Statistics and Sampling Techniques

• Statistics is the art and science of gathering, organizing, analyzing, and making inferences from numerical information obtained from an experiment. Cool side note: There is evidence that statistical methods the 17^{th century.}
• Numerical information obtained from an experiment is referred to as data.
• Descriptive statistics is concerned with collection, organization and analysis of data.
• Inferential statistics is concerned with making generalizations or predictions from the data collected.
• Probability vs Statistics
  • A probability problem is solved by examining the experiment, knowing all outcomes, and computing the chances of an event based on this information.
  • A statistician will conduct an experiment a number of times, observing the results and predicting the outcomes.
  • If there is a box of marbles
  • The probability expert will open the box, count the marbles and predict what will happen.
  • The statistician will sample marbles from the box and predict what the contents of the box are.
• When performing statistics, the universal set is the population
• A subset of the population is called the sample
• Making decisions bout a population based upon a sample can be inaccurate
  • But it is frequently too costly
  • Or too expensive
  • Or not possible
  • To examine the entire population.
• Think about quality control, crash testing, ....
  • Would it be a good thing to test every car produced in a crash test?
  • Would it be possible to ask every person their opinion on who should be president?
  • Would it be feasible to check every can of soup for spoilage
• So we need to employ sampling techniques.
• There is a note that different symbols are used to represent the same statistical measurement for populations and samples
  • mean , x̄ for sample, μ for population
  • standard deviation, s for a sample, σ for a population
• When sampling, a goal is to obtain an unbiased sample or a sample that is representative of the entire population.
• There are a number of techniques employed to construct samples.
• Question: Should we build a hockey arena?
  • Biased Sampling or Purposive sampling.
    • At worst, pick items in the population which will give the answer you want
    • This is not a valid sampling technique if useful information about the population is required..
    • This can be done unintentionally, for example consult the experts in a given field.
    • Hockey Arena : Ask only students involved in athletics, or only students involved in a single degree program.
    • Hockey Arena : Or ask the people who run the athletic programs, they know the most about constructing and maintaining athletic venues and about the population of people who would use such a venue.
    • The results will be biased.
  • Convenience Sampling
    • Sample the portion of the population that is easy to reach.
    • Such a sample is most likely to be biased and unreliable.
    • The book states it is sometimes better than nothing.
    • Hockey Arena: Ask the first 10 people to exit McComb.
    • Hockey Arena: Ask the first 10 people to show up at Baron-Forness in the morning.
  • Random Sampling
    • A sample is conducted in such a way that each member of the population is equally likely to be drawn.
    • Assign a number to each item.
    • Select a set of these numbers using a random number generator.
    • You need to be sure that the sample is truly random.
    • How could we do this for the Hockey Arena?
  • Systematic Sampling
    • Sample every n^th item on a list or production line.
    • Starting point should be random.
      • Select the sampling interval (n) (every 100 items)
      • Select a random number between 0 and 99 (s)
      • Select items s,n+s, 2n+s, ... for the entire population
    • Make sure that the technique does not correspond to some element in the manufacturing process.
    • How would we do this with the Hockey Arena?
  • Cluster Sampling
    • Called area sampling
    • Break an area into smaller areas or clusters.
    • Sample all items located within a random sample of cluster.
    • Sometimes a random sample of items within the cluster is sampled.
    • Breaking a city into blocks and randomly selecting 20 blocks, t hen sampling all residents of those blocks.
    • Selecting 10 boxes of bolts and counting the defective bolts in those boxes.
    • How could we do this for the Hockey Arena?
  • Stratified Sampling
    • This method is used when there are divisions within the population
    • And the opinions of all sub-groups are desired.
      • Break the population into sub groups or strata
      • Every element of the population must fit in exactly one strata.
      • perform a sample on EVERY strata.
      • The number of samples in a given strata is based on the proportionality of the strata to the total population.
    • Examples (Classify each type of sampling):
      • All registered vehicles in the state of Georgia are classified according to type: subcompact, compact, mid-size, full-size, SUV and truck. A random sample of vehicles from each category is selected.
      • Every 10^th iPod coming off an assembly line is checked for defects.
      • A state is divided into counties. A random sample of 12 counties is selected. A random sample from each of the 12 counties is selected.
      • All customers at the Eggstra Good Breakfast Restaurant are asked if their should be a law limiting cholesterol intake.
      • The first 10 people to walk by this room are asked a question.