UNIT-II Types of Sampling and Estimates

Types of Sampling

Sampling is the process of selecting a subset of individuals from a statistical population to estimate characteristics of the whole population. The two primary categories are non-probability sampling and probability sampling.

Non-Probability Sampling

In non-probability sampling, items for the sample are selected based on the judgment of the researcher, convenience, or other non-random criteria. Every member of the population does not have a known or equal chance of being selected.

• Definition: A sampling technique where the odds of any member being selected for the sample cannot be calculated.
• Common Uses: Pilot studies, qualitative research, and situations where random sampling is impractical.
• Common Mistakes: Assuming the sample represents the entire population when severe selection bias is present.

Probability Sampling

In probability sampling, every element in the population has a known, non-zero probability of being selected. This allows researchers to calculate sampling error and make statistically valid inferences about the population.

• Definition: A sampling method where each population member has a determinable chance of selection.
• Real-World Applications: National censuses, large-scale socio-economic surveys, and quality control testing.

Probability Sampling & Simple Random Sampling

Simple Random Sampling (SRS) is the most fundamental probability sampling technique where every possible sample of size n has the exact same probability of being selected.

Simple Random Sampling with Replacement (SRSWR)

In SRSWR, once an element is selected from the population, it is put back (replaced) before the next draw. Therefore, the same population element can be chosen more than once in a single sample.

• Characteristics: Population size remains constant across all draws. The probability of selecting any given unit remains 1 / N at every draw.

Simple Random Sampling Without Replacement (SRSWOR)

In SRSWOR, once an element is selected, it is removed from the population and cannot be chosen again. Each element can appear only once in the sample.

• Characteristics: The population size decreases by 1 with each consecutive draw. This is the most commonly used method in practical survey sampling.

Definition and Procedure of Selecting a Sample

Selecting a sample requires a clear definition of the target population and a systematic approach to ensure fairness and accuracy.

Definition: A sample is a smaller, manageable subset of a larger population, chosen to represent the characteristics of the entire group.

Procedure of Selecting a Simple Random Sample

1. Define the Target Population: Clearly identify the complete group of units about which information is desired.
2. Construct a Sampling Frame: Create a comprehensive list of all distinct sampling units in the population (e.g., a student roster, a registry of businesses).
3. Assign Identification Numbers: Number every unit in the sampling frame sequentially from 1 to N.
4. Select Random Numbers: Use a random number table or a computer-generated random number generator to select n distinct numbers.
5. Draw the Sample: Match the selected random numbers to the corresponding units on the sampling frame to form the final sample.

Estimates of Population Mean and Total

Sample estimates (statistics) are used to infer unknown population parameters. Let N be the population size, n be the sample size, and y_i be the observed value of the i-th unit in the sample.

Estimate of Population Mean

The sample mean (ȳ) is an unbiased estimator of the population mean (μ).

ȳ = (1 / n) × ∑ y_i (summed from i = 1 to n)

To estimate the total population mean across the entire population, the sample mean serves as the direct point estimator.

Estimate of Population Total

The population total (Y) is the sum of values for all units in the population. It is estimated by multiplying the sample mean by the total population size N.

Ŷ = N × ȳ = N × ((1 / n) × ∑ y_i)

Exam-Oriented Note: Always remember that the multiplier for the population total estimator is the total population size N, not the sample size n. Distinguish clearly between sample values (n, y_i) and population parameters (N, μ, Y).