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Understanding Sampling Distributions and the Central Limit Theorem in Probability

This lesson explores the concept of sampling distributions, a theoretical framework that enables the calculation of sample statistics' probabilities. By understanding how to generalize from samples to populations, we can identify sampling error and evaluate the confidence we have in our estimates. We discuss the mean and standard deviation of sampling distributions, as well as the Central Limit Theorem, which highlights how the sampling distribution becomes more normal as the sample size increases. Practical examples illustrate the determination of sample likelihood using z-scores.

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Understanding Sampling Distributions and the Central Limit Theorem in Probability

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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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