Dennis:
Example A is a mistaken interpretation of a confidence interval for a mean.
Unfortunately, this is is a very common misinterpretation.
What you have described in Example A is a _prediction_ interval for
an individual observation. Prediction intervals rarely get taught except
(maybe)
in the context of a regression model.
Jon
At 03:11 PM 9/26/01 -0400, you wrote:
>as a start, you could relate everyday examples where the notion of CI seems
>to make sense
>
>A. you observe a friend in terms of his/her lateness when planning to meet
>you somewhere ... over time, you take 'samples' of late values ... in a
>sense you have means ... and then you form a rubric like ... for sam ... if
>we plan on meeting at noon ... you can expect him at noon + or - 10 minutes
>... you won't always be right but, maybe about 95% of the time you will?
>
>B. from real estate ads in a community, looking at sunday newspapers, you
>find that several samples of average house prices for a 3 bedroom, 2 bath
>place are certain values ... so, again, this is like have a bunch of means
>... then, if someone asks you (visitor) about average prices of a bedroom,
>2 bath house ... you might say ... 134,000 +/- 21,000 ... of course, you
>won't always be right but .... perhaps about 95% of the time?
>
>but, more specifically, there are a number of things you can do
>
>1. students certainly have to know something about sampling error ... and
>the notion of a sampling distribution
>
>2. they have to realize that when taking a sample, say using the sample
>mean, that the mean they get could fall anywhere within that sampling
>distribution
>
>3. if we know something about #1 AND, we have a sample mean ... then, #1
>sets sort of a limit on how far away the truth can be GIVEN that sample
>mean or statistic ...
>
>4. thus, we use the statistics (ie, sample mean) and add and subtract some
>error (based on #1) ... in such a way that we will be correct (in saying
>that the parameter will fall within the CI) some % of the time ... say, 95%?
>
>it is easy to show this via simulation ... minitab for example can help you
>do this
>
>here is an example ... let's say we are taking samples of size 100 from a
>population of SAT M scores ... where we assume the mu is 500 and sigma is
>100 ... i will take a 1000 SRS samples ... and summarize the results of
>building 100 CIs
>
>MTB > rand 1000 c1-c100; <<< made 1000 rows ... and 100 columns ... each
>ROW will be a sample
>SUBC> norm 500 100. <<< sampled from population with mu = 500 and sigma = 100
>MTB > rmean c1-c100 c101 <<< got means for 1000 samples and put in c101
>MTB > name c1='sampmean'
>MTB > let c102=c101-2*10 <<<< found lower point of 95% CI
>MTB > let c103=c101+2*10 <<<< found upper point of 95% CI
>MTB > name c102='lowerpt' c103='upperpt'
>MTB > let c104=(c102 lt 500) and (c103 gt 500) <<< this evaluates if the
>intervals capture 500 or not
>MTB > sum c104
>
>Sum of C104
>
> Sum of C104 = 954.00 <<<< 954 of the 1000 intervals captured 500
>MTB > let k1=954/1000
>MTB > prin k1
>
>Data Display
>
>K1 0.954000 <<<< pretty close to 95%
>MTB > prin c102 c103 c104 <<< a few of the 1000 intervals are shown below
>
>Data Display
>
>
> Row lowerpt upperpt C104
>
> 1 477.365 517.365 1
> 2 500.448 540.448 0 <<< here is one that missed 500 ...the
>other 9 captured 500
> 3 480.304 520.304 1
> 4 480.457 520.457 1
> 5 485.006 525.006 1
> 6 479.585 519.585 1
> 7 480.382 520.382 1
> 8 481.189 521.189 1
> 9 486.166 526.166 1
> 10 494.388 534.388 1
>
>
>
>
>
>_________________________________________________________
>dennis roberts, educational psychology, penn state university
>208 cedar, AC 8148632401, mailto:[EMAIL PROTECTED]
>http://roberts.ed.psu.edu/users/droberts/drober~1.htm
>
>
>
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___________
----------------------------------------------- | \
Jon Cryer, Professor Emeritus ( )
Dept. of Statistics www.stat.uiowa.edu/~jcryer \ \_University
and Actuarial Science office 319-335-0819 \ * \of Iowa
The University of Iowa home 319-351-4639 \ /Hawkeyes
Iowa City, IA 52242 FAX 319-335-3017 |__________ )
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"It ain't so much the things we don't know that get us into trouble.
It's the things we do know that just ain't so." --Artemus Ward
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