Sampling Distributions

• Basic properties of the sample mean ȳ

  • Identical to population mean μ

  • Variance of a sample mean σ^2 / n. The standard deviation of the sample mean is the population standard dev (σ) divided by sqrt(n). It’s called the standard error of the mean

  • As a sample size increases, the standard error of the mean decreases

• σȳ = σ/sqrt(n)

• Z = (Ȳ - μ) / (σ * 1/sqrt(n))

• Central Limit Theorem: all random samples of the same size n (when large enough), the distribution of Z is approx. normal with mean 0 and variance 1.

• Note: when something asks for the probability of “at most two” sum the probabilities for 0, 1, and 2

• To flip (Z < -#), it becomes (Z > #)

• Two types of statistical inference: estimation, hypothesis testing

• T distribution has the same general shape as the normal distriution

  • The sample size must be greater than or equal to 25 to do the t distribution without the normal assumption

• The standard deviation is greater than one, increases when sample size decreases

• Alpha = 1 - (confidence) divided by 2

• T sub alpha, n-1

• S is the same as sigma

• Plus or minus the average mean with the tscore times S/sqrt(n) for the distribution

• This tscore times S/sqrt(n) is the margin of error

• The degree of freedom for a normal distribution is infinity