BONUS ActivityMATH 250 Elements of StatisticsFall 2016 DUE DATE: 10/04/2016 General Instructions: Please place your name above, then complete the following question feel for the activity before continuing. Make sure to save this Excel file often using the filena submit your answers to this activity by attaching your Excel file through the completion link in Blackboard. Use the area to the near right in this Excel worksheet when calculating any v calculate values must be shown in the spreadsheet in order to receive full credit. Overview: Introduction: The activity is based on an old experiment coming from the days when probability w cards, or whatever could be found. We will analyze the paths taken by marbles as th pegs. Similar probability experiments are frequently simulated in various museums a The Price is Right game show. In the 1870s, Sir Francis Galton created a device he called a quincunx for studying p with a chute at the top. The chute was filled with marbles which were dispensed into forcing the marbles to change direction, the choice of direction being (theoretically) r for catching the marbles. Galton's original sketch of the quincunx, shown at the right, illustrates his setup and after completing their journey. Notice how the pegs are arranged as alternating rows there is a peg in the next row. This falling marble will strike a peg in each row as it pr any arrangement of five objects in a rectangle, one object at each corner and one in quincunx to study and explain the final distribution of marbles among the bins. This activity's goal is to have you describe, mathematically and probabilistically, the p through a quincunx that has one more bin than number of rows of pegs. This activit First you will use a specific probability distribution to model the possible outcomes o randomness. Second, you will use your model to predict the outcome of the experim to replicate the experiment allowing you to compare your theortical results with an ac Beginning Analysis: Using the link to the right, visit the sitehttp://www.mathsisfun.com/data/quincunx.htm marble through a quincunx. You can set the number of rows of pegs, number of ma left at a pin. (For our purposes in this course we will investigate primarily the probab go left. For a deeper understanding of binomial distributions, you may choose to ch questions below with this different setting) After clicking on the Start button, you will it lands in a bin. Do note that there are often a number of paths a marble can take t bottom keeps track of the number of marbles (frequency values) that have fallen in e \"Maximum Speed\" button to increase the speed of the process. For simplicity, let's look at a case with only 3 rows of pegs and hence 4 bins at the b beginning its descent. Suppose we choose to number the bins 0 to 3 from left to rig Then for a marble to make it into bin 0 the marble would have to go left at each pin i hit.) Assuming a fair board, then since the probability is 0.5 at each pin hit, the prob (0.5)=1/8 or 0.125. Notice that this is just the mulitplication rule of probability at work probability 1/8 . But for the marble to land in bin number 1 (the second bin from the left), there are th namely: bounce left, bounce left, bounce right bounce left, bounce right, bounce left bounce right, bounce left, bounce left Notice that each of these paths has one right bouncethis is why we choose to labe For simplicity, let's look at a case with only 3 rows of pegs and hence 4 bins at the b beginning its descent. Suppose we choose to number the bins 0 to 3 from left to rig Then for a marble to make it into bin 0 the marble would have to go left at each pin i hit.) Assuming a fair board, then since the probability is 0.5 at each pin hit, the prob (0.5)=1/8 or 0.125. Notice that this is just the mulitplication rule of probability at work probability 1/8 . But for the marble to land in bin number 1 (the second bin from the left), there are th namely: bounce left, bounce left, bounce right bounce left, bounce right, bounce left bounce right, bounce left, bounce left Notice that each of these paths has one right bouncethis is why we choose to labe number of right bounces. Since each of these three paths leading to bin number 1 h 3/8--the addition rule of probability at work. Similarly you should be able to show tha the rightmost bin is 1/8. Hence the probability distribution table for this small quincun at the right. Now that you hopefully understand the idea of the quincunx and how to analyze the move on to the exercises for this project. *Citation NOTE: This activity is taken and slightly adjusted from an activity published by Addison Wesley http://occawlonline.pearsoned.com/bookbind/pubbooks/triola_awl/chapter4/essay1/ 1. Suppose the six bins at the bottom of a five row quincunx are numbered from 0 to 5. any single marble landing in bin number n is answered through the use of a binomia of "success"is . You may want to go back and reread portions of Section 5-3 in the produce your argumentreflect on what has to happen for the marble to land in bin how you define \"success", S, and \"failure", F, as well as why p = .. 2. Using what your have learned from the resources of this course in regard to binomia distribution table to the right for the case n = 7 and p = . You are required to use th probability values within the table. 3. Using your probability distribution created in #2, predict how many marbles would lan the quincunx one at a time. Calculate these predictions at the right. 4. Use the applet simulation at http://www.mathsisfun.com/data/quincunx.html (see the number of rows to 7, to determine the number of marbles that fall in each bin when t performed. State the results of the simulation at the right. Compare with your predic Discuss why your predictions and the simulation agree/disagree. The results are different because the results from question 3 are ba questions are based off real data that came from putting the experim 5. Suppose you could build a quincunx that would accommodate any number of pegs, would see shape-wise in the distribution of the marbles into the bins as the number pass through? I think the shape would still be the same, because even when puttin marbles tend fo fall into the middle and not on the sides. I would pr pegs, rows,bins, and marbles you choose. (You may want to also look at what happens if the quincunx is not "fair"--that is if a m 60% chance of going right. Are you surprised by the resulting distribution of marbles predictable and useful for analyzing certain distributions and two-outcome situations NAME: Courtney Zuk te the following questions. NOTE: Read the entire document below to get a file often using the filename "yournameBonusActivity". Once complete, ough the completion link in the Unit 2 Bonus Activity assignment description et when calculating any values/statistics/parameters. Methods/work to e full credit. he days when probability was first studied with the use of coins, marbles, pegs, aths taken by marbles as they fall through vertical boards consisting of rows of ated in various museums around the world and in games of chance like \"Plinko\" on d a quincunx for studying probability. The device was made up of a vertical board which were dispensed into an array of pegs. The pegs acted as obstructions, ction being (theoretically) random. At the bottom of the quincunx was a set of bins ht, illustrates his setup and shows a possible arrangement of marbles in their bins rranged as alternating rows so that between two pegs in one row e a peg in each row as it progresses. In fact, the word quincunx refers to at each corner and one in the middle. Galton's idea was to use the bles among the bins. y and probabilistically, the possible resting places for a marble passing f rows of pegs. This activity combines a number of aspects of statistics. el the possible outcomes of a particular experiment, one with its roots in he outcome of the experiment. Lastly, this project will employ simulation theortical results with an actual experimental outcome. un.com/data/quincunx.html to view an animation that simulates the path of a ows of pegs, number of marbles (balls), and the probability for the ball to go to the http://www.mathsisfun.com/data/quincunx.html stigate primarily the probability at 0.50, meaning equal probability to go right or to ons, you may choose to change the probability and answer the same set of on the Start button, you will see the marble bounce downward from peg to peg until paths a marble can take to find a way to the same bin and that the graph at the values) that have fallen in each bin. You may want to use the \"Fastforward\" or ocess. and hence 4 bins at the bottom. The picture at the right shows the marble e bins 0 to 3 from left to right (yes three pin rows lead to four bin possibilities!) have to go left at each pin it hits (that is zero movements to the right from any pin .5 at each pin hit, the probability of landing all the way to the left is (0.5)(0.5) n rule of probability at work. In fact, any unique path through the pegs will have n from the left), there are three paths starting from the top which end in this bin, is is why we choose to label the bins starting at 0, each bin label corresponds to Bin 0 Bin 0 1 2 3 Bin 1 Bin 2 Bin 3 Probability 1/8 3/8 3/8 1/8 nx and how to analyze the associated probabilities with each bin, you are ready to ity published by Addison Wesley at er4/essay1/ are numbered from 0 to 5. Make a case (argue) that the probability distribution of rough the use of a binomial distribution process where n is 5 and the probability p ortions of Section 5-3 in the text and draw a picture of the quincunx to help you or the marble to land in bin number 0, Bin number 3, etc. Don't forget to mention why p = .. course in regard to binomial distributions, produce the binomial probability You are required to use the Excel function =BINOM.DIST( ) to create the Bin 0 1 2 3 4 Marbles 0.008 0.055 0.164 0.273 0.273 ow many marbles would land in each of the 8 bins if 200 marbles were sent down the right. ata/quincunx.html (see the top link above next to the Overview region), setting the that fall in each bin when this trial of 200 marbles experiment is actually Compare with your predictions from exercise 3 above. Do they agree exactly? sagree. ults from question 3 are based off just an equation, where the results in this me from putting the experiment to practice. date any number of pegs, bins and marbles you wanted. What do you think you to the bins as the number of bins increases and a "near" infinite number of marbles because even when putting in the highest number of rows on the website (19), the ot on the sides. I would predict the results would be the same no matter how many e. nx is not "fair"--that is if a marble only has, say, a 40% chance of going left and a lting distribution of marbles? This is why binomial distributions are so very nd two-outcome situations.) 5 6 7 0.164 0.055 0.008 Bin 0 1 2 3 4 5 6 7 Marbles 2 11 33 55 55 33 11 2 Bin 0 1 2 3 4 5 6 7 Marbles 0 12 28 49 64 37 10 0 .com/data/quincunx.html