23 May TMGT361
Assignment VI Instructions
Lecture/Essay
Randomness
What does random mean? It means it wasn’t controlled, that mere chance cause it (whatever it is). Random is not the same as merely being mixed, or disorganized, or no discernable pattern. If I take different densities of different sizes of pebbles and I put those pebbles in a bottle and I gently shake the bottle, the pebbles, due to their size and shape and density and how I shake the bottle will settle. Though I might be able to discern any pattern, though a different mix of pebbles or a different type of shaking could result in a near infinite settlings, the settling wasn’t random. If I use all the same size and shape and density of pebbles, and I shake the bottle enough that every pebble has an equal chance of being in every position—now I achieve, or for all practical purposes, achieve a random arraignment of pebbles.
Randomness is not achieved unless every possible outcome (every pebble location, every coin or die toss outcome, every card that can be dealt) that can occur has an equal change of occurring. If any outcome has a greater chance of occurring than another outcome (e.g., heads more than tails on a coin toss) we know instinctively that something is not right. The reason a coin toss is considered fair is because it is random, i.e., there is no impartiality or bias in the event, heads and tails had an equal and random probability of occurring.
There is an argument that noting is truly random, that if we enough information we could predict everything perfectly. Philosophically this may be true but we live in a real world where multiple forces and sequences of events affect things to the point where we can’t predict coin tosses (using fair coins and tossing the coins in a random fashion). Numerous, mostly undetectable and/or uncontrollable forces and imperfections influence every process and outcome. We cannot make a perfect circle nor a perfect drill bit. We cannot perfectly repeat every shot of the basketball. We cannot make two perfectly identical nails or ballot moves.
Do not think that every mistake or error is random; most are not; they have a cause, a special, assignable, or attributable cause (even though we may be unaware of the cause). We cannot do anything about random error (which may be termed common cause or expected error). The oldest and core part of quality as a profession or discipline is to reduce non-random error. Therefore, it would be very useful to separate random errors (which you cannot control or eliminate) from non-random errors. Sometimes the cause of the error leads to an obvious and clear mistake. Often, mistakes are not clear. Often we don’t know something is a mistake. Think of the seemingly random nature of people getting sick before we knew there were germs and how to pasteurize. There was a cause and we didn’t know it. Similarly, most actions, products, and processes are affected by random and non-random error.
Statistics have greatly helped us separate special cause from common cause error. With statistics, we can be certain of the probability that something is random or not—we may not know or every know the cause of the non-random error or how to prevent it, but we can still identify it. We will learn about statistical tests in a future lecture. Many tables, charts, and graphs that we use to make quality decisions are based on statistics and the probabilities of outcomes. Without, the table (or the math calculations that created the table) we wouldn’t know if 2 heads in row is unusual enough that we label it as non-random. Two heads in a row happens 25% of the time. If we labeled this as non-random, we would have made a bad conclusion most of time. Five heads in a row only happens due to randomness 3.125% of the time. Are we willing to label a 3% random occurrence as non-random. If I flip a coin 5 times and get 5 heads are you going to accuse me of cheating? If one hundred people each toss 5 coins, the probability is that 3 of those people will toss 5 heads. Do you accuse those three of cheating? Ten heads in a row naturally, randomly occurs slightly less than 1 time in a 1,000. If the card or lottery player makes a 1 in a 1,000 or 1 in a billion win, was it cheating? Maybe. What we would do determine (usually statistically, based on probabilities) if the outcome was unusual and then see if the cause can be determined to be random or assignable. Sometimes we just go with the odds. If I am monitoring a manufacturing process and something happens (goes wrong) that only randomly happens 1 in a 1,000 times, the other 999 times it happens it will not be random. If I take action (which takes time and money and slows production) to fix thing, I will only being trying to fix something that cannot be fixed (for random error cannot be changed) 1 in a 1,000 times.
Note the decisions have to be made concerning the above, decisions that require judgment and non-quantitative reasoning.
· What is an error, mistake, or any other occurrence that I should be concerned about. We often do not realize what we should be concerned about.
· How do I detect an error or occurrence of interest?
· The previous bullet is often answered by—how extreme does my result have to be before I label it as non-random, e.g., 1 in 10, 1 in 1,000? When do I call the coin tosser or card player a cheat? How certain do I want to be that something unusual (cheating or other non-random error) is occurring.
Initial Post
See the general assignment instructions for information about the quality and quantity expectations and evaluation criteria.
VI. Sampling and measurement. Write up the results of the following, answering the questions, discussing any results and interpretations.*
a. Get a deck of standard playing cards. Let the deck be the lot or population (n=52; don’t use the jokers). Shuffle the deck 7 times. Assume the king of hearts (the suicide king) is a defect. A good deck has no defects; a bad deck has at least one defect. What is the proportion defective based on what you know about the deck?
Pretend you don’t know anything about the population (the deck, because in real life you don’t; you only can make estimates of the population based on samples). Based only on the sample do the following (remember, you don’t know what is in the deck, you don’t know if there is a suicide king or not). Draw a random sample of 4 cards. Did you get a defect? Do you have a good or bad deck?
Return the sample of 4 to the deck (sampling with replacement) and draw another sample of 4 and repeat: for this sample, do you have a good deck or not?
b. Assume a good deck is 25% of each suit (out of 52 cards, there are 13 each of hearts, spades, diamonds, and clubs). Deal 4 cards. Did you get 25% of each suit? (You can just focus on one suit if you want; you will learn the same thing.) Why or why not? Basing your decision only on the sample, do you have good deck or not?
Put the 4 cards back and reshuffle. Repeat the above. Do this for a total of 10 samples. Average the 10 sample proportions for all 4 suits. What can you conclude about a single sample compared to the population? What can you conclude about the average of sample proportions compared to a single sample? What can you conclude about the average of sample proportions compared to the population?
Draw a sample of 3 cards. From this sample only is it possible to get 25% of each suit? Replace the sample of 3 and draw a sample of 1. Repeat with a sample of 5. Is it possible to get 25% of each suit? What conclusions can you draw from this?
With replacement, draw a sample of 10, 26, 51, and 52 (the entire population). What was the proportion of the suits for each sample? What can you conclude about the results?
c. Shuffle the deck and put it face down. Randomly remove one card and keep it face down so that you do not know what card was removed and which ones remain. Deal a sample of 4 cards. Do you have a good deck (25% of each suit)? With replacement, deal samples of 25 and 51. What can you conclude about the results?
Do all the preceding yourself (you can use someone else’s cards :). The point is for you to experience first-hand how random error can and will make the sample results, e.g., the cards you dealt, be different from the population. With any wisdom and life experience at all, you already know this. But you might not have thought about it the way we need to in this class. Note that you are being prepared to understand the concept and perform a test (judge or conclude) to see if the results you expect (from knowing about the population) are similar or not to what you got in the sample.
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