Understanding Randomness

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Chapter 11

• Understanding Randomness

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What is Randomness?

• Some things that are random
• Rolling dice
• Shuffling cards
• Lotteries
• Bingo
• Flipping a coin

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Randomness

• Why do we need randomness?
• We use randomness in our data collection to give
  a fair and accurate picture of the world.
• Drawing conclusions from data relies on
  randomness in data collection

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Randomness

• How do we use randomness in Statistics?
• Simulations (Chapter 11)
• Sampling and Surveys (Chapter 12)
• Experiments (Chapter 13)
• Very important for any statistical Inference

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Simulations

• Identify the component to be repeated.
• Decide how the outcome will me modeled.
• Decide how a trial will be simulated.
• State clearly what the response variable is.
• Run several trials. (as many as possible)
• Analyze the response variable.
• State conclusions in the context of the problem.

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Simulations

• A simulation consists of a sequence of random
  outcomes that model a situation.
• Example Suppose we have a basketball player who
  is an 80 free-throw shooter. How many shots can
  she make in a row without missing?

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Example - simulations

• Component the most basic event we are
  simulating a single free-throw (foul shot)
• Trial the sequence of events we want to
  investigate shooting until a miss
• Response number of shots made before the miss

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Example - Simulations

• To perform one trial
• Use a sequence of random digits (from a random
  number table)
• She makes 80 of her free-throw, so let the
  digits 0 - 7 represent shots that are made
• Let the digits 8 and 9 represent shots that are
  missed

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Example - Simulations

• Use the Random Digits Table to perform a trial.
• Row 1 and Column 1 of table gives
• 96299 07196
• In this trial she misses her first shot.
• Trial 2
• Row 2 and Column 1 of table gives
• 71622 35940
• Stop at 9 (her first miss) she takes 7 shots
  before her first miss
• Trial 3 etc.

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Algorithm for Using the Random Number Table

• We have a population of size N we want to take a
  sample size n
• Number all of the individuals in the population
• lt10 in pop everyone gets a one digit number
• lt100 in pop everyone gets a two digit number
• Go to the table and write down numbers by
• 1s (if individuals are labeled with one digit)
• 2s (if individuals are labeled with two digits)

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Algorithm Continued

• Throw out numbers that do not correspond to an
  individual in the population
• Throw out repeats (we dont want to sample the
  same person twice!)
• First n numbers are the sample

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Example - Selecting a Random Sample

• Population 30 companies
• Sample 4 companies
• Number the companies 01, 02, , 09, 10, 11, 12,
  , 30
• Go to random number table and write down numbers
  by twos
• Throw out 00 and 31 through 99
• Throw out repeats
• First 4 numbers are our sample

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Example Selecting a Random Sample

• Random Numbers69051 64817 87174 09517 84534
  06489 87201
• By twos69 05 16 48 17 87 17 40 95 17 84 53 40
  64 89 87 20
• Throw out 00 and 31 through 9905 16 17 17 17 20
• Throw out repeats05 16 17 20
• Sample 05, 16, 17, 20