AP®︎ Statistics: Topic 4.1 Flashcards

The probability distribution of x̄ for all possible random samples of a fixed size n from the same population; a distribution of sample means, not individual observations

μ_x̄ = μ, for random sampling from a population with mean μ

σ_x̄ = σ/√n, when sampled values are independent

Data should be collected using a random sample so probability methods can describe how x̄ varies from sample to sample

When sampling without replacement from a finite population, require n/N ≤ 0.10, equivalently N ≥ 10n, so observations are approximately independent

If the population distribution is normal, then the sampling distribution of x̄ is normal for any sample size

If the population is not normal, the sampling distribution of x̄ becomes approximately normal as n increases, provided sampled values are independent

For this course, if the population is nonnormal, n ≥ 30 is usually enough for the sampling distribution of x̄ to be approximately normal; extremely skewed populations may need much larger n

z = (x̄ - μ)/(σ/√n)

In repeated random samples of size n from the population, the average of the resulting sample means is μ

In repeated random samples of size n, the sample mean typically varies by about σ/√n from the population mean

The probability is the long-run proportion of random samples of size n whose sample means satisfy the stated condition

As n increases, σ_x̄ decreases; sample means from larger samples are less variable, and the reduction follows 1/√n

Population distribution: values for all individuals, with mean μ and SD σ; sample data distribution: values in one sample, with mean x̄ and SD s; sampling distribution of x̄: distribution of sample means from repeated samples, with mean μ_x̄ and SD σ_x̄