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Unformatted text preview: Home Page Title Page Contents JJ II J I 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 Home Page Title Page Contents JJ II J I 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 JJ II J I 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’ Home Page Title Page Contents JJ II J I 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 JJ II J I 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 b μ was defined as ∑ n i =1 y i n . • Define the estimator as e 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 . Home Page Title Page Contents JJ II J I 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 e μ ∼ G μ, σ √ n (2) • This is called the sampling distribution for e μ . Home Page Title Page Contents...
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This note was uploaded on 04/18/2010 for the course STAT 231 taught by Professor Cantremember during the Spring '08 term at Waterloo.
 Spring '08
 CANTREMEMBER
 Statistics

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