SAMPLING DISTRIBUTIONS
Simple Random Sample A simple random sample X 1 , , X n , taken from a population represented by a random variable X with mean  and standard deviation  , has the following characteristics.  Each X i , i  1,, n , is a random variable that has the same distribution as X, and thus the same mean  and standard deviation  . The X i ’s are independent random variables implying the following identity:



V  X 1    X n   V  X 1     V  X n    2     2  n 2



The sample mean, X where X 

X1    X n , is a continuous random variable n

with mean  X   and standard deviation  X  of the mean. Sampling Distribution of the Mean



, known as the standard error

n

Result 1: If X is Normal, then X is Normal and Z  of the sample n.

X 

 n , regardless of the size

Result 2: If n is large enough, n  30 , then X is approximately normal and X  Z , regardless of the distribution of the population random variable X.



n

Sampling Distribution of the Proportion For a binomial random variable X with parameters n (sample size) and  (probability of X success or population proportion), the sample proportion is given by p  . n The sample proportion p is a continuous random variable with mean  p   and standard deviation  p 

 1    n .

Result 3: If n is large enough, n  5 and n1     5 , then p is approximately p  normal and Z  .  1    n

Why? We want to use the sample mean X to estimate the population mean  . Likewise, we want to use the sample proportion p to estimate the population proportion  .

Confidence Interval Interpretation Consider a 1     100% confidence interval estimate for the mean based on a sample of size n . For   .05 the interpretation of a 95% confidence interval is: “If we were to take all possible samples of size n ,