# Lecture15 - lecture 15 Condence Intervals I lecture 15...

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lecture 15, Confidence Intervals I 1 / 26 Confidence Intervals z -Based Confidence Intervals for a Population Mean: σ Known t -Based Confidence Intervals for a Population Mean: σ unknown t Distribution t -Based Confidence Intervals A Brief Summary lecture 15, Confidence Intervals I Outline 1 Confidence Intervals 2 z -Based Confidence Intervals for a Population Mean: σ Known 3 t -Based Confidence Intervals for a Population Mean: σ unknown t Distribution t -Based Confidence Intervals 4 A Brief Summary

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lecture 15, Confidence Intervals I 2 / 26 Confidence Intervals z -Based Confidence Intervals for a Population Mean: σ Known t -Based Confidence Intervals for a Population Mean: σ unknown t Distribution t -Based Confidence Intervals A Brief Summary lecture 15, Confidence Intervals I Sampling Distribution of Sample Mean ( review) When the population size N is infinity or large, if the population is normally distributed, or if the population is not normally distributed but the sample size n is large, then the sampling distribution of ¯ X is (exactly or approximately) normal with mean μ and standard deviation σ ¯ X σ n . The standard deviation of the sampling distribution of sample means is also called the standard error (for short, SE) of the sample mean
lecture 15, Confidence Intervals I 3 / 26 Confidence Intervals z -Based Confidence Intervals for a Population Mean: σ Known t -Based Confidence Intervals for a Population Mean: σ unknown t Distribution t -Based Confidence Intervals A Brief Summary lecture 15, Confidence Intervals I Empirical rule for the sample mean I Empirical rule for the sample mean: (a) about 68% of all possible sample means are within one standard deviation σ ¯ X of μ (b) about 95% of all possible sample means are within two σ ¯ X of μ (c) about 99.7% of all possible sample means are within three σ ¯ X of μ Typically, firstly, μ is unknown, and secondly, there’s only one sample How to make use of this rule? Can we say something about μ based on the sample?

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lecture 15, Confidence Intervals I 4 / 26 Confidence Intervals z -Based Confidence Intervals for a Population Mean: σ Known t -Based Confidence Intervals for a Population Mean: σ unknown t Distribution t -Based Confidence Intervals A Brief Summary lecture 15, Confidence Intervals I Confidence Intervals Interval Estimates Interested in population characteristics (parameters) Too expensive or impossible to obtain complete data of the population A sample is taken; sample provides useful information about the population: e.g., sample mean is a point estimate of population mean but the information is imperfect e.g., sample mean is unlikely to be exactly equal to the population mean Point estimate doesn’t provide information about the size of the estimation error In this chapter, we will study confidence intervals for a population mean interval estimates
lecture 15, Confidence Intervals I 5 / 26 Confidence Intervals z -Based Confidence Intervals for a Population Mean: σ Known t -Based

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