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Sampling Distribution and Parameter Estimates

Introduction to Sampling Distribution

• Scenario: Assessing approval of a proposed law to ban possession of assault rifles in Placid Coast, California.

• Target Population: All adult residents.

• Sampling Frame: Voter registration list of approximately 20,000 registered voters.

Sampling Process

• To estimate the population parameter of approval, a random sample of 100 individuals is selected from the voter registration list.

• Variable Considered: Attitudes toward the proposed law, classified as binomial (approve/disapprove).

• First sample results: 48 approve, 52 disapprove, indicating 48% approval.

• A second sample shows 51% approval; a third shows 52% approval. These statistics are represented graphically.

Analyzing Sample Statistics

• Multiple Samples: By drawing more samples of 100 voters, we can plot various approval percentages on a summary graph.

• Sampling Distribution: Results form a normal or bell-shaped curve, highlighting the range of estimates.

Understanding Sampling Error

Concept of Sampling Error

• Probability theory aids in understanding how sample statistics relate to the population parameter.

• Distribution of Sample Statistics: Many independent samples will cluster around the true population parameter.

Estimating Sampling Error

• Standard Error Formula:

  [ S = \sqrt{\frac{P(1-P)}{n}} ]where:

  • S = Standard Error

  • n = Sample size (number of cases)

  • P = Proportion of successes in the population.

Example Calculation

• If 50% approval exists in the population, substituting values to calculate standard error:[ S = \sqrt{\frac{0.5 \times 0.5}{100}} = 0.05 ]

• This indicates a standard error of 5%, implying a reasonable estimate of where sample percentages may lie in relation to the true parameter.

Confidence Levels and Intervals

Confidence Levels Defined

• Researchers often infer probability about a sample's representativeness:

  • 68% Confidence: The estimate is likely within one standard error of the population parameter.

  • 95% Confidence: Within two standard errors.

  • 99.9% Confidence: Within three standard errors.

Sample Size Impact on Standard Error

• Inverse Function: Increasing sample size decreases standard error, leading to tighter clustering of sample statistics around the population parameter.

• Example: To reduce standard error from 5% (n=100 samples) to 2.5%, the sample size must be increased to 400.

Estimating Population Parameters

• When the true parameter is unknown, researchers substitute sample estimates into the formula, allowing estimation of population parameters and expected error degrees.