Re: small deviation, great effect

[Jim Stapleton <stapleton@stt.msu.edu>]

> Dear All:
> Can anybody please indicate a real-life example where even small
> deviation from normality in a very large sample produces a noticeable
> detrimental effect?
> Thank you
> Xao Ping
> R@R Pharmakinetics
> Taiwan 


Let F(x) = .999 phi(x) + .001 G(x), where G is the cdf which puts mass
1  at 1 million. phi is the std normal cdf. F and phi are
"close", depending of course on how "closeness" is
measured.
 Let X be a std. normal cdf with prob .999 and be 1 million with
prob. 0.001.    X has cdf F(x). 
.
For small samples from F (say < 100) the sample is unlikely to contain
the 1 million, and the sample mean will be small . However, for n = k
(1000), with k not close to zero, the probability that the sample with
contain the 1 million is approximately 1 - exp(-k) , and the mean will be
at least (1 million)/(1000 k) = 1000/k. 

Jim Stapleton


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Jim Stapleton
Professor and Graduate Director
Dept. of Statistics and Probability
Michigan State University
517-355-9678