Lesson 4.pdf - Lesson 4 Confidence Intervals for Means What...

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Lesson 4: Confidence Intervals for Means What is CI and why? We talked about two different methods of point estimation namely likelihood estimator and methods of moments estimator, for unknown parameters. For example we know that sample mean ࠵? is an unbiased estimator for unknown population mean ࠵? when the sample is coming from a normal population. In fact, ࠵? is an unbiased estimator for unknown population mean ࠵? regardless the distribution from which we sampled. But depending on the sample we get, sample mean may or may not be close to the population mean. Instead of giving a single value as the estimate (e.g. ࠵? for population mean ࠵? ) we give range of values to reflect the uncertainty of just using a subset of the population. Recall from STAT 318… Central Limit Theorem Suppose we have a random sample of size n from a population with mean ࠵? and standard deviation ࠵? . Then, CLT says that when n is large, sample mean ࠵? ! has an approximately normal distribution with mean ࠵? and standard deviation ࠵? / ࠵? . Note: i. If the parent population is Normal then the distribution of ࠵? ! is exactly Normal for any sample size. ii. If the parent population is not Normal then the accuracy of the approximation depends on the sample size n (usual cutoff is 30) . Empirical Rule (68-95-99.7 Rule) In a normal distribution with mean ࠵? and standard deviation ࠵? . Z-score If ࠵? ~ ࠵? ( ࠵? , ࠵? ! ) then ࠵? = ! ! ! ! is called the z-score and normally distributed with mean 0 and standard variance 1 (i.e: ࠵? ~ ࠵? ( 0 , 1 ) ) Ø About 68% of the data falls within ࠵? of ࠵?
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