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Sampling and Sampling Distributions

Population

The totality of all possible outcomes or observations of a survey of experiment is a population

Sample

Any subset of a population

Finite population

Consists of a finite number of elements

Infinite population

One that consists of an infinitely (countable and uncountable) large number of elements.

Random Sampling

Simple random sample (SRS)

A set of members taken from a given population is called a sample of size .

A simple random sample (SRS) of members is a sample that is chosen such that every subset of observations of the population has the same probability of being selected.

Sampling from an Infinite Population

Simple random sample: Infinite population

Let be a random variable with certain probability distribution .

Let be independent random variables each having the same distribution as . Then is called a random sample of size from a population with distribution .

The joint probability function of () is given by:

Sampling Distribution of Sample Mean

Statistic

Suppose a random sample of observations () has been taken. A function of is called a statistic.

Sample mean \bar{X}, sample variance S^{2}.

Sampling Distribution

The probability distribution of a statistic is called a sampling distribution

Theorem 6

Related to the center and spread of sampling distribution.

For random samples of size taken from an infinite population with mean and variance , the sampling distribution of the sample mean has mean and variance .

The expectation of the sample mean is equal to the population mean -

In the long run, does not introduce any systematic bias as an estimator of , thus can serve as a valid estimator of it.

For an infinite population, as gets larger and larger, , the variance of becomes smaller and smaller: the accuracy of the estimator becomes better.

As:varianceaccuracy(of sample mean)(of sample mean in estimating population mean)(and by definition, standard error)Using the Law Of Large Numbers

Standard error

The spread of the sampling distribution is described by its standard deviation (also known as standard error).

It is denoted .

The standard error can be understood intuitively by interpreting it as such: it describes how much tends to vary from sample to sample of size .

Law of Large Numbers (LLN)

If are independent random variables with the same mean and variance , then for any

It is increasingly likely that is close to , as gets larger.

Central Limit Theorem

Central Limit Theorem (CLT)

If is the mean of a random sample of size taken from a population having mean and finite variance , then as :

Equivalently, this means:

The CLT states that, under rather general condiitions, for large , sums and means of random samples drawn from a population follows the normal distribution closely. (If the random sample comes from a normal population, is normally distributed, regardless.)

Rule of thumb

The mean of a large number of independent samples will have an approximately normal distribution.

  • If population is symmetric with no outliers, good approximation to normality appears after as few as 15-20 samples.
  • If population is moderately skewed, such as exponential or , then it can take between 30-50 samples before getting a good approximation
  • If population is extremely skewed, CLT may not be appropriate even with a lot of samples.

Other Sampling Distributions

Distribution

\chi^{2} Distribution

Let be a standard normal random variable. A random variable with the same distribution as is called a random variable with one degree of freedom.

Let be independent and identically distributed standard normal random variables. A random variable with the same distribution as is called a random variable with degrees of freedom.

We denote a random variable with degrees of freedom as .

Properties of the distribution:

  1. If , then , and
  2. For large , is approximately
  3. If are independent random variables with degrees of freedom respectively, then is a random variable with degrees of freedom.
  4. The distribution is a family of curves, each determined by degrees of freedom . All density functions have a long right tail.

long right-tail ### Sampling distribution of $\frac{(n-1)S^{2}}{\sigma^{2}}$

The sampling distribution of the random variable has little practical application. Thus, the sampling distribution of when is considered instead.

Theorem

If is the variance of a random sample of size taken from a normal population having the variance , then the random variable:

has a distribution with degrees of freedom.

Distribution (Student’s distribution)

t-Distribution

Suppose and . If and are independent, then

follows the -distribution with degrees of freedom.

Properties:

  • distribution with degrees of freedom is denoted
  • distribution approaches as parameter . When , we can replace it by .
  • If , then and for .
  • Graph of distribution is symmetric about the vertical axis and resembles the graph of the standard normal distribution.

t-distribution, with v degrees of freedomf(t)

Theorem 15

If are independent and identically distributed normal random variables with mean and variance then

follows a distribution with degrees of freedom.

Distribution

F-Distribution

Suppose and are independent. Then the distribution of the random variable

is called a distribution with degrees of freedom.

Properties:

  • The distribution with degrees of freedom is denoted by
  • If , then
  • If , then . This follows immediately from the definition of the distribution.
  • Values of distribution can be found in the statistical tables or software. The values of interests are such that
  • It can be shown

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