Unit 6: Sampling Distributions

Unit 6: Sampling Distributions

• This unit covers the basic concepts of sampling distributions, including their usage, the sampling distribution of the mean, the central limit theorem, and its applications.

Basic Concepts

• A population encompasses all items or objects of interest.
• A sample is a portion of the population selected for analysis.
• A parameter is a summary measure describing a population characteristic. Examples include population mean and standard deviation.
• A statistic is a summary measure calculated from a sample. Examples include sample mean and standard deviation.

Example

• In 2007, 5.21% of a population experienced a certain kind of depression.
• A year later, a sample of 5000 individuals was selected, and 6% experienced the depression.
• Population: All individuals in the 2007 population
• Sample: The 5000 individuals selected in the later survey.
• Parameter: The overall 5.21% rate in the 2007 population.
• Statistic: The 6% rate in the sampled group.

Statistical Inference

• Statistical inference involves drawing conclusions about population parameters from sample statistics.
• This process typically follows these steps:
  • Collecting data from a population.
  • Selecting a sample from the collected data.
  • Computing a statistic from the sample.
  • Making various statements about population parameters using the sample statistic.

Why Sampling Distributions?

• Inferential statistics aims to draw conclusions about population parameters.
• A statistic's value changes with each sample.
• A statistic acts as a random variable with a probability distribution; this distribution is called a sampling distribution.
• The sampling distribution's probability distribution changes with changes to the population parameter, thus the sample statistic reflects information about the population.

Review

• Population Mean: μ = ΣXi / N (where μ is the population mean, ΣXi is the sum of all values in the population, and N is the population size)
• Sample Mean: x̄ = ΣXi / n (where x̄ is the sample mean, ΣXi is the sum of values in the sample, and n is the sample size)

Developing a Sampling Distribution

• Example: A population (N=4) with ages (18, 20, 22, 24).
• Possible samples of size 2 (with replacement).
• Calculating sample means: 18, 19, 20, ..., 24

Developing a Sampling Distribution

• Describing the population distribution (μ, σ) in age example given in the slides.
• Creating the sample means distribution from the possible samples.
• Observing that the distribution of sample means is not the same as the population distribution.

Summary Measures of the Sampling Distribution

• Calculating the mean (μx̄) and standard deviation (σx̄) of the sampling distribution of sample means for the age example discussed in the slides.

Comparing the Population with its Sampling Distribution

• Comparing the distributions (e.g., population distribution of age versus the sampling distribution of the mean of ages in samples of size 2, and of sample means of size n =6) visually to assess the effect of sample size.

Sampling Distribution of the Sample Mean

• The mean of a sampling distribution of sample means equals the population mean (μx̄ = μ).
• The standard deviation of a sampling distribution of sample means is calculated as σx̄ = σ/√n (where σ is the population standard deviation and n is the sample size).

Shape of the Sampling Distribution

• Shape depends on whether the population has a normal distribution.
• If the population is normal, the sampling distribution is normal regardless of sample size.
• If the population is not normal, the central limit theorem (CLT) applies; for large sample sizes (n ≥ 30), the sampling distribution of sample means is approximately normal.

Effect of Sample Size

• Larger sample sizes lead to closer approximations of the population distribution for the sampling distribution of sample means.
• Larger sizes also reduce the spread of the sampling distribution

How Large is Large Enough?

• A rule-of-thumb is n ≥ 30 for many cases.
• For approximately symmetric populations, n = 15 might suffice before relying on the approximation to normality when utilizing CLT.
• The normal population distribution gives a normal sampling distribution of the mean irrespective of sample size

Central Limit Theorem (CLT)

• The Central Limit Theorem (CLT) allows us to describe the sampling distribution of means from a population that doesn't have a normal distribution. The distribution will tend towards a normal distribution as sample size increases.

Example- ACT Scores

• A specific ACT score example demonstrating how to calculate the probability under a sampling scenario, and utilizing the CLT in a problem.

Exercises

• Example problems illustrating applications of concepts and use of the normal distribution and or central limit theorem (CLT) calculations.

This quiz covers the fundamental concepts of sampling distributions, including the sampling distribution of the mean and the central limit theorem. It also explores definitions of population, sample, parameter, and statistic through practical examples. Test your understanding of these key statistical concepts!