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Unformatted text preview: stat 226 homework 5 Answer Key 1. Why?  Why do we need to do this???? While ¯ x as a point estimate is unbiased (on average it will estimate the population mean correctly) it will rarely equal the value of the unknown population mean μ . Our estimate often will be close to μ but the estimate does not provide any information on how close it will be to μ . A confidence interval allows us not only to incorporate the variability of our sample estimator ¯ x (how much does ¯ x usually vary about the unknown population mean μ ) but also to attach a level of confidence. We are able to make a statement how confident we can be that the population mean μ will be contained in the obtained confidence interval. In general, a point estimator is never considered very informative without providing any information of how much variability is associated with this estimator. 2. Critical Values. (a) z * = 1 . 02 (b) z * = 1 . 75 (c) z * = 2 . 17 (d) As the level of confidence increases so does the value of z * . This should make sense intuitively because z * contributes to the width of confidence interval. A larger z * value will cause the interval to get wider which in turn yields more confidence that the interval indeed captures the unknown population mean. 3. Confidence Intervals. (a) ¯ x ± z * σ √ n ⇒ 50 ± 1 . 02 10 √ 25 ⇒ 50 ± 2 . 04 ⇒ (47 . 96 , 52 . 04) (b) ¯ x ± z * σ √ n ⇒ 50 ± 1 . 75 10 √ 25 ⇒ 50 ± 3 . 50 ⇒ (46 . 50 , 53 . 50) (c) ¯ x ± z * σ √ n ⇒ 50 ± 2 . 17 10 √ 25 ⇒ 50 ± 4 . 34 ⇒ (45 . 66 , 54 . 34) (d) As the confidence level increases, the width of the interval also increases. The only thing that changes is the critical value, which gets larger with higher confidence levels (See 2.(d))....
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This note was uploaded on 05/05/2010 for the course STAT 226 taught by Professor Abbey during the Fall '08 term at Iowa State.
 Fall '08
 ABBEY

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