Probability: Part 2 Sampling DistributionsSampling Distribution(cont.)exampleMean of Sampling Distrib.Central Limit TheoremSlide 7Sample LikelihoodSlide 9Probability: Part 2Sampling DistributionsWed, March 17th 2004Sampling DistributionA theoretical distribution that allows us to calculate probability of our sample stats–Can then generalize from sample  popEx) 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.)How much confidence should we have in our sample estimate of the pop parameter?Sampling distribution – gives probabilities of all possible sample values–Found by taking all possible random samples of size N from pop, compute their means plotexampleCan do this for all possible combinations of N=2 (w/replacement) and calculate ybar each time:ybar f2 1 (1 way to get ybar=2, 2 then 2)3 2 (could pull 2 then 4, or 4 then 2)4 3 etc…5 46 37 28 1 …if you plot this distribution  it is your sampling distribution!Mean of Sampling Distrib.Sampling 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 meansCentral Limit TheoremAs N increases, sampling distribution has less variability & looks like a normal curveAs N increases, mean of samp distribution = mean of populationUsually when N> 30 sampling distrib will be normal(cont.)Given 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As N increases, standard error decreases & we’ll be more confident in our sample estimateSample LikelihoodUse z scores, now to find the likelihood of a sample mean (rather than an individual score)1st find mean & standard errorFor IQ test, what is prob of group of 9 students has mean >= 112?Pop mean = 100, y = 151st, need samp distrib mean & standard error(cont.)Ybar (m in lab) = 100Ybar (x or s in lab) = 15 / sqrt (9) = 5Z = ybar -  / ybarZ = 112-100 / 5 = 2.4Use unit normal table to find probability of z=2.4, p = .0082So very unlikely (.0082) to get a sample of 9 students w/average IQ of 112 from pop with  =