Chapter 11 Chapter11 Presentation 1213 Understanding Randomness Copyright 2009 Pearson Education Inc 1 What Are Random Events If we know what events could happen but can t predict in any one case what will happen and knowing what just happened tells us nothing about what will happen next i e events are independent of each other then events are happening at random Are Random Events always equally likely Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 2 Why Be Random Statisticians don t think of randomness as the annoying tendency of things to be unpredictable or haphazard Statisticians use randomness as a tool Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 3 Using Randomness To Our Advantage One way statisticians use randomness is to help them select a representative sample from a larger population they want to study We can use randomness to help us simulate reality A simulation mimics reality by using random numbers to represent outcomes of real events Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 4 In Class Simulation Suppose a basketball player is a 75 free throw shooter meaning that in the long run they make about 75 of their free throws and miss about 25 Suppose we have this player attempt 50 free throws Would it surprise you if they made all 50 free throws If they only made 10 free throws would you begin to doubt that they were really a 75 free throw shooter What about 15 What about 20 Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 5 75 Free Throw Shooter So out of 50 attempts what is the smallest number of made free throws that would not surprise you Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 6 Simulating 50 Free Throws We can simulate this situation with 2 coins There are 4 equally likely outcomes when you flip 2 coins 1 25 Missed Free Throw 2 3 75 Made Free Throw 4 Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 7 Simulating 50 Free Throws Cont In groups of 3 simulate 50 free throws 2 people flip one coin each 1 person records made or missed and total attempts After 50 free throw attempts determine the number of made free throws Send one of your team members to the podium computer to enter your number of made free throws After all teams are finished we will examine the results Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 8 Summary of Results Number of teams Minimum made free throws Maximum made free throws Average made free throws Average made free throws Standard deviation of made free throws Shape of the distribution of made free throws Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 9 Would it make sense to use the 68 9599 7 rule for this phenomenon based on our class data Histogram Goodness of Fit Test Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 10 Regardless of your answer on the previous page use the 68 95 99 7 rule to help you decide what would be an unusually small number of made free throws out of 50 attempts for a 75 free throw shooter Chapter11 Presentation 1213 How Copyright does 2009 this match the guess you made earlier Pearson Education Inc 11 How Realistic Was Our Simulation Are there some real life factors we have failed to incorporate into our simulation The more realistic we want to make our simulation the more complicated it gets to properly simulate the outcome of interest Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 12 Simulations and Statistics Many of the things simulated in this chapter are simple enough to find an exact average standard deviation distribution etc We used these simple situations for illustration purposes However computer simulations are necessary in situations where exact answers are either too complex or impossible to determine For example How do different combinations of weather conditions and speed impact aircraft landings How many jobs are created or terminated due to fluctuations in interest rates Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 13 Simulations and Statistics Cont Most real life situations people want to simulate are influenced by many different factors The person building the simulation model must understand these many factors and how they interact and have access to good simulation software These many factors can be difficult to define and realistically simulate Some simulation models take years to make them realistic e g flight simulators Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 14 It s Not Easy Being Random If you want to do real simulations on a computer you need random numbers not coins to flip It s surprisingly difficult to generate random values even when they re equally likely Computer algorithims have become a popular way to generate random numbers Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 15 Modern Day Random Number Generators Web sites are available to generate random integers in a range of values the user specifies One such web site is http www random org They claim their randomness comes from atmospheric noise not computer algorithms Chapter11 Presentation 1213 Copyright 2009 Pearson Education Inc 16
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