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 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

Definition of sample If X 1,X 2,X 3, …,X n 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.

The sample mean E[X]=E[(X 1 +X 2 + … X n )/n]=(1/n)(E[X 1 ]+E[X 2 ]+ … E[X n ]) = μ Var(X)=Var{ (X 1 +X 2 + … X n )/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

The central limit theorem Let X 1,X 2, …, X n 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

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

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.

The sample variance

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

Implications If X 1,X 2,X 3, …,X n, is a sample from a normal population having mean μ and variance σ2, then

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:

Homework #5 Problem 8,10,,15,23,28