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UT Knoxville STAT 201 - Chapter 11

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1Chapter11 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 11 Understanding RandomnessChapter11 Presentation 1213Copyright © 2009 Pearson Education, Inc.2What 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 1213Copyright © 2009 Pearson Education, Inc.3Why 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 1213Copyright © 2009 Pearson Education, Inc.4Using 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 1213Copyright © 2009 Pearson Education, Inc.5In 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 1213Copyright © 2009 Pearson Education, Inc.6“75%” Free Throw Shooter So, out of 50 attempts, what is the smallestnumber of made free throws that would notsurprise you?Chapter11 Presentation 1213Copyright © 2009 Pearson Education, Inc.7Simulating 50 Free Throws We can simulate this situation with 2 coins. There are 4 equally likely outcomes when you flip 2 coins:75% - Made Free Throw25% - Missed Free Throw1234Chapter11 Presentation 1213Copyright © 2009 Pearson Education, Inc.8Simulating 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 1213Copyright © 2009 Pearson Education, Inc.9Summary 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 1213Copyright © 2009 Pearson Education, Inc.10Would it make sense to use the 68-95-99.7 rule for this phenomenon based on our class data?Histogram?Goodness of Fit Test?Chapter11 Presentation 1213Copyright © 2009 Pearson Education, Inc.11Regardless 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.How does this match the guess you made earlier?Chapter11 Presentation 1213Copyright © 2009 Pearson Education, Inc.12How 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 1213Copyright © 2009 Pearson Education, Inc.13Simulations 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 1213Copyright © 2009 Pearson Education, Inc.14Simulations 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 1213Copyright © 2009 Pearson Education, Inc.15It’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 1213Copyright © 2009 Pearson Education, Inc.16Modern 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


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UT Knoxville STAT 201 - Chapter 11

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