Stat 321 – Lecture 19 Central Limit Theorem

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

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.

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

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

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.

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

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 = ( )/5.11 =  92.6% of boats were overweight…

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

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

Number Bet yp(y) -$ $ E(Y) = SD(Y) = 5.76

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?

1000 spins About 5% positiveAbout 38% positive