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.

Stat 321 – Lecture 19Central 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 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.

Previously

Rules for Expected ValueE(X+Y) = E(X) + E(Y)

Rules for VarianceV(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 X1, …, Xn 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. X X

X

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 = (159.57-167)/5.11 = -1.45 92.6% of boats were overweight…

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

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

y p(y)

-$1 .9737

$35 .0263

E(Y) = -.0526

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% positive About 38% positive