Sampling & Sampling Distributions

What is the goal of statistical inference?

To estimate population parameters using sample statistics and evaluate if a sample is representative of a population.

What are the two key concepts used to make inferences from samples?

1. Sampling distributions & Central Limit Theorem
2. Null hypothesis significance testing (NHST)

Why do we want to assess the probability of samples?

Because samples vary, leading to sampling error.

Why don’t we directly study populations?

Populations are usually too large and hard to access completely, so we rely on samples.

What is the “sampling problem”?

There are virtually infinite unique samples you could draw from a population, and we need to know how likely our specific sample is.

What is sampling error?

The variability in sample means from sample to sample due to random chance.

What is a sampling distribution?

A theoretical distribution of sample means obtained from all possible samples from a population.

What does the Central Limit Theorem (CLT) state (for N ≥ 30)?

Sample means follow a normal distribution

• Mean of sample means equals population mean (μ)

• Variance of sample means = σ² / N

• Standard error = σ / √N

Why is the CLT important?

It allows us to use properties of the normal distribution to assess probabilities of sample means.

What is the formula for z-score of a sample mean (𝑧̄)?

𝑧̄ = (𝑥̄ − μ) / (σ / √N)

What do we use the z̄ score for?

To calculate the probability of obtaining a given sample mean.

What’s the key idea of sampling distributions summarized?

They help us assess the likelihood of obtaining certain sample means and allow statistical inference.

What is the probability of a sample mean ≥ 105 when μ = 100, σ = 15, N = 40?

Calculate z̄ = (97 − 100) / (15 / √40)

What is the probability of a sample mean ≤ 97 when μ = 100, σ = 15, N=40?

Calculate z̄ = (97 − 100) / (15 / √40)