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Review for Midterm

This review covers topics such as sampling framework, inference, conditional probability, uniform and continuous distributions, model links, and the bootstrap method. It provides an overview of chapters 1-6 and includes a response to student's questions.

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Review for Midterm

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

  2. 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

  3. 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

  4. 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.

  5. Conditional Probability • Example: Urn [3 Red and 5 Green] SWOR • Let R1 be event that the first draw is red • Let G2 be event that the second draw is green • P(A|B) = P(A and B)/P(B) where A and B are events (i.e. sets of sample space outcomes) P(R1|G2) = ? =P(G2 | R1) * P(R1) / P(G2) =(5/7)*(3/8) / P(G2) P(G2) = P(G2 and R1) + P(G2 and R1’) = P(G2| R1)P(R1) + P(G2| R1’)P(R1’) = 5/7 * 3/8 + 4/7 * 5/8 = 5/8 So P(R1|G2) = 3/7

  6. 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 =

  7. 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

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

  9. The bootstrap - bare bones • A statistic t(x1,x2,…,xn) estimates parameter  • Need: SD of t(), since it is precision of estimate. • Method: Re-Sample (x1,x2,…,xn) 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 .

  10. 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

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