a. What is P(x1 , , x2 . , x3 . ,…, xn . ),
that is, the probability that only the first observation is smaller than the median and all
others exceed the median?
b. What is the probability that only x2 is smaller
than the median and all other n21 observations
exceed the median?
c. What is the probability that exactly one of the
’s is less than ?
d. What is P( , y2), where y2 denotes the second
? Hint: , y2 occurs if either all n of
the observations exceed the median or all but
one of the xi
e. With yn21 denoting the second largest xi
, what is
P( . yn21)?
f. Using the results of parts (d) and (e), what is
P(y2 , , yn21)? What does this imply about
the confidence level associated with the interval (y2, yn21)? Determine the interval and associated confidence level for the data given in
Suppose we have obtained a random sample
x1,…, xn from a continuous distribution and wish
to use it as a basis for predicting a single new observation xn11 without assuming anything about the
shape of the distribution. Let y1 and yn denote the
smallest and largest, respectively, of the n sample
a. What is P(xn11 , x1)?
b. What is P(xn11 , x1 and xn11 , x2), that is, the
probability that xn11 is the smallest of these
c. What is P(xn11 , y1)? What is P(xn11 . yn)?
d. What is P(y1 , xn11 , yn), and what does this
say about the prediction level associated with
the interval (y1, yn)? Determine the interval and
associated prediction level for the curing time
data given in Exercise 83
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