Chapter 9

Chapter 9: Sampling Distributions

9.1 Introduction to Sampling Distributions

  • Definition: A sampling distribution is created through the process of sampling.

  • Methodology: Utilizes rules of probability, expected value, and variance to derive the sampling distribution.

9.2 Types of Sampling Distributions

  • Sampling Distribution of the Mean

    • Involves repeated sampling and measuring the mean of samples.

  • Sampling Distribution of a Proportion

    • Focuses on the ratio of successes to the total sample size.

  • Sampling Distribution of the Difference Between Two Means

    • Considers the differences in means from two independent samples.

9.3 Sampling Distribution of the Mean

  • Example: Rolling a fair die infinitely results in a random variable (X) defined as the number of spots on any throw.

    • Probability Distribution for X: P(1) = 1/6, P(2) = 1/6, ..., P(6) = 1/6

    • Mean (μ) and Variance (σ²) can be calculated.

9.4 Sampling Distribution of Two Dice

  • Sampling distribution is derived from all possible combinations of two dice rolls.

  • Sample Size (n = 2): 36 possible outcomes; 11 unique means (e.g., x̄ = 3.5 occurs frequently).

9.5 Construction of Sampling Distributions

  • The sampling distribution of the sample mean is classified and organized based on the outcomes of the sample.

  • Standard Error: The standard deviation of the sampling distribution, calculated as σ/√n.

9.6 Central Limit Theorem (CLT)

  • Definition: The mean of a random sample approaches a normal distribution as sample size increases.

  • Critical Points:

    • If the population is normal, the sample mean is normal regardless of sample size.

    • If the population is not normal, a sample size of n ≥ 30 is generally adequate for normal approximation.

9.7 Sampling Distribution of the Sample Mean

  • The relationship between the sample mean and the population parameters.

  • Formula: Z = (X̄ - μ) / (σ/√n)

9.8 Finite Population Correction

  • When dealing with finite populations, a correction factor is applied to the standard error.

  • If N (population size) is at least 20 times larger than n (sample size), the correction can often be ignored.

9.9 Application of Sampling Distributions

  • Example 9.1(a): Sparkling water bottles mean amount with realized values.

    • µ = 2.2L, σ = 0.3L; Probability that X > 2 is computed.

  • Example 9.1(b): Mean of four bottles. Assessed for probabilities exceeding 2.0L.

9.10 Probability and Inference

  • Example 9.2: Analysis of political support over a sample size of 300 regarding voter representation.

    • Probability that more than 50% would vote for a candidate based on current support levels.

    • Uses p-hat to determine probabilities and validate claims.

9.11 Difference of Two Means

  • Derived from two independent normal populations.

  • Conditions for normality and calculation of mean and standard deviation when comparing sample means.

9.12 Statistical Inference

  • Knowledge of the parameter includes probability statements about sample statistics.

  • Transition from assessing the population parameters to sampling from the population for better statistical inferencing.

  • Next Steps: Use the established sampling distribution to infer about the population parameters.