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Chap7 - STATISTICS 211 PROF EMANUEL PARZEN Chapter 7 ONE...

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STATISTICS 211 PROF EMANUEL PARZEN Chapter 7 ONE SAMPLE, TWO SAMPLE STATISTICAL METHODS STATISTICAL INFERENCE PARAMETERS , p μ Our Data Modeling Strategy has VALIDATION action, phase, problem 3 whose goal is to find parameters of probability models that are VALID or PLAUSIBLE. This means that for these parameter values observed data has higher probability of being observed. This chapter provides a handbook or summary of methods used in applied statistics to learn knowledge from data about the following parameters: (one sample continuous variable Y: parameter mean \mu) (one sample, 0-1 valued variable: probability p) (two independent samples, 0-1 variables: difference of probabilities) (two independent samples, 0-1 variables: pooled p, 2 by 2 table) (two independent samples, continuous variable Y: difference of means) (two independent samples, continuous variable Y: equal variances) (one sample, continuous variable: parameter \sigma) Bi-variate continuous paired data (X,Y): difference of means Bi-variate 0-1 valued: difference of probabilities, 2 by 2 table- ------------------------------------------------------------------------------------- #(ONE SAMPLE, CONTINUOUS VARIABLE Y:PARAMETER Mean μ ) Compute from sample: SAMPLE SIZE n , MEAN ( M Y , STANDARD DEVIATION S For HYPOTHESIS TEST 0 0 : H μ μ = COMPUTE ( SAMPLE MEAN;TRUE MEAN Z TEST STATISTIC ( 29 ( 29 ( ( 29 ( 29 0 0 ; M Y Z M Y SE M Y μ μ - = ( ( SE M Y n σ = if known [ ] 2 VAR Y σ = ( ( SE M Y S n = if unknown σ is estimated by S “SAMPLING DISTRIBUTION” of ( ( 0 ; Z M Y μ ASSUMING 0 μ TRUE MEAN ( 0,1 NORMAL Z exact if Y Normal and σ known ( 0,1 NORMAL Z approximate by central limit theorem, σ known ( 0,1 NORMAL Z approximate if 40 n , σ estimated by S ( 1 degrees of freedom STUDENT n - distribution if 40 n < , and σ estimated by S “Rejection Region” for hypothesis 0 0 : H μ μ = is interval of values of ( ( 0 ; Z M Y μ with (1) low specified probability under distributions above, and (2) maximum

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possible probability under specified appropriate alternative hypothesis of form 0 0 0 , , μ μ μ μ μ μ < . “ACCEPTANCE REGION” of values ( ( 0 ; Z M Y μ to accept 0 0 : H μ μ = at significance level .05 (specified probability of TYPE I ERROR: rejecting 0 H assuming 0 H is true) ( ( ( ( 0 .025; ; .975; Q Z Z M Y Q Z μ if alternate hypothesis 0 μ μ ( ( ( 0 .05; ; Q Z Z M Y μ if alternate hypothesis 0 μ μ < ( ( ( 0 ; .95; Z M Y Q Z μ if alternate hypothesis 0 μ μ Confidence 95% confidence interval, defined as interval of values 0 μ that satisfy: Accept 0 0 : H μ μ = at significance level 5%, is computed by computing endpoints of confidence interval from endpoint function (also called “confidence quantile of parameter” ) which has several formulas: If ( ( 0 ; Z M Y μ has ( 0,1 NORMAL distribution confidence interval endpoint function is ( ( ( ( ( ; TRUE MEAN ; Q P M Y SE M Y Q P Z μ = + When Y is Normal, sample size n<40, and we use S to estimate \sigma it is standard practice to use more accurate formula for confidence interval endpoints based on fact first discovered in 1904 by W.S. Gossett, whose day job was Guinness beer brewer: ( ( 0 ; Z M Y μ has ( 1 STUDENT t n - distribution ( ( ( ( ( ( ; ; 1 Q P M Y SE M Y Q P Student n μ = + - ( ( SE M Y n σ = if σ known S n = if σ estimate by S Two sided 95% CI can be expressed ( ( .025; .975; Q Q μ μ μ
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Chap7 - STATISTICS 211 PROF EMANUEL PARZEN Chapter 7 ONE...

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