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Home Page Title Page Contents Page 1 of 77 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit STAT231: STATISTICS Paul Marriott [email protected] July 8, 2008

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Home Page Title Page Contents Page 2 of 77 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Chapter 4: Statistical Inference In this chapter we will look at the problem of inference We have a mathematical structure which quantifies sample error Use probability theory to understand some fundamental sam- pling distributions Look at the idea of a confidence interval (i) mathematically (ii) computationally and (iii) in the context of a PPDAC in- vestigation Look at the idea of a hypothesis test
Home Page Title Page Contents Page 3 of 77 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Sample error and sampling distribution Want to understand sample error a ( S ) - a ( P Study ) associated with the attribute of interest a ( · ) . Sample Study Population Sampling protocol Inference Think of sample error as being ‘random’

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Home Page Title Page Contents Page 4 of 77 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Example: Hospital discharges Suppose that the study population is N = 393 short stay hospitals Response variate is x i which is the number of patients dis- charged from the i th hospital in January 1968. In this example we will assume that we have the knowledge of the whole population. Take different, and independent, samples from this population and make estimates of population attributes based on these samples. We can then compare each estimate to other independent esti- mates, and also compare all estimates to the actual population values.
Home Page Title Page Contents Page 5 of 77 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Example: Hospital discharges Use model Y = μ + R, R G (0 , σ ) . (1) Thus the model for taking n independent units (i.e. the com- plete sample) is given by Y i = μ + R i , R i G (0 , σ ) independently, with i = 1 , . . . , n . The maximum likelihood estimate μ was defined as n i =1 y i n . Define the estimator as μ = n i =1 Y i n where note we have replaced the realisations y 1 , . . . , y n with the random variables, Y 1 , . . . , Y n .

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Home Page Title Page Contents Page 6 of 77 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Sampling distribution From properties of Gaussian distributions we would expect μ G μ, σ n (2) This is called the sampling distribution for μ .
Home Page Title Page Contents Page 7 of 77 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Simulation: Hospital discharges Sampling dist: n=8 sample mean

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