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Clase 3

Clase 3. Parameters and statistics. Population: hypothetical set of N “source” observations (N very large) Sample: a set of n observations from the source. Residuals Degree of Freedom. Other parameters. Mode and median. Central limit effect.

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Clase 3

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  1. Clase 3

  2. Parameters and statistics • Population: hypothetical set of N “source” observations (N very large) • Sample: a set of n observations from the source. Residuals Degree of Freedom

  3. Other parameters • Mode and median

  4. Central limit effect • Produces a tendencyfor real error dist. tobe “normal like” • Robustnesstononnormalityforlargepopulations of means • Errors can beconsidered as linear combinationsordifferent error sources: e=a1e1+a2e2+… • Willtendtonormality as e • (ej. roll thebones)

  5. Statistical Dependence • P(y3|y1)=p(y3) • More thanone variable: p(y1,y2)=p(y1)p(y2|y1)=p(y2)p(y1|y2) • Forindependence: p(y1,y2)=p(y1)p(y2) • Forthesamephenomena: considerrandomdrawing, independence IID. Normal distribution (parent) NIID

  6. Measure of L.D. and correlation • Covariance (L.D.) • Correlationcoeff. • Autocorrelation (rk vs. k isthe autocorrelationfcn.)

  7. L.D. Linear regression, least squares example.

  8. t Student • σ unknown • Substitute s for σ in the normal deviate • t has a Student’s distribution with ν=n-1 degrees of freedom. 1908, W.S. Gosset Guiness Brewery

  9. Random Sampling If the observations can be regarded as a random sample from some population of a given mean and standard deviation ver pp. 58

  10. Random sampling FortwoIndependentlydistributed (random) variables yA and yB: Does not depend on the form of the distribution. seeEx. pp. 58

  11. t-Student • Randomsampling + NIID, n observationsfrom a parentdist. (,σ2) • Samplemeans normal (, σ2 /n) • s2distributedindependently of , as χ2 (chi-squared) scaledwith n-1 D.O.F. • t isdistributed as t-Student, n-1 D.O.F. Fig. 2.12 pp. 45

  12. Chi-square • NIID • Then • Forthevariancewithknown mean

  13. F (Fisher) distribution • Two populations with σ1, σ2, normal, samples n1 and n2. Sample variances s1, s2 (DOF n1,2-1) • The ratio What’s all that for?? ex. Fertilizer pp. 78; randomization

  14. Summary… • Experimentsshouldbecomparative • Genuinereplication (many times) • Blocking (pairing) toavoiderrors • Randomizationsshouldbepart of theexperiment • Onlychecking original data can preventbad data • Wachoutfor IID, exchangeability and linear dependenceviolations

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