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UW-Madison STAT 371 - STAT 371 Discussion 6

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STAT371 DISCUSSION 6 October 13, 20021. Statistical estimation• The estimate of population meanµ is sample mean ¯y• The estimate of population standard deviation σ is sample stan-dard deviation s.2. The standard error of the mean isSE¯y=s√nwhich is a measure of the reliability or precision of ¯y as an estimate ofµ: the smaller the SE, the more precise the estimate.Consider: What is the distincton between standard error and standarddeviation?3. Confidence interval• The construction of confidence interval:If the sample size is n, sample mean is ¯y, and the standard erroris SE¯y, then the (1 −α)% confidence interval for µ is constructedas follows:¯y ± tα2s√nwhere the criticl value tα2is determined from Student’s t distri-bution with df = n − 1. For instance, if α = 10, then the 90%confidence interval is ¯y ± t0.5SE¯y.• The interpretation of a confidence interval:Suppose the 95% confidence interval of µ is (a,b), which of thefollowing statement is true?– Pr{a< µ <b}=95%– We are 95% confidence that the population mean µ is betweena and b.– Pr{a< ¯y <b}=95%1– We are 95% confidence that the sample mean ¯y is between aand b.– If we take 100 samples from the population and construct 10095%confidence intervals. Then there will be 95 confidenceintervals containing µ.– Pr{the next sample will give us a confidence interval thatcontains µ}=0.954. Planning a study to estimate µTo get a desired standard error, the sample size should be:n ≥ (Guessed SDDesired SE)2Exercise:Y follows a normal distribution with mean 20 and standand deviation 2.Take a sample from the population and get these data:19.10672 20.49547 19.20281 16.81740 19.18170 19.44320 18.34311 19.5148119.22503 25.53221 20.24905 18.80119 21.46908 sx• get sample mean¯Y and sample standard deviation s• the sample error• Pr{19<¯Y <22}• the 90% confidence interval for µ• if we want the standard error to be less than 0.1, how large should thesample


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UW-Madison STAT 371 - STAT 371 Discussion 6

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