Lecture07 - 2 May 2003 Biostatistics 6650--L7 1 Todays...

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Unformatted text preview: 2 May 2003 Biostatistics 6650--L7 1 Todays Schedule Estimation of the population mean Central limit theorem Interval estimation Mean of continuous data known unknown( t-distn) Binomial Proportion Poisson Sample size for CIs Variance( chi-square distn) HO: Geigy Bin CI/ Rosner T8 2 May 2003 Biostatistics 6650--L7 2 Estimation of Goal: Make a statement about the population mean Sample statistics , Median, mode, trimmed mean, range, s 2 , percentiles Natural estimator is , is a random variable. It can be shown that the E( ) from a sample of size n will equal . Thats good on average, but it doesnt say anything about a given sample. Can we describe a region where we think is based on our sample? X X X X 2 May 2003 Biostatistics 6650--L7 3 Estimation of If 2 is the variance of an individual X, what is the variance of ? n i i=1 n 2 i i=1 n 2 i i=1 2 2 2 Var{X}=Var{ X /n} =(1/n )Var{ X } =(1/n ) Var(X ), independent =(1/n )n = /n X 2 May 2003 Biostatistics 6650--L7 4 Estimation of Var( ) = 2 /n , Logical that variability in sample means should decrease with the sample size Standard Deviation of or Standard Error of the Mean(SEM) or Sampling distribution of X ~ N( , 2 ) ~N( , 2 /n ) Standardize : X X X Z n - = / , estimate with / SE n s n = X X X 2 May 2003 Biostatistics 6650--L7 5 Central Limit Theorem Central Limit Theorem If X is from an unknown distribution with mean and variance 2 , then for n large the sample mean will have an approximate normal distribution with mean and variance 2 /n. X ~ ?( , 2 ) N( , 2 /n) Extremely important for statistical inference. Often dont know the distributional form, but suspect non-normality Applies also to the sample sum in most cases n&gt;25 is adequate for approximations to two decimal places-- Lindgren, Stat Theory, 3rd Ed X : &amp; 2 May 2003 Biostatistics 6650--L7 6 CLT: Distn of Sample Means www.rci.rutgers/~cabrera/211/clt.gif n=1 n=5 n=50 n=500 2 May 2003 Biostatistics 6650--L7 7 Central Limit Theorem Think about X with a Binomial distribution X is the sum of n independent 0/1 variables the CLT applies to sums Normal approx to Binomial(and Poisson) is a special case of CLT 2 May 2003 Biostatistics 6650--L7 8 Interval Estimation: Continuous X When is known , an approximate 100*(1- )% confidence interval for the population mean is given by: Or by: Where, Z 1- /2 is 100*(1- )th percentile of the standard normal distribution 1 1 2 2 ME to ME, ME=margin of error [ / ] to [ / ] X X X Z n X Z n --- +- + 1 2 / X Z n - 2 May 2003 Biostatistics 6650--L7 9 Interval Estimation: Continuous X Development of 95% CI for .95 Pr( 1.96 1.96) Pr( 1.96 1.96) / Pr( 1.96( / ) 1.96( / )) Pr( 1.96( / ) 1.96( / )) Pr( 1.96( / ) 1.96( / )) Pr( 1.96( / ) Pr( 1....
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This note was uploaded on 06/17/2011 for the course BME 6650 taught by Professor Multipleinstructors during the Spring '03 term at Mayo Clinic College of Medicine.

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Lecture07 - 2 May 2003 Biostatistics 6650--L7 1 Todays...

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