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Unformatted text preview: ISYE 2028 A and B Lecture 10 Sampling Distributions and Test Statistics Dr. Kobi Abayomi March 13, 2009 1 Introduction: The context for Confidence Intervals and Hypothesis Testing: Sampling Distributions for Test Statistics Here is a (non-exhaustive) illustration of the population—sample dichotomy that is the center of what we are studying in this introductory course. Population Sample Random Variable Statistic Population Mean, Expectation Sample Mean Parameter Estimate μ x We make assumptions or define a population to ”fit” observed data. Our data is information about events we wish to speak—or gain inference about. The natural framework is that of an experiment: the population composes our assumptions about what might happen; the sample data compose what we actually observe. Our beliefs about what we see—that is the sample distribution, are related to our general assumptions— that is the population distribution. We have canonical population models in our overview of random variables. Bernoulli, Bi- nomial, Poisson, Normal, Exponential, etc. characterize types of experiments; we use these characterizations to make statements about data. 1 Bernoulli distribution to model simple events that can either happen or not. Like whether a coin turns up heads or not. Binomial distribution to model sums or totals of Bernoulli events. Like whether a coin turns up heads k times in n tosses. Poisson distribution to model Binomial type events when the probability of any event is very low, and the number of events is very high. Like the number of soldiers who are kicked in the head in a military campaign in 18th century France. Exponential distribution to model continuous, positive events like waiting times, or time to failure. Normal distribution to model averages of events, or events where the outcomes are contin- uous, or when we just don’t know any better (ha!). Chi-Square distribution to model squared deviations, sums of squared deviations, and squared normal random variables. Moving on, we use our these canonical random variables, to make statements about observed data. The setup is almost always this: we compare observed data to an expected value under our assumptions. This comparison yields a test statistic . We then use our probability model (i.e. our fundamental assumption about population for the data) to make a probabilistic statement about the population parameter. In general, a test statistic looks like this: TestStatistic = observed value- expected value standard error (1) In general the ”observed value” will be some statistic or function of data. The ”expected value” will be some parameter , the population correspondence of the statistic. We call statistics used in this context – to estimate population parameters – estimators . A popular notation is to use ˆ θ , read ”theta-hat”, as an estimator of the population parameter θ . We have already been exposed to one such estimator: ˆ μ = x – the sample mean....
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