Sampling Distributions Definition: A sampling distribution is the probability distribution of a statistic, such as the sample mean or sample proportion For continuous distributions, will consist of specification of the shape of the distribution, expected value (mean) and standard deviation

Example: From the population 2, 4, 4, 5, 6, 9 draw samples of size n = 2 (2 + 4)/2 = 3 (2 + 5)/2 = 3.5 (2 + 6)/2 = 4 Etc. What is P(x = 3)? How many samples are possible?

There are 15 ways to take six things two at a time Hence P(3) = 2/15; P(3.5) = 1/15, etc. Homework assignment: complete the sampling distribution and find E(x ) and standard deviation of the sampling distribution Recall that

Sampling Distribution of x-bar When sampling from a normally distributed population x is normally distributed with

This last term is called the standard error of the mean A standard error is the standard deviation of the sampling distribution of a statistic Be careful to distinguish Standard deviation of a population or sample Standard error: the standard deviation of the probability distribution of a statistic

Examples: Customer purchases are normally distributed with μ = $100 and σ = $12 What is the probability that a randomly selected receipt is between $97 and $103? What is the probability that a sample of 36 receipts will have sample mean between 97 and 103?

The Central Limit Theorem (Loosely) For sufficiently large samples, the sampling distribution of x is approximately normal, no matter what distribution the population has and

Allows use of normal probability procedures even when population is skewed, bimodal or otherwise strange “Sufficiently large” is quite modest: we usually require n  30