Review for Midterm Including response to student’s questions Feb 26.

Sampling Framework Population of Interest Sometimes know dist’n model (shape) Usually don’t know parameters Sample A set of n numbers where n is the sample size - usually drawn at random from the population (like “tickets from a hat”). Random Sampling may be with or without replacement i.e. SWR vs SWOR

Inference Goal To use sample data to estimate population parameters. Example: use to estimate  But, would like accuracy of estimate. If unbiased, accuracy is just SD of, estimated by

Sampling Distribution of Approx Normal (CLT) Expected Value of is  (the population mean) SD of is  is  / called standard error (  is the population SD and n is the sample size) Usually,  and  must be estimated from the sample, using and s.

Conditional Probability P(A|B) = P(A and B)/P(B) where A and B are events (i.e. sets of sample space outcomes) P(R 1 |G 2 ) = ?=P(G 2 | R 1 ) * P(R 1 ) / P(G 2 ) =(5/7)*(3/8) / P(G 2 ) P(G 2 ) = P(G 2 and R 1 ) + P(G 2 and R 1 ’) = P(G 2 | R 1 )P(R 1 ) + P(G 2 | R 1 ’)P(R 1 ’) = 5/7 * 3/8 + 4/7 * 5/8 = 5/8 So P(R 1 |G 2 ) = 3/7 Example: Urn [3 Red and 5 Green] SWOR Let R 1 be event that the first draw is red Let G 2 be event that the second draw is green

Uniform Distributions Discrete P(X=x) = 1/n x=1,2,3,…,n Mean = (n+1)/2 SD = Continuous for 0<x<c and 0 otherwise. Mean = c/2 SD =

Model Links Waiting time for kth success - neg. bin. Waiting time for rth event - gamma Waiting time for first success - geom. Waiting time for first event - exponential Number of events during time - Poisson Time between successive events - exp

Shape of Gamma family Parameters ,   = 1 -> exponential  large -> normal  moderate -> right skew  contracts or expands scale. Mean =  SD =   Determining reasonable ,  (Use Mean&SD)

The bootstrap - bare bones A statistic t(x 1,x 2,…,x n ) estimates parameter  Need: SD of t(), since it is precision of estimate. Method: Re-Sample (x 1,x 2,…,x n ) many times and compute t() each resample. Then compute SD of resample values of t(). Result - an estimate of the precision of t() as an estimate of .

Overview of Ch 1-6 Ch 1 - “Distribution” - tables and graphs Ch 2 - Probability Calculus - counting rules, conditioning Ch 3&4 - Models and Connections Ch 5 - CLT and sampling distribution of a statistic Ch 6 - Estimators, Estimates, and the Bootstrap