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Distributions of sampling statistics

Distributions of sampling statistics. Chapter 6 Sample mean & sample variance. Sample vs. population. A population is a large collection of items that have measurable values associated with the experimental study

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Distributions of sampling statistics

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  1. Distributions of sampling statistics Chapter 6 Sample mean & sample variance

  2. Sample vs. population • A population is a large collection of items that have measurable values associated with the experimental study • A proper sampling technique is adopted to select items, that is so called a sample, from the large collection in order to draw some conclusions about the population. • From selecting the small one to prospecting the large whole

  3. Definition of sample • If X1,X2,X3,…,Xn are i.i.d. variables with the distribution F, then they constitute a sample from the distribution F. • The population distribution F is usually not specified completely. And sometimes it is supposed that F is specified up to a set of unknown parameters. • The parametric inference problem emerges. • A statistics is a random variable whose value is determined by the sample data and used to inference the supposed parameter.

  4. The sample mean • E[X]=E[(X1+X2+…Xn)/n]=(1/n)(E[X1]+E[X2]+…E[Xn])=μ • Var(X)=Var{(X1+X2+…Xn)/n} • =(1/n2)(nσ2 )=σ2/n • c.f. population mean & variance: μ,σ2 • If the sample size n increases, then the sample variance of X will decrease. See fig. 6.1

  5. The central limit theorem • Let X1,X2, …, Xn be a sequence of i.i.d. random variables • each having mean μ and variance σ2 • The sum of a large number of independent random variables has a distribution that is approximately normal See example 6.3b and p.206 the binomial trials

  6. Approximate distribution of the sample mean • See example 6.3d, 6.3e

  7. How large a sample is needed? • If the underlining population distribution is normal, then the sample mean will also be normal regardless of the sample size. • A general thumb is that one can be confident of the normal approximation whenever the sample size n is at least 30.

  8. The sample variance

  9. Joint distribution of A chi-square distribution with n degree of freedom A chi-square distribution with 1 degree of freedom A chi-square distribution with n-1 degree of freedom

  10. Implications • If X1,X2,X3,…,Xn, is a sample from a normal population having mean μ and variance σ2, then

  11. Sampling from a finite population • A binomial random variable • E[X]=np, σ2=np(1-p) • If is the proportion of the sample that has a special characteristic and equal to X/n, then By approximation:

  12. Homework #5 • Problem 8,10,,15,23,28

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