XII Sampling Distributions of Statistics

• The Idea:
  • sample characteristics are statistics

    
    
    

  • population characteristics are parameters

    
    
    
    

  • GOAL: we want to draw inferences about population parameters from sample statistics

    
    
    

  • sample statistics serve as estimates of population parameters

    
    
    
    
    

  • thus, we need to do everything in our power to obtain reliable estimates of population parameters with minimal error

    
    
    

  • today's lecture is about quantifying the reliability (or precision) of sample estimates by examining their sampling distributions

• Can create a sampling distribution (2 methods - theoretical and formula based)

• Theoretical
  • start with a population with mean

(ex. avg salary of all business grads)

• take many samples of size N from the population and calculate the mean of each sample
  
  
  
  
  
  
  
  
  
• make a histogram of the sample means

  
  
  
  
  
  
  
  
  
  

• this distribution of sample means is the Sampling Distribution of the Mean (SDM)

• the mean of the SDM =

  
  
  
  
  

• the SD of the SDM (a.k.a. Standard Error of Mean) =

  
  
  
  
  
  
  

• Example creating a SDM (theoretical approach)

• interested in average number of movies attended in the US in 1 year. (underlying population parameters: m = 6.1 and s = 1.4)

• we take a very large number of samples (1000) of size 100 from the population

 

• we calculate means for each of the 1000 samples and make a histogram of the means (SDM)

  
  
  
  
  
  
  
  
  
  
  
  
  

   

• NOTE:

• Sampling Distributions are theoretical
  
  
• we do not directly obtain them via this procedure (i.e. repeatedly sampling from the population - "brute force" method)
  
  
• instead we use formulas to obtain their parameters (e.g., ) from a single sample
  
  
• the formula approach is a theoretically-based shortcut to the long "brute-force" approach
  
  

Formula Approach (create SDM from a single sample)

• 3 Questions:
  
  • will all s equal m
    
    
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  • what will mean of SDM be?
    
    • =

      
      

  • what will SD of SDM be?
    
    • =
      
      
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• draw SDM

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

• symbols
  

 

Sample

Population

SDM

Mean:

SD:

• Properties of SDM
  • How does size of s and N affect the precision of estimate?

   

   

  • example:

	s = 2	s = 20
N = 36	=	=
N = 100	=	=

• is ____________ than s
  
• ______________as s increases
  
• ______________as N increases

• indicates precision of estimate's of m
  
  
  • if large, not precise
    
    
  • if small, is precise

• thus, our sample estimates more accurate and less variable if
  we ____________N and _____________ s.
  
  
  • i.e. small standard error () means sample statistic is ____________ reliable estimate of population parameter

• We can create a sampling distribution for any sample statistic, so we can estimate their reliability.