Elements of statistics

Elements of Statistics—Fall 2014
DUE DATE:  09/30/14
NAME: Victor Eugene Otero
General Instructions: Please place your name above, then complete the following questions. NOTE: Read the entire document below to get a feel for the activity before continuing. Make sure to save this Excel file often using the filename “yournameBonusActivity”. Once complete, submit your answers to this activity by attaching your Excel file through the completion link in the Unit 2 Bonus Activity assignment description in Blackboard. Use the area to the near right in this Excel worksheet when calculating any values/statistics/parameters. Methods/work to calculate values must be shown in the spreadsheet in order to receive full credit.
The activity is based on an old experiment coming from the days when probability was first studied with the use of coins, marbles, pegs, cards, or whatever could be found. We will analyze the paths taken by marbles as they fall through vertical boards consisting of rows of pegs. Similar probability experiments are frequently simulated in various museums around the world and in games of chance like “Plinko” on The Price is Right game show. In the 1870s, Sir Francis Galton created a device he called a quincunx for studying probability. The device was made up of a vertical board with a chute at the top. The chute was filled with marbles which were dispensed into an array of pegs. The pegs acted as obstructions, forcing the marbles to change direction, the choice of direction being (theoretically) random. At the bottom of the quincunx was a set of bins for catching the marbles. Galton’s original sketch of the quincunx, shown at the right, illustrates his setup and shows a possible arrangement of marbles in their bins after completing their journey. Notice how the pegs are arranged as alternating rows so that between two pegs in one row there is a peg in the next row. This falling marble will strike a peg in each row as it progresses. In fact, the word quincunx refers to any arrangement of five objects in a rectangle, one object at each corner and one in the middle. Galton’s idea was to use the 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 possible resting places for a marble passing through a quincunx that has one more bin than number of rows of pegs. This activity combines a number of aspects of statistics. First you will use a specific probability distribution to model the possible outcomes of a particular experiment, one with its roots in randomness. Second, you will use your model to predict the outcome of the experiment. Lastly, this project will employ simulation to replicate the experiment allowing you to compare your theortical results with an actual experimental outcome.
Beginning Analysis:
Using the link to the right, visit the sitehttp://www.mathsisfun.com/data/quincunx.html to view an animation that simulates the path of a marble through a quincunx. You can set the number of rows of pegs, number of marbles (balls), and the probability for the ball to go to the left at a pin. (For our purposes in this course we will investigate primarily the probability at 0.50, meaning equal probability to go right or to go left. For a deeper understanding of binomial distributions, you may choose to change the probability and answer the same set of questions below with this different setting) After clicking on the Start button, you will see the marble bounce downward from peg to peg until it lands in a bin. Do note that there are often a number of paths a marble can take to find a way to the same bin and that the graph at the bottom keeps track of the number of marbles (frequency values) that have fallen in each bin. You may want to use the “Fastforward” or “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 bottom. The picture at the right shows the marble beginning its descent. Suppose we choose to number the bins 0 to 3 from left to right (yes three pin rows lead to four bin possibilities!) Then for a marble to make it into bin 0 the marble would have to go left at each pin it hits (that is zero movements to the right from any pin hit.) Assuming a fair board, then since the probability is 0.5 at each pin hit, the probability of landing all the way to the left is (0.5)(0.5)(0.5)=1/8 or 0.125. Notice that this is just the mulitplication rule of probability at work. In fact, any unique path through the pegs will have probability 1/8 . But for the marble to land in bin number 1 (the second bin from the left), there are three paths starting from the top which end in this bin, 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 bounce—this is why we choose to label the bins starting at 0, each bin label corresponds to number of right bounces. Since each of these three paths leading to bin number 1 has probability 1/8, the probability of landing in this bin is 3/8–the addition rule of probability at work. Similarly you should be able to show that the probability of landing in bin number 2 is 3/8 and of the rightmost bin is 1/8. Hence the probability distribution table for this small quincunx with 3 rows is given in the probablity distribution table at the right.
Bin Probability
0 1/8
1 3/8
2 3/8

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