1. 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

2. 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?

3. 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

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

5. 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

6. 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?

7. 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

8. 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