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Chapter 18 – Central Limit Theorem

Chapter 18 – Central Limit Theorem. Dice simulator. http://www.stat.sc.edu/~ west/javahtml/CLT.html Let’s roll a single die a few times and see what happens What do you think would happen if we rolled 2 dice? Would we expect the same distribution? Now let’s try 3 dice and then 5 dice.

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Chapter 18 – Central Limit Theorem

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  1. Chapter 18 – Central Limit Theorem

  2. Dice simulator • http://www.stat.sc.edu/~west/javahtml/CLT.html • Let’s roll a single die a few times and see what happens • What do you think would happen if we rolled 2 dice? • Would we expect the same distribution? • Now let’s try 3 dice and then 5 dice

  3. NFL Data Minitab data set

  4. Sampling Distribution of a Mean Rolling dice simulation 10,000 individual rolls recorded Figure from DeVeaux, Intro to Stats

  5. Sampling Distribution of a Mean Roll 2 dice 10,000 times, average the 2 dice Figure from DeVeaux, Intro to Stats

  6. Sampling Distribution of a Mean Rolling 3 dice 10,000 times and averaging the 3 dice Figure from DeVeaux, Intro to Stats

  7. Sampling Distribution of a Mean Rolling 5 dice 10,000 times and averaging Figure from DeVeaux, Intro to Stats

  8. Sampling Distribution of a Mean Rolling 20 dice 10,000 times and averaging Once again, as sample size increases, Normal model appears Figure from DeVeaux, Intro to Stats

  9. Central Limit Theorem • The sampling distribution of any mean becomes more nearly Normal as the sample size grows. • The larger the sample, the better the approximation will be • Doesn’t matter what the shape of the distribution of the population is! (uniform, symmetric, skewed…) • Observations need to be independent and collected with randomization.

  10. CLT Assumptions • Assumptions: • Independence: sampled values must be independent • Sample Size: sample size must be large enough • Conditions: • Randomization • 10% Condition • Large enough sample

  11. Which Normal Model to use? The Normal Model depends on a mean and sd Sampling Distribution Model for a Mean When a random sample is drawn from any population with mean µ and standard deviation σ, its sample mean y has a sampling distribution with: Mean: µ Standard Deviation:

  12. Example: CEO compensation 800 CEO’s Mean (in thousands) = 10,307.31 SD (in thousands) = 17,964.62 Samples of size 50 were drawn with: Mean = 10,343.93 SD = 2,483.84 Samples of size 100 were drawn with: Mean = 10,329.94 SD = 1,779.18 According to CLT, what should theoretical mean and sd be? Example from DeVeaux, Intro to Stats

  13. Normal Models Binomial µ = npσ = Sampling distribution for proportions Sampling distribution for means SD(y) =

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