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 X i are independent random variables and every X i 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) IF X and Y are independent

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

7. The Central Limit Theorem Let X 1, …, X n be independent and identically distributed random variables, each with mean  and variance  2. Then if n is sufficiently large, has (approximately) a normal distribution with E( ) =  and V( ) =  2 /n.

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

9. 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…

11. Roulette Total winnings vs. average winnings Find P(X > 0) Exact sampling distribution with n = 2 -101.27699.4983.2244 Exact sampling distribution with n =3 -1-1/31/31.1458.3963.3543.1063

12. 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

14. Number Bet yp(y) -$1.9737 $35.0263 E(Y) = -.0526 SD(Y) = 5.76

15. 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?

16. 1000 spins About 5% positiveAbout 38% positive