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Central Limit Theorem: Understanding Sampling Distributions

Learn about the Central Limit Theorem, sampling distributions, and unbiased estimators. Examples include weight analysis and roulette simulations. Office hours 1-3 pm today.

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Central Limit Theorem: Understanding Sampling Distributions

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  1. Stat 321 – Lecture 19 Central Limit Theorem

  2. Reminders • HW 6 due tomorrow • Exam solutions on-line • Today’s office hours: 1-3pm • Ch. 5 “reading guide” in Blackboard • Ignore page numbers

  3. Definitions • A statistic is any quantity whose value can be calculated from sample data. • A simple random sample of size n gives every sample of size n the same probability of occurring. Consequently, the Xi are independent random variables and every Xi has the same probability distribution. • As a function of random variables, a statistic is also a random variable and has its own probability distribution called a sampling distribution. • When n is small, we can derive the sampling distribution exactly. In other cases, we can use simulation to investigate properties of the sampling distribution. • A statistic is an unbiased estimator if E(statistic) = parameter.

  4. Previously • Rules for Expected Value E(X+Y) = E(X) + E(Y) • Rules for Variance V(X+Y) = V(X) + V(Y) IFX and Y are independent

  5. Moral • It is often possible to find the distribution of combinations of random variables like sums and averages • What about the sample mean…

  6. The Central Limit Theorem • Let X1, …, Xn be independent and identically distributed random variables, each with mean m and variance s2. Then if n is sufficiently large, has (approximately) a normal distribution with E( ) = m and V( ) = s2/n.

  7. Example • Ethan Allen October 5, 2005 Are several explanations, could excess passenger weight be one?

  8. Weights of Americans • CDC: mean = 167 lbs, SD = 35 lbs • Want P(T > 7500) for a random sample of n=47 passengers • Equivalent to P(X>159.57) • Sampling distribution should be normal with mean 167 lbs and standard deviation 5.11 lbs • Z = (159.57-167)/5.11 = -1.45 • 92.6% of boats were overweight…

  9. Roulette • Total winnings vs. average winnings • Find P(X > 0) • Exact sampling distribution with n = 2 -1 0 1 .27699 .4983 .2244 • Exact sampling distribution with n =3 -1 -1/3 1/3 1 .1458 .3963 .3543 .1063

  10. Empirical Sampling Distributions • Starts to get very cumbersome to do this for large n so will use simulation instead Approximately 35% of samples have a positive sample mean

  11. Number Bet y p(y) -$1 .9737 $35 .0263 E(Y) = -.0526 SD(Y) = 5.76

  12. Number bet • What does CLT predict for n = 50 spins? • Approximately 47% of samples have positive average? Only 36% Increases to 49% with large n?

  13. 1000 spins About 5% positive About 38% positive

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