+ AP Statistics: Chapter 11 Pages 258-269 Rohan Parikh Azhar Kassam Period 2 + Goals Be able to recognize random outcomes in a real-world situation Be able to recognize when a simulation might usefully model random behavior in the real-world Know how to perform a simulation either by generating random numbers on a computer or calculator, or by using some other source of random values, such as dice, a spinner, or a table of random values Be able to describe a simulation so that others could repeat it Be able to discuss the results of a simulation study and draw conclusions about the questions being investigated + Terms Random: an event is random if we know what outcomes could happen, but not which particular values will happen Random numbers: random numbers are hard to generate. Nevertheless, several internet sites offer an unlimited supply of equally random values Simulation: a simulation models random events by using random numbers to specify event outcomes with relative frequencies that correspond to the true real-world relative frequencies we are trying to model. Simulation component: the most basic situation in a simulation iin which something happens at random + Terms Outcome: an individual result of a component of a simulation is considered the outcome Trial: the sequence of several components representing events that we are pretending will take place Response variable: values of the response variable record the results of each trial with respect to what we are interested in + Steps in Conducting a Simulation Identify the component to be repeated Explain how you will model the outcome Explain how you will simulate the trial Clearly state the response variable Analyze the response variable State your conclusion in the context of the problem + + Using the Calculator To select a random number: 1) Hit MATH 2) Hit PRB menu 3) Hit 5:randInt( randInt(0,1): selects a random number between 0 and 1 randInt(1,6): selects a random number between 1 and 6, which is similar to rolling a dice randInt(1,6,2): selects two random numbers between 1 and six, very similar to rolling two dice + Example A cereal box manufacturer advertises that they put a picture of a famous athlete in each box. Tiger Woods accounts for 20%, Lance Armstrong for 30%, and the rest Serena Williams. How many boxes of cereal does one have to buy in order to get all three pictures. Step 1: Assign numbers, 0-9, to each athlete based on percentage. TW: 0,1; LA: 2,3,4; SW: 5,6,7,8,9 Step 2: Set up a random simulation and begin picking numbers between 1-10 continuously until all three groups are picked Step 3: Run multiple trials and average the number of boxes the trial takes. EX: 137 2554251 1123428 82320 4553241 On average, it takes 5.8 ((3+7+7+5+7)/5) boxes to obtain all 3 pictures. + Things to Remember Don’t overstate your case: a simulation isn’t real, so don’t stretch the data from the simulation Model the outcome chances accurately: Do not overlook key points from the data or situation Run enough trials: makes the data accurate and useable + Homework Problem #13 You take a quiz with 6 MC questions. After studying, you assume you have an 80% chance to get an individual question correct. What are the chances of getting all the questions right? 1) Assign the correct result to numbers 0-7 and incorrect to 7 & 8 2) Run 20 trials, picking 6 numbers randomly each time. If all 6 are between 0-7, all would be correct. Otherwise, you got a question wrong. 3) 915467 392782 320105 892310 704390 862973 289016 963091 312835 60067 831496 569675 326814 428944 266874 963488 274420 361605 209827 217258 All correct occurs 5 out of 20 times… 25% chance to get all right + Homework Problem #31 4 couples at a party decide to play a game. If each of the 8 people write their name on a slip, what is the chance that every person will be paired with someone other than who they came with? 1) Assign each couple 1-2, 3-4, 5-6, 7-8… ignore 0,9 2) Randomly select 4 groups of two numbers as a trials… run 11 trials 3) [8,4 1,4 6,3 6,7] [3,2 7,4 3,6 8,6] [2,4 3,2 3,8 2,5] [3,6 7,3 4,5 4,8] [2,4 2,8 4,1 3,8] [4,5 1,2 2,8 2,5] [3,5 3,6 4,6 2,5] [2,7 4,5 2,4 3,4] [1,5 1,2 2,4 5,8] [3,7 1,6 3,7 1,2] [1,8 1,5 1,7 6,8] This simulation shows that each person does not pair up with a new person 4/11 times, so 7/11 times of the time results in each person paired up with a new person. This equates to 36.4% of the time.