Sampling Distribution Theory

1
Sampling Distribution Theory

• Sections 5.3 and 5.4

2
Population

• Population with distribution described by
  probability function f(x).
• If the population is discrete f(x) is a pmf. If
  the population is continuous f(x) is a pdf.
• Statisticians are interested in making inferences
  about f(x).
• What is the mean of the population.
• What is the variance of the population.
• What is the proportion of yess in the population?

3
Collect Data

• Take a random sample of size n. Use this sample
  to make inferences about the population itself.
• What is a random sample?
• Xi is the ith observation from the population
• Since each Xi is chosen from the population, the
  distribution of Xi is the same as that of the
  population. (f(x) is the pmf/pdf of Xi.)
• The Xi are independent.
• Definition. A random sample is a collection of
  independent identically distributed random
  variables X1 , X2 , , Xn with common pmf/pdf
  f(x).

4
Statistic

• A statistic is any quantity whose value can be
  calculated from sample data.
• Prior to obtaining data, there is uncertainty
  about what value the statistic will take.
• Thus a statistic is a random variable.
• When the statistic is calculated for a data set,
  you have a realization of the random variable.

5
Notation

• Use capital letters to denote data and statistics
  before a sample is taken. (These are random
  variables).
• X1 , X2 , , Xn means get a sample of size n.
• means get a sample and calculate sample
  mean.
• Use lowercase letters to denote data obtained and
  statistics calculated from that data. (These are
  realizations of random phenomenon.)
• x1 , x2 , , xn is a list of data.
• is a sample mean calculated from data.

6
Sampling Distribution

• Since a statistic is a random variable, it has a
  distribution ( a pmf or pdf).
• This distribution is usually called a sampling
  distribution to emphasize that the statistic
  varies depending on the sample of data collected.
• There are two ways to determine a sampling
  distribution.
• Thru simulation.
• Thru probability.

7
Distribution of Sample Mean thru Simulation

• Population year of pennies. Estimate the mean
  of this population by taking a random sample and
  computing sample mean.
• Sketch histogram of population. Give mean and
  variance.

8
Simulation of sampling distribution.

• Estimate the sampling distribution for the sample
  mean of samples of size 5 via 500 simulations.
  Give mean and variance.
• Estimate the sampling distribution for the sample
  mean of samples of size 10 via 500 simulations.
  Give mean and variance.
• Estimate the sampling distribution for the sample
  mean of samples of size 25 via 500 simulations.
  Give mean and variance.

9
Distribution of the sample mean thru probability.

• Let X be the number of packages mailed by a
  randomly selected customer at a certain shipping
  facility. Suppose the distribution of X is
• Consider a random sample of size n2 customers,
  and let be the sample mean. Obtain the pmf
  of

10
Results.

• Let X1, X2, Xn be a random sample from a
  population with mean m and variance s2. Consider
  
• Mean and Variance
• The expected value of the sample mean is m.
• The variance of the sample mean is s2/n.
• Distribution (Shape)
• If the random sample comes from a normal
  population, then the sample mean is itself
  normally distributed.
• If the random sample comes from a non-normal
  population, and the sample size is sufficiently
  large, then the sample mean is approximately
  normal. The larger the value of n, the better
  the approximation. (This is called the Central
  Limit Theorem.)

11
Concluding Example

• Suppose X1, X2, X25 are a random numbers (that
  is a random sample from a population that is
  distributed uniformly on the interval 0,1.)
• Note the mean of a uniform 0,1 r.v. is 1/2 and
  the variance is 1/12
• Calculate
• Calculate
• What if the sample size were 3 instead of 25?