Sampling distributions for sample means and the central limit theorem

Learning objectives

State the mean μ and the standard deviation σ/√n of the sampling distribution of a sample mean, and explain how the sample size affects its variability

Check the conditions for the sampling distribution of a sample mean — a random sample, and the 10% condition when sampling without replacement — and justify a normal model either from a normal population or from a sample large enough for the central limit theorem, commonly n at least 30

Calculate and interpret the probability that a sample mean falls in a given interval, and contrast it with the corresponding probability for a single observation

State the mean μ₁ − μ₂ and the standard deviation √(σ₁²/n₁ + σ₂²/n₂) of the sampling distribution of a difference between two sample means, check its conditions, and calculate a probability for the difference