Population Sampling and Error Analysis Study Notes

Population and Sample Statistics

• Population Statistics

  • Population parameters are summary descriptors of variables of interest. These include proportions, means, and variances.

  • Denoted as:

  • Mean: $\mu$

  • Variance: $\sigma^2$

• Sample Statistics

  • Descriptors of the same relevant variables calculated from sample data.

  • Used as estimators of population parameters and form the basis for inferences about the population.

  • Denoted as:

  • Mean: $\bar{X}$

  • Variance: $v-2$

• Sampling Frame

  • Defined as the list of cases in the target population from which the sample is drawn.

  • Ideally a complete and accurate list of members of the population.

  • In practice, discrepancies often arise between the sampling frame and the desired population.

• Sampling Variability

  • Refers to the phenomenon where sample statistics differ from one sample to another, leading to variability in estimates.

Types and Causes of Errors

1. Sampling Errors

   • Arise from the process of taking a sample.

   • Cause sample results to differ from the results of a complete census.

2. Non-sampling Errors

   • Errors not related to the act of sampling and may occur even in a full census.

Random Sampling Error

• Defined as the deviation between the sample statistic and the population parameter.

• Caused solely by the randomness in the selection of a sample.

• This is the only error accounted for in calculating the margin of error in confidence statements.

Bad Sampling Methods

• Convenience Samples: Samples that are easy to access but not representative of the population.

• Voluntary Response Samples: Involve individuals self-selecting to participate, often leading to bias.

• Incomplete Sampling Frame: Can result in under-coverage, where certain groups are omitted from sampling.

Non-sampling Errors Categories

A. Processing Errors
- Errors arising from mistakes in mechanical tasks such as arithmetic calculations or data entry.
B. Poorly Worded Questions
- Questions that are slanted to favor one particular response over others, leading to biased results.
C. Response Error
- Inaccurate responses due to factors such as lying, poor memory, or misunderstanding of questions.
D. Nonresponse Error
- Occurs when selected individuals do not respond or provide data in the sample.

Characteristics of a Good Sample

• A good sample is unbiased towards the population, avoiding systematic errors.

• Bias: Refers to the consistent deviation of the sample statistic from the population parameter in the same direction across multiple samples.

• Variability: Describes the degree to which sample statistics fluctuate when multiple samples are taken. High variability may indicate that the results are unreliable.

• A well-designed sampling method should have both small bias and small variability.

Strategies to Improve Sampling Quality

• To reduce bias, implement random sampling methods.

• To reduce variability, use a larger sample size when utilizing Simple Random Sampling (SRS).

• The variability can be minimized by sufficiently increasing the sample size.

Types of Sampling Methods

1. Probability Sampling

   • Essential principle: every member of the population has a known non-zero chance of being selected.

   • This type of sampling allows for estimates of precision.

2. Nonprobability Sampling

   • Based on arbitrary, subjective selection methods without a defined chance of selection for each member.

Simple Random Sampling (SRS)

• Consists of selecting $n$ individuals from the population such that every possible set of $n$ individuals has an equal chance of being the selected sample.

• Probability of selection formula: $\text{Probability of selection} = \frac{\text{Sample Size}}{\text{Population Size}}$

Complex Probability Sampling Methods

• Systematic Sampling:

  • Involves selecting members of a population at fixed periodic intervals.

  • Starts by choosing a random starting point, then every $k$-th member is chosen.

  • Example: For a population list of 1,000 people and a desired sample size of 100, every 10th person would be selected ( ext{k} = 1000/100 = 10).

  • Best used when the population list (like a phone directory) has no hidden patterns.