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Chapter 11

Chapter 11. Understanding Randomness. What is Randomness?. Some things that are random: Rolling dice Shuffling cards Lotteries Bingo Flipping a coin. Randomness. Why do we need randomness? We use randomness in our data collection to give a fair and accurate picture of the world.

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Chapter 11

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  1. Chapter 11 Understanding Randomness

  2. What is Randomness? • Some things that are random: • Rolling dice • Shuffling cards • Lotteries • Bingo • Flipping a coin

  3. 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

  4. 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

  5. It’s Not Easy Being Random

  6. It’s Not Easy Being Random (cont.) • It’s surprisingly difficult to generate random values even when they’re equally likely. • Computers have become a popular way to generate random numbers. • Even though they often do much better than humans, computers can’t generate truly random numbers either. • Since computers follow programs, the “random” numbers we get from computers are really pseudorandom. • Fortunately, pseudorandom values are good enough for most purposes.

  7. It’s Not Easy Being Random (cont.) • There are ways to generate random numbers so that they are both equally likely and truly random. • The best ways we know to generate data that give a fair and accurate picture of the world rely on randomness, and the ways in which we draw conclusions from those data depend on the randomness, too.

  8. A Simulation • A simulation consists of a collection of things that happen at random. • The most basic event is called a component of the simulation. • Each component has a set of possible outcomes, one of which will occur at random.

  9. A Simulation (cont.) • The sequence of events we want to investigate is called a trial. • Trials usually involve several components. • After the trial, we record what happened—our response variable. • There are seven steps to a simulation…

  10. 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.

  11. 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?

  12. 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 • Statistic: Find the mean number of shots made.

  13. 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

  14. Example - Simulations • Use the Random Digits Table to perform a trial. • Row 1 and Column 1 of table gives • 5965 2913 • In this trial she misses her second shot. • Trial #2 • Row 2 and Column 1 of table gives • 5801 9383 • In this trial she also misses her second shot. • Trial #3 • Row 3 and Column 1 of table gives • 1223 5344 3649 • In this trial she does not miss a shot. • Trial #4 etc.

  15. 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 • <10 in pop: everyone gets a one digit number • <100 in pop: everyone gets a two digit number • Go to the table and write down numbers by • 1’s (if individuals are labeled with one digit) • 2’s (if individuals are labeled with two digits)

  16. Algorithm Continued • Throw out numbers that do not correspond to an individual in the population • Throw out repeats (we don’t want to sample the same person twice!) • First n numbers are the sample

  17. 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

  18. Example – Selecting a Random Sample • Random Numbers69051 64817 87174 09517 84534 06489 87201 • By twos:69 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}

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