Samplind distributions

Sampling Distributions

Overview

  • Sampling distributions are critical in understanding how statistics behave in repeated sampling.

Key Topics

  • Sampling variability and distributions

  • Sampling distribution of a mean and Central Limit Theorem (CLT)

  • Sampling distribution of the sample proportion

  • Finite Population Correction (FPC)

  • Bias and noise

Descriptive vs Inferential Statistics

  • Descriptive Statistics:

    • Describe, show and summarize data in a meaningful way.

  • Inferential Statistics:

    • Infer or predict properties of a population based on data from a sample.

Statistical Estimation

  • Estimating an unknown mean (𝜇) by selecting a simple random sample:

    • Use the sample mean to estimate the population mean.

    • Different samples yield different sample means, showing sampling variability.

    • Goal: Understand what happens with many samples to capture this variability.

Sampling Variability

  • The variability in the values derived from repeated samples.

Two Main Areas of Inferential Statistics

  1. Estimating Parameters:

    • Point estimate: Average wage in the UAE

    • Range of likely true population mean values.

  2. Hypothesis Testing:

    • Compare the proposed hypothesis with observed data to see if results are significant (90%, 95%, or 99% confidence).

Statistical Estimation Examples

  • Average salary estimation for junior data scientists:

    • Point estimate and range example with computed values based on sample data.

Toward Statistical Inference

  • Understanding the sampling distribution of possible statistic values is essential for statistical inference.

  • The sampling distribution includes all possible values and their frequencies.

Sample and Sampling Distribution

  • Distinction between:

    • Sample Distribution: Distribution of observed or measured individual observations.

    • Sampling Distribution: Theoretical distribution of a sample statistic (mean).

  • Standard Deviation vs. Standard Error:

    • Standard deviation users sample spread, while standard error measures the estimate's precision.

Developing a Sampling Distribution

  • Assume a population size N = 4, with random variable X as the number of business meetings.

    • The random variables are independent and identically distributed (i.i.d).

Sampling Distribution Properties

  • Estimation of sample mean:

    • Defined by all sample means.

Standard Error of the Mean

  • Indicates variability in sample means:

    • Depends on sample size (n).

Characteristics of Normal Distribution and CLT

  • If the population is normal, sampling distributions will also be normal:

    • Sample mean is an unbiased estimator of the population mean.

  • Central Limit Theorem states that sample means are approximately normally distributed, even if the population is not.

Application Examples

  • Cereal Packet Weight Example:

    • Demonstrating sample means and probabilities using Z-scores.

  • Insurance Industry and the Law of Large Numbers:

    • The theorem assists in assessing risks over large numbers.

Bias and Noise in Estimation

  • Bias: Difference in accuracy of estimates.

  • Variability: Spread of the sampling distribution influenced by sample design.

  • Aim for low bias and low variability using random sampling and appropriate sample sizes.

Exercises and Applications

  • Problems to assess understanding of sample proportion distributions and probabilities in varying scenarios (e.g., tax increase vote support).

  • Statistical methodologies to determine required sample sizes based on desired probabilities.