STATS Module 7

• Population Probability Distribution


  The probability distribution of the population data

  
• Suppose there are only five students in an advanced statistics class and the midterm scores of these five students are 70, 78, 80, 80, 95
• x


  The score of a student
• Population Frequency and Relative Frequency Distributions

  • 70
  • 78
  • 80
  • 80
  • 95

  
• Population Probability Distribution


  • 0.2
  • 0.2
  • 0.4
  • 0.4
  • 0.2

  
• Population mean = 403/5 = 80.6
• Sampling Distribution


  The probability distribution of the sample statistic x

  
• The total number of possible samples of 3 scores each that can be selected, without replacement, from the population of 5 scores is 10

  
• All possible samples of 3 scores and their means

  • ABC (76.0)
  • ABD (79.3)
  • ABE (84.3)
  • ACD (76.7)
  • ACE (81.7)
  • ADE (84.3)
  • BCD (79.3)
  • BCE (82.3)
  • BDE (82.3)
  • CDE (85.0)
• Frequency and Relative Frequency Distributions of the sample means

  • 76.0 (1/10)
  • 79.3 (2/10)
  • 81.7 (1/10)
  • 82.3 (2/10)
  • 84.3 (2/10)
  • 85.0 (1/10)
  • 76.7 (1/10)
  • 79.3 (1/10)

  
• Sampling error


  The difference between the value of a sample statistic and the value of the corresponding population parameter

  
• Sampling error = x - μ, assuming the sample is random and no non-sampling error has been made

  
• Non-sampling errors


  Errors that occur in the collection, recording, and tabulation of data

  
• Reasons for non-sampling errors

  • Non-random sample
  • Questions not fully understood
  • Respondents give false information
  • Poll taker makes a mistake

  
• Nonsampling error = 82.33 - 80.60 = 1.73, of which only 1.07 is due to sampling error

  
• Mean of the sampling distribution of x

  Equal to the population mean μ

  
• Standard deviation of the sampling distribution of x


  σ/√n, where σ is the population standard deviation and n is the sample size

  
• The formula for the standard deviation of the sampling distribution of x holds true if the sample size is small compared to the population size (n/N ≤ 0.05)

  
• Finite population correction factor


  √(1 - n/N), used when n/N > 0.05
• The spread of the sampling distribution of x is smaller than the spread of the population distribution

  
• The standard deviation of the sampling distribution of x decreases as the sample size increases

  
• Sampling from a normally distributed population


  The sampling distribution of the sample mean x is also normally distributed, with mean μ and standard deviation σ/√n

  
• The shape of the sampling distribution of x is normal when the population is normal, regardless of sample size

  
• Sampling from a population that is not normally distributed

  According to the central limit theorem, for a large sample size (n ≥ 30), the sampling distribution of x is approximately normal, irrespective of the shape of the population distribution

  
• About 68.26% of the sample means will be within one standard deviation of the population mean

  
• About 95.44% of the sample means will be within two standard deviations of the population mean

  
• Central limit theorem


  If the sample size is large (n ≥ 30), the shape of the sampling distribution of the sample mean is approximately normal

  
• Sample size


  Large (n ≥ 30)

  
• The shape of the sampling distribution of the sample mean is approximately normal

  
• Sample mean


  The mean of a sample

  
• Sampling distribution of the sample mean


  The probability distribution of all possible sample means

  
• About 68.26% of sample means will be within one standard deviation of the population mean

  
• About 95.44% of sample means will be within two standard deviations of the population mean

  
• About 99.74% of sample means will be within three standard deviations of the population mean

  
• Rational agents


  Economic agents who are able to consider the outcome of their choices and recognise the net benefits of each one
• Producers act rationally by


  Selling goods/services in a way that maximises their profits
• Workers act rationally by


  Balancing welfare at work with consideration of both pay and benefits
• Governments act rationally by


  Placing the interests of the people they serve first in order to maximise their welfare
• Rationality in classical economic theory is a flawed assumption as people usually don't act rationally
• If you add up marginal utility for each unit you get total utility