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Inference: Probabilities and Distributions Feb 28 - 29, 2012 A funny little thing called probability • As we noted earlier, when we take a sample and conduct a study we want to generalize (or infer) the results of our study to the wider population our sample is drawn from. • If 40 percent of our sample say they will vote Conservative we would like to estimate that this is the situation among the general population • However, we know there is a chance that if we had drawn two separate samples and done two simultaneous studies, we might have gotten different results for each sample • If the variation of results between samples is too great, then we cannot generalize our results to the wider population. • Therefore we need to know about probability to estimate the chance that our results will vary from the actual situation in the population at large. Because there is always a probability we could be wrong… • We always state our confidence interval and our margin of error • For example, a pollster might tell us there is a 95% chance that our survey result is accurate plus or minus 3% (meaning there is a 95% chance that the real support for the Conservatives among the general population is between 37% and 43%) • 95% is our confidence interval • +/- 3% is our margin of error • More will be said about these terms in later weeks In the Long-Run • The law of probability is based on a regularly documented observation that while chance can produce erratic results over the short-term or when small numbers are looked at, it generates regular and predictable outcomes over the long-term and as numbers increase • Random – Outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions (this is not a pattern). • Probability – The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. Examples of Commonly Known Probable Long-term Results • Coin Tossing – Fifty/Fifty • Stockmarket returns – Reversion to the mean • The fall of cards in a game (if you are able to count them properly and quickly enough) Randomness • Perfect Randomness is very rare and the ability to select numbers totally at random (so that each and every other number had just as good a chance of being selected and no pattern can ever be predicted) is valuable Probability Models • Sample Space “S” of a random phenomenon is the set of all possible outcomes • An Event is an outcome or a set of outcomes of a random phenomenon (the roll of the dice, the flip of the coin, etc.) • A Probability Model is a mathematical description of a random phenomenon consisting of – A sample space S – A way of assigning probability to events As in figure 10.2 in the book: There are 36 possible combinations if you roll two standard dice. If we wanted to define a sample space S for (5) it would be comprised of the four possible ways to roll 5 (i.e. the four “events” that result in 5 A ={ roll 1 & 4, roll 2 & 3, roll 3 & 2, roll 4 &1} Graphics: Moore 2009 Some Formal Probability Rules • A probability is a number between 0 and 1 – An event with a probability of 0 ought never to occur, An event with a probability of 1 out to always occur • All possible outcomes together must equal 1 • If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. Eg. If one event occurs in 40% of cases, and the other in 25% and the two cannot occur together then the probability of one or the other occurring is 65% • The probability that an event does not occur is 1 minus the probability that it does occur Discrete vs. Continuous Models • Discrete Probability Models – Assume that the sample space is finite – To assign probabilities list the probabilities of all the individual outcomes (must be between 1 and 0 and add up to 1).The probability of an event is the sum of the outcomes making up the event. – Think of our dice example: What is the probability of rolling a five? • • • • • roll 1 & 4 = 1/36 roll 2 & 3 = 1/36 roll 3 & 2 = 1/36 roll 4 &1 = 1/36 Total Probability = 4/36 = 1/9 = 0.111 • Continuous Probability Models – Assign probabilities as areas under a density curve (such as the normal curve) – The area under the curve and above any range of values is the probability of an outcome in that range – This is what we did in chapter 3! Break Time Sampling Distributions • Some Key Words – Parameter: a number that describes the population. We often can only speculate on this as we only have data for a sample. – Statistic: is a number that can be computed from the sample data without making use of any unknown parameters. We often use statistics to estimate parameters. • The Law of Large Numbers: – Draw observations at random from any population with finite mean µ . – As the number of observations drawn increases, the mean xof the observed values gets closer and closer to the mean µ of the population . Two types of distribution of variables (be careful) • The Population Distribution of a variable is the distribution of values of the variable in the population • The Sampling Distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population (in other words, how a statistic varies in many samples drawn from the same population) For those who like math Suppose that X is the mean of a SRS of size “n” drawn from a large population with mean and standard deviation Then the sampling distribution of and standard deviation n X has mean The Central Limit Theorem • In fact life gets better still • As we saw earlier, the mean of a sampling distribution will approach the mean in the population if you draw a big enough sample often enough • Something better happens with the shape of this sampling distribution – Even if the population distribution is not normal, when the sample is large enough, the distribution of the mean changes shape so as to approach normal (provided the population has a finite standard deviation). Bottom Line and Caution • If we can compute the average for a large random sample we have a decent guess as to what the average is in the population • The average of a sample is generally a better guess of the average of the population than any one case in the population. In other words, based on a survey of incomes, I can tell you with a reasonable level of certainty what the average income of Canadians is. I cannot tell you just from that what the income of any specific Canadian is.