# Phil: Statistical Hypothesis Testing and Mixing / Top-level Draw Essay

Submitted By juicetime
Words: 1333
Pages: 6

Introduction

Objective – The objective of this project was to simulate a small raffle drawing and analyze the effect of various mixing/drawing strategies on the randomness of the ticket numbers that are drawn.

Experiments – Forty raffle tickets (cut from a common template) were numbered 1 – 40 and dropped sequentially into a baseball cap. Four experiments were conducted based on the following mixing/drawing strategies:

A. Short Mixing/ Top-Level Draw. With this strategy, the top 2/3 of the tickets were mixed for 5 seconds. The subsequent drawing was done from tickets in the top of the stack.

B. Short Mixing/ Mixed-Level Draw. This was the same mixing strategy as above. This time, however, the drawing was from varied “random” levels and locations within the hat instead of simply from the top. The objective was to try to achieve more randomness in the draw to compensate for the small amount of mixing.

C. Medium Mixing/ Top-Level Draw. With this strategy, all levels of the tickets were mixed for 15 seconds. The subsequent drawing was done from tickets in the top of the stack.

D. Long Mixing/ Top-Level Draw. With this strategy, all levels of the tickets were mixed for 45 seconds. The subsequent drawing was done from tickets in the top of the stack.

Analyses – Two sets of analyses were conducted. The first involved analyzing the randomness of the complete drawing of all 40 raffle tickets. The second involved analyzing the randomness of the first 15 raffle tickets.

Control of Variability - In conducting this experiment, we attempted to control the variability in several ways. First, while cutting the ticket templates out, we tried to keep each ticket the same size by cutting straight down the dividing lines of the paper. Inconsistently sized tickets may distort the data of the random drawing. Also, we made sure to fully and properly fold the tickets to assure that none will stick out strangely and cause a problem with the data. We made sure to follow the experiment guideline of sequentially dropping four tickets into the hat by grouping each four close together than dropping them in still close together. With mixing the tickets, we made sure to allow the exact amount of time for mixing each set of tickets and mixed the right area of the ticket pile. Also with drawing tickets, we made sure to draw at a steady pace and list each number in its proper order and draw from the correct places.

Randomization – Within the experiment, we employed randomization in many areas. While sequentially dropping four tickets into the hat, we did not aim exactly for one place to drop the tickets. Simply, we dropped the tickets in the hat and they landed in close, but random places. Also, in regards to mixing, we randomly shuffled tickets within the hat without regard to what ticket we were touching. With drawing tickets from the hat, aside from the certain places where we were supposed to draw from, the tickets chosen were at random. In one of the steps we were also supposed to choose tickets in random places within the hat. Randomization appeared useful in several places within the experiment and much of our data shows the benefit of this process. Our data should not have any distorted errors from the lack of use of randomization.

First Analyses – Full Permutations

Overview – In the first set of analyses, 40 tickets were drawn from four different drawing/mixing strategies. The null hypothesis for all these strategies was that the resulting permutations of the 40 tickets would be random. These permutations were then compared with other permutations that could have potentially been drawn. It is assumed that tickets that are less mixed will result in a negative bias meaning that higher ticket numbers will most likely be drawn earlier. This assumption can be tested with the correlation coefficient value. It is expected that as the mixing times increase, the correlation coefficient will decrease towards 0