# Consider the situation described in Exercise 83. a. What is P(x1 , , x2 . , x3 . ,…, xn . ), that…

Consider the situation described in Exercise 83.

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

xi

’s is less than ?

d. What is P( , y2), where y2 denotes the second

smallest xi

? Hint:  , y2 occurs if either all n of

the observations exceed the median or all but

one of the xi

’s does.

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

Exercise 83.

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

observations.

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

three observations?

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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