Re: st: Re: Confidence Interval vs Confidence in Judgement

[Steven Samuels <sjhsamuels@earthlink.net>]

> "the smaller the t-value,the smaller is the confidence interval"


That is right. The multiplier for the standard deviation for P level  
confidence is the (1-P)/2 percentile of the t-distribution. The lower  
P is, the shorter is this percentile.  Suppose the degrees of freedom  
for t is df 50. Then the multipliers for the SD for some confidence  
levels from 70% to 99%  are:
 foreach p of numlist .7 .8 .9 .95 .99 {
  2. di "Confidence Level = " 100*`p'   ". Multiplier = " invttail 
(50,(1-`p')/2)
  3. }
Confidence Level = 70. Multiplier = 1.0472949
Confidence Level = 80. Multiplier = 1.2987137
Confidence Level = 90. Multiplier = 1.675905
Confidence Level = 95. Multiplier = 2.0085591
Confidence Level = 99. Multiplier = 2.6777933


As you can see, a longer interval increases the confidence  
(probability) that the interval includes the true value. A shorter  
interval comes with lower confidence. Ninety-five percent CI's are so  
sometimes so long as to be uninformative. In such cases I would  
accept the trade-off of lower confidence (90%, 80%) to gain a shorter  
interval. Although I don't have the reference, I believe that John  
Tukey advocated the display of CI's with several levels of  
confidence. Such a display leaves the choice of confidence to the  
reader.
-Steve
On Oct 10, 2008, at 5:13 PM, Victor M. Zammit wrote:

> The statement that I use to compute my confidence interval is the  
> usual
> ttest, ie the difference between the mean of the random sample and  
> the mean
> of the population, multiplied by the square root of the sample size  
> and the
> product is divided by the standard deviation of the random  
> sample.Because
> the result is normalised,you could infer from it the confidence  
> interval,for
> the particular degree of freedom.
> The problem that I am having is that the smaller the t-value,the  
> smaller is
> the confidence interval,but the higher is the confidence in your  
> judgement
> call (claim),which seems to be counter-intuitive for me.
> Victor M. Zammit
>
>
>
> -- Original Message ----- 
> From: "Steven Samuels" <sjhsamuels@earthlink.net>
> To: <statalist@hsphsun2.harvard.edu>
> Sent: Friday, October 10, 2008 6:20 PM
> Subject: Re: st: Re: Confidence Interval vs Confidence in Judgement
>
>
> > Victor, show us the statements that you are using to compute your
> > confidence interval.
> >
> > -Steve
> > On Oct 10, 2008, at 12:11 PM, Victor M. Zammit wrote:
> >
> > > Dear Stata users,
> > > I am reproducing a t-table,for degrees of freedom, from 1 to 30,and
> > > after
> > > taking 40,000 random samples  of obs.,from  2 to 31,each time  
> > > from an
> > > infinite normally distributed population ,and repeated the whole
> > > process for
> > > 10 times,my ttable has converged pretty much to that of Fisher and
> > > Yates.
> > > The program is very simple and I would be very glad to reproduce it
> > > to any
> > > one interested.
> > > But having established the various confidence intervals associated
> > > with the
> > > t-values for the degrees of freedom indicates above,I am finding it
> > > counter-intuitive, that the closer the t-value is to 0,and hence
> > > the closer
> > > you are to being correct in your judgement,the smaller the resulting
> > > confidence interval.Obviously,I am confusing high confidence with  
> > > wide
> > > confidence interval.
> > > I would like to know of other terminology that would make the
> > > concept less
> > > counter-intuitive.
> > > I thank you in advance,
> > > Victor M. Zammit
> > >
> > > *
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