Probability: Part 2 Sampling Distributions Wed, March 17 th 2004

Sampling Distribution wA theoretical distribution that allows us to calculate probability of our sample stats –Can then generalize from sample  pop Ex) pop of 2,4,6,8  y = 5 (mu = pop mean) Draw random sample of N=2 from that pop and get 4 and 6, ybar = 5 (pretty good representation of pop mean!)… but if we drew 8 and 8, ybar = 8 (not so good) The difference betw sample estimate and population parameter = sampling error

(cont.) wHow much confidence should we have in our sample estimate of the pop parameter? wSampling distribution – gives probabilities of all possible sample values –Found by taking all possible random samples of size N from pop, compute their means  plot

example wCan do this for all possible combinations of N=2 (w/replacement) and calculate ybar each time: ybarf 21(1 way to get ybar=2, 2 then 2) 32(could pull 2 then 4, or 4 then 2) 43etc… …if you plot this distribution  it is your sampling distribution!

Mean of Sampling Distrib. wSampling distribution also has a mean and std dev: –  ybar = mean of samp distrib = pop mean –Standard deviation of samp distrib is called the standard error:  ybar =  y / sqrt N …where  y is standard dev of pop (sigma) Represents average distance between pop & sample means

Central Limit Theorem wAs N increases, sampling distribution has less variability & looks like a normal curve wAs N increases, mean of samp distribution = mean of population wUsually when N> 30 sampling distrib will be normal

(cont.) wGiven this, we’ll use the sampling distribution to find out how probable (or improbable/unusual) our 1 sample happens to be –Is it a good representation of the pop or not? Use probability to determine wAs N increases, standard error decreases & we’ll be more confident in our sample estimate

Sample Likelihood wUse z scores, now to find the likelihood of a sample mean (rather than an individual score) w1 st find mean & standard error For IQ test, what is prob of group of 9 students has mean >= 112? Pop mean = 100,  y = 15 1 st, need samp distrib mean & standard error

(cont.)  Ybar (m in lab) = 100  Ybar (  x or s in lab) = 15 / sqrt (9) = 5 Z = ybar -  /  ybar Z = / 5 = 2.4 Use unit normal table to find probability of z=2.4, p =.0082 So very unlikely (.0082) to get a sample of 9 students w/average IQ of 112 from pop with  = 100