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Probability: Part 2 Sampling Distributions Wed, March 17th 2004 Sampling Distribution A 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.) 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 plot example Can do this for all possible combinations of N=2 (w/replacement) and calculate ybar each time: ybar f 2 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 4 6 3 7 2 8 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 means Central 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 estimate Sample Likelihood Use z scores, now to find the likelihood of a sample mean (rather than an individual score) 1st find mean & standard error For IQ test, what is prob of group of 9 students has mean >= 112? Pop mean = 100, y = 15 1st, 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 = 112-100 / 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