Homework Problem 12, Due Wednesday March 5.
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Consider the not uncommon situation where you may 
have 25 iid observations  X_i , i=1,..,25, which you 
think are approximately normally distributed, but 
you are worried that the actual distribution may have 
heavier tails. A reasonable test statistic to test 
this is 

 | max_i X_i - min_i X_i | /                   (*)
      (sqrt( sum_{i=1}^n (X_i-Xbar)^2 )/24)

where Xbar is the sample mean of the 25 observations X_i, and 
the denominator is the usual sample standard deviation.  This 
statistic is sometimes called a studentized range (but it is 
not the "Tukey studentized range" arising in Multiple Comparisons 
Theory and treated in the R function ptukey), and can easily 
be seen to be invariant under affine changes of variable 
X_i -> a X_i + b.

FIND THE SIGNIFICANCE LEVEL OF THE HYPOTHESIS TEST WHICH
REJECTS WHEN THE TEST STATISTIC (*) IS AT LEAST 5.

Since this is a relatively rare event and involves a 
25-dimensional integral, it is hard to evaluate exactly.
So this exercise is to find the probability as accurately 
as you can by simulation. 

You can get only a crude evaluation of the probability by 
doing a direct simulation. 

So you should try to speed up the simulation to get 
acceptable accuracy within a feasible sample size. I suggest 
that you simulate from N(0,1) and EITHER try the method 
of control variates with control variable  .04* sum_i |X_i|^3
(because the exact mean of this variable is easy to find for
standard normal, and large values of the variable are 
associated with large values of the statistic(*)) 
OR try the method of importance sampling OR BOTH. 
(For the importance sampling distribution, we want 
to simulate from a distribution with tails something like 
the normal but heavier, but which is also easy to simulate. 
I suggest something like a symmetrized [ie multiplied by 
+ or - 1 with equal probability] r.v. with the distribution 
of the square root of a Gamma density with shape parameter 
less than but close to 0.5.  (Another possiblity is to 
simulate from a density which is proportional to Normal(0,1)
on (-b,b) and to Normal(0,sig^2) on the complement of (-b,b),
where b and c and the probability weight pwt assigned to 
(-b,b) are parameters.

NOTE: here as in the previous problem it is hard to get 
much of a speedup by any available method !!