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Probability: Part 2 Sampling Distributions

Probability: Part 2 Sampling Distributions. Wed, March 17 th 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)

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Probability: Part 2 Sampling Distributions

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

  2. 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

  3. (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

  4. 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!

  5. 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

  6. 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

  7. (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

  8. 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

  9. (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

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