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# Introduction to Mathematical Statistics STAT 311

UW

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This 2 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 311 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/205084/stat-311-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.

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Date Created: 09/17/15

Stat 311 Approximate con dence intervals for the expectation Let X1 Xn be independent identically distributed random variables with expectation M and variance 02 The method of moments77 estimator for M is just the sample mean that is X X ngm 77 A somewhat tedious calculation shows that n EZXi 7 502 n 71W 1 i1 and the method of moments suggests the estimator for the variance where S is called the sample standard deviation The law of large numbers implies the consistency of both of these estimators that is lirn PX7m gt60 lirn PSZ702 gt60 for each 6 gt 0 The consistency of S and the central limit theorem in turn imply that X7M SW lirn Pa S b 1 ii ba e 2dxltIgtb7ltIgta so in particular 7 S 7 S PX7 7lt ltX 7 a 7u7 v If we select a so that 2ltIgta 7 1 95 that is a 1960 then we say that the interval P7a 2ltIgta71 lta SW S 7 S 7 S 7 S X7 7 X 7 X719607 X 19607 a MW W W is the 95 con dence interval for M For a 90 con dence interval select a so that 2ltIgta 7 1 90 ie a 1645 For a 97 con dence interval select a so that 2ltIgta 7 1 97 ie a 217 Of course if the value of the standard deviation 039 is known then the con dence interval is given by X7ainXai f V General principle for con dence intervals Find a function of the data and the parameter hX1 Xn whose distribution does not depend on the unknown parameter at least approximately For example if X17 Xn are exponentially distributed with parameter A then the distribution of X1 Xn does not depend on A In particular7 one doesnt need to estimate the variance to calculate a con dence interval for the parameter of an exponential distribution For a C con dence interval7 nd ac and be such that O Pac lthX1th9 ltbcm 2 Note that ac and be are not uniquely determined by They are usually selected so that the resulting con dence interval is as short as possible Solve the inequality in the probability in 2 to obtain PlX1Xn lt 9 lt uX1Xn m For example7 if hX1 7XmA X1 Xn and Pac lt hX1 7XmA lt b0 then b0 0 ac A 7 lt lt 100 X1Xn X1Xn determines the con dence interval P Problems 1 A sample of 60 meteorites found in the Arizona desert was weighed The sample mean was 58 grams7 and the sample standard deviation was 20 grams Compute an approximate large sample 95 con dence interval for the population mean E0 A sample of 1300 UW students are asked if the school colors should be changed to purple and pink 960 respond that the colors should be changed Based on this data7 construct the 95 con dence interval for the true fraction of UW students who believe that the colors should be changed 9 Verify the identity in 1 for n 3 7 Each of 40 students in a chemistry class performs 20 repetitions of an experiment measuring the percentage of iron in an ore sample Assuming that the measurements are normally distributed with expectation equal to the correct percentage7 each student calculates a 95 con dence interval for the percentage based on his or her data When the results are collected7 two of the students discover that their con dence intervals dont even overlap A student in the course who is also taking Stat 311 claims that this result is not surprising Do you agree Explain Support your explanation with calculations

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