Lecture 6: Sampling & Sampling Distributions

What is sampling in statistics?

The process of selecting a subset (sample) from a population to estimate characteristics of the whole.

Why do we use samples?

It’s more practical, cost-effective, and quicker than studying an entire population.

What is a statistic in sampling?

A numerical value calculated from sample data (e.g. sample mean x̄).

What is a sampling distribution?

The distribution of a statistic (like the mean) over many samples from the same population.

What is the mean of the sampling distribution of the sample mean?

It equals the population mean: μx̄ = μ

What is standard error of the mean?

The standard deviation of the sample mean distribution: SE = σ / √n

What is SE = σ / √n?

SE is standard error; σ is population standard deviation; n is sample size.

What happens to SE when sample size increases?

Standard error gets smaller — estimates become more precise.

What is the Central Limit Theorem (CLT)?

For large n, the sampling distribution of the sample mean approaches a normal distribution, even if the population isn’t normal.

When does the CLT apply?

Typically when n ≥ 30 or if the population is already normal.

What is the Law of Large Numbers?

As sample size increases, the sample mean x̄ gets closer to the population mean μ.

What does CLT allow us to do?

Use normal distribution to calculate probabilities for sample means.

What is the Z-score for a sample mean?

Z = (x̄ - μ) / (σ / √n)

What does Z = (x̄ - μ) / (σ / √n) mean?

It tells how many standard errors the sample mean is from the population mean.

What is the finite population correction?

Used when sampling without replacement from a small population: SE = σ / √n × √[(N - n)/(N - 1)]

When is finite population correction needed?

When sample size is more than 5% of the population (n > 0.05N)

What is the sampling distribution of the sample proportion p̂?

The distribution of p̂ values from many samples.

What is the mean of p̂?

Mean = population proportion P

What is the standard error of the sample proportion?

SE = √[P(1 - P)/n]

What is the Z-score for p̂?

Z = (p̂ - P) / SE

When is p̂ approximately normal?

When nP ≥ 5 and n(1 - P) ≥ 5 (success/failure rule).

Why is sampling distribution important?

It helps us understand the variability of sample statistics and make inferences.