Chapter 7: Sampling and Sampling Distributions

7.1 Overview

• In this chapter, we integrate concepts from previous chapters:

  • Population and sample statistics (Chapters 1-3)

  • Probability distributions (Chapters 4-6)

• Goal: Make inferences about populations using samples.

7.2 Selecting a Sample

Key Definitions

• Element: Entity from which data is collected.

• Population: Entire collection of elements of interest.

• Sample: Subset of the population.

• Sampled Population: The population from which a sample is drawn.

• Frame: List of elements for selecting a sample.

Purpose of Sampling

• Samples are selected to gather data answering research questions about a population.

• Sample results yield estimates of population characteristics; they may be good approximations with proper methodologies.

Sampling from Finite Populations

• Defined by lists such as membership rosters or inventory numbers.

• Simple Random Sample: Each possible sample of size n has the same probability of selection.

  • With Replacement: Each element can be selected more than once.

  • Without Replacement: More common; each element can only appear once in the sample.

  • Random numbers often aid in sample selection during large projects.

Example: Selecting Students

• 900 applicants for admission to St. Andrew's College:

  • Assign random numbers to each applicant.

  • Select 30 smallest random numbers for the sample.

Sampling from Infinite Populations

• Situations where obtaining a full list of elements isn’t feasible.

• Examples: ongoing manufacturing processes, bank transactions.

• Random samples need to fulfill the conditions:

  • Each element is from the population of interest.

  • Element selection is independent.

7.3 Point Estimation

Definition

• Statistical Inference: Inferring population characteristics from sample data.

• Point Estimation: Inferring population parameters using sample statistics.

• Point estimates are single values (e.g., (\bar{x}) for population mean (\mu)).

Key Questions

• When does a sample statistic accurately estimate population parameters?

• When does it produce precise estimates?

• What methods can facilitate these assessments?

Example: College Applications

• Estimating average SAT and housing preferences of 900 applicants using a sample of 30.

• Calculated point estimates:

  • (\bar{x} = 1684) - Sample mean.

  • (s = 85.2) - Sample standard deviation.

  • (\bar{p} = 0.67) - Sample proportion wanting on-campus housing.

Population Parameters (full population data)

• Population mean SAT score: (\mu = 1697)

• Population standard deviation: (s = 87.4)

• Proportion wanting on-campus housing: (p = 0.72)

7.4 Sampling Distribution of (\bar{x})

Sampling Distribution

• Sampling distribution of (\bar{x}): Probability distribution of all possible sample means from repeated sampling.

• Key features:

  • Expected Value: (E[\bar{x}] = \mu)

  • Unbiased if the expected value equals the population parameter.

Central Limit Theorem (CLT)

• For large sample sizes, the sampling distribution approximates a normal distribution:

  • Holds true regardless of the population's distribution.

  • For most scenarios, n (\geq 30) ensures a normal approximation.

Example with College Applications

• Problem: Calculate the probability that an estimate from 30 applicants falls within +/- 10 of the actual mean SAT score (estimate between 1687 and 1707).

• Standard error of mean computed, z-scores calculated for the probabilities.

7.5 Sampling Distribution of (\bar{p})

Population Proportion Estimation

• Sampling distribution of (\bar{p}): Probability distribution for the sample proportion.

• Expected value: (E[\bar{p}] = \rho)

Example: Housing Preferences

• For St. Andrew’s College, estimating the probability that a sample of 30 students reflects the true proportion of those wanting housing within +/- 0.05 of population proportion (0.72).

7.6 Properties of Point Estimators

Key Properties

1. Unbiasedness: (E[\hat{\theta}] = \theta) (estimate equals population parameter).

2. Consistency: As sample size increases, estimator approaches the true parameter value.

3. Efficiency: Among unbiased estimators, preferred based on lowest variance.

Summary and Practice

• Analyze multiple point estimators by their unbiasedness, consistency, and efficiency to find the best estimator for a given parameter.