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Statistics : Statistical Inference Krishna.V.Palem Kenneth and Audrey Kennedy Professor of Computing Department of Computer Science, Rice University 1

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Contents Summary of Statistics Learnt so Far Statistical Inference Central Limit Theorem and its implications Estimation theory Interval Estimation What is Confidence Interval? Tutorial 2

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Statistical Inference The process of making guesses about the truth from a sample Sample (observation) Make guesses about the whole population Truth (not observable) Population parameters Sample statistics *hat notation ^ is often used to indicate “estitmate” 3 Source: K. Cobb, Stanford

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4 Statistical Inference Population (parameters, e.g., and ) select sample at random Sample collect data from individuals in sample Data Analyse data (e.g. estimate ) to make inferences

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5 How close is Sample Statistic to Population Parameter ? Population parameters, e.g. and are fixed Sample statistics, e.g. vary from sample to sample How close is to ? Cannot answer question for a particular sample Can answer if we can find out about the distribution that describes the variability in the random variable

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Contents Summary of Statistics Learnt so Far Statistical Inference Central Limit Theorem and its implications Estimation theory Interval Estimation What is Confidence Interval? Tutorial 6

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The Central Limit Theorem: If all possible random samples, each of size n, are taken from any population with a mean and a standard deviation , the sampling distribution of the sample means (averages) will: 1. have mean: 2. have standard deviation: 3. be approximately normally distributed regardless of the shape of the parent population (normality improves with larger n). 7

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What is it really saying? (1) It gives a relationship between the sample mean and population mean This gives us a framework to extrapolate our sample results to the population (statistical inference); (2) It doesn’t matter what the distribution of the original data is, the sample mean will always be Normally distributed when n is large. This why the Normal is so central to statistics 8

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Example: Toss 1, 2 or 10 dice (10,000 times) Toss 1 dice Histogram of data Toss 2 dice Histogram of averages Toss 10 dice Histogram of averages Distribution of data is far from Normal Distribution of averages approach Normal as sample size (no. of dice) increases 9

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Central Limit Theorem (3) It describes the distribution of the sample mean The values of obtained from repeatedly taking samples of size n describe a separate population The distribution of any statistic is often called the sampling distribution

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