural and other seasonal data, which may not be available in a census.

Selecting the right sample is crucial as errors can invalidate results.
Two Categories of Sampling Techniques:
1. Probability Sampling: Each member of the population has a known probability of being selected into the sample.
2. Nonprobability Sampling: There is bias in the selection process, and no recognized probability exists for selection.

Definition: A method where every member of the population has a known, non-zero chance of being included in the sample, ensuring representativeness.
Examples of Probability Sampling:
- Family Income and Expenditure Survey (FIES)
- Labor Force Survey
- Quarterly Survey of Establishments
- Opinion polls (e.g., Social Weather Stations, Pulse Asia)

Simple Random Sampling (SRS):
- Each possible sample has an equal chance of being picked.
- All members have an equal chance of being included, including possibilities of selection with or without replacement.
- Requires a listing of all population elements (sampling frame).
Systematic Sampling:
- Elements are selected at uniform intervals from a randomized list.
- Example: If sampling size is 4 from a population of 20, divide into groups of size k, where k = N/n.
Stratified Sampling:
- The population is divided into strata based on shared characteristics.
- Samples are chosen from each stratum, ensuring representation from all groups.
- Considerations: Need to verify if the strata differences are important for the study.
Cluster Sampling:
- The population is divided into clusters; clusters are randomly sampled, and all elements within selected clusters are surveyed.
- Useful to minimize geographic disparity in data collection.

Definition: Methods where not all members of the population have a chance of being selected.
Types of Nonprobability Sampling:
1. Haphazard or Accidental Sampling: Unsystematic selection method; not representative.
2. Convenience Sampling: Selection based on ease of access/availability.
3. Volunteer Sampling: Participants volunteer for participation.
4. Purposive Sampling: Expert selected representative sample based on subjective judgment.
5. Quota Sampling: Selection based on predetermined quotas.
6. Snowball Sampling: Existing participants recruit further participants.

Population: Total set of observations or elements within a specific data set.
- Example: All students enrolled for a certain program.
Sample: A subset derived from the population for analytical purposes.
- Example: A group of 200 students selected from a larger population.

Parameter: Numerical measure describing a whole population.
- Example: Average height of all students at a school, represented as 65 inches.
Statistic: Numerical description derived from a sample.
- Example: Average height obtained from a sample of 50 students, represented as a sample statistic.

FNRI-DOST surveyed 14 million adults and found 80% are at risk of hypertension.
- Parameter: Percentage of all adults at risk.
- Statistic: 80% from the sample.
Researcher sampling 100 deaths found average age of 73.
- Parameter: Mean age from data of all women.
- Statistic: Mean from the sample group.
For drug Capvex, a random sample of 300 patients showed 250 healed within 10 weeks.
- Parameter: Proportion of all patients healed.
- Statistic: Proportion from the sample, represented as rac{250}{300} = 0.833.

Definition: A distribution depicting the frequency of statistics observed across all random samples drawn from a given population.
Example Calculation:
- Population: {2, 4, 9, 10, 5}
- List of samples of size 3: {2,4,9}, {2,4,10}, …,{4,10,5} with their corresponding means calculated.

Mean of Sampling Distribution: Equal to the population mean m or ext{EV} = ext{μ}.
Variance of Sampling Distribution:
- ext{Var}(ar{X}) = rac{σ^2}{n} for finite population.
- ext{Var}(ar{X}) = rac{σ^2}{N} for infinite population.
Standard Deviation of Sampling Distribution:
- σ_{ar{X}} = rac{σ}{ ext{√n}} for finite population.

If random samples of size n are taken from a population with mean μ and standard deviation σ, the sampling distribution of the mean approaches normality as n increases (usually $n ≥ 30$ is sufficient).

Height of Pupils Example: Mean if 50 pupils show height of 120 cm, convert to z-score and find associated probabilities.
Average Time in Shower Example: Using mean and sample size to calculate probabilities of exceeding a specific mean time.
Mean NAT Scores: Calculate the probability of a mean falling within a defined range using z-scores.

Mean and variances calculated for various examples demonstrate resonance between sample means and population parameters.