266 Some Elementary Statistical Inferences(a)P(Y 1 <ξ 0. 5 <Y 5 ).(b)P(Y 1 <ξ 0. 25 <Y 3 ).(c)P(Y 4 <ξ 0. 80 <Y 5 ).4.4.26.ComputeP(Y 3 <ξ 0. 5 <Y 7 )ifY 1 <···<Y 9 are the order statistics of a
random sample of size 9 from a distribution of the continuous type.4.4.27.Find the smallest value ofnfor whichP(Y 1 <ξ 0. 5 <Yn)≥ 0 .99, whereY 1 <
···<Ynare the order statistics of a random sample of sizenfrom a distribution of
the continuous type.
4.4.28.LetY 1 <Y 2 denote the order statistics of a random sample of size 2 from
a distribution that isN(μ, σ^2 ), whereσ^2 is known.
(a)Show thatP(Y 1 <μ<Y 2 )=^12 and compute the expected value of the
random lengthY 2 −Y 1.(b)IfXis the mean of this sample, find the constantcthat solves the equation
P(X−cσ < μ <X+cσ)=^12 , and compare the length of this random interval
with the expected value of that of part (a).4.4.29.Lety 1 <y 2 <y 3 be the observed values of the order statistics of a random
sample of sizen= 3 from a continuous type distribution. Without knowing these
values, a statistician is given these values in a random order, and she wants to select
the largest; but once she refuses an observation, she cannot go back. Clearly, if she
selects the first one, her probability of getting the largest is 1/3. Instead, she decides
to use the following algorithm: She looks at the first but refuses it and then takes
the second if it is larger than the first, or else she takes the third. Show that this
algorithm has probability of 1/2 of selecting the largest.4.4.30.Refer to Exercise 4.1.1. Using expression (4.4.10), obtain a confidence
interval (with confidence close to 90%) for the median lifetime of a motor. What
does the interval mean?4.4.31.LetY 1 <Y 2 <···<Yndenote the order statistics of a random sample of
sizenfrom a distribution that has pdff(x)=3x^2 /θ^3 , 0 <x<θ, zero elsewhere.(a)Show thatP(c<Yn/θ <1) = 1−c^3 n,where0<c<1.(b)Ifnis 4 and if the observed value ofY 4 is 2.3, what is a 95% confidence interval
forθ?4.4.32.Reconsider the weight of professional baseball players in the data filebb.rda.
Obtain comparison boxplots of the weights of the hitters and pitchers (use the R
codeboxplot(x,y)wherexandycontain the weights of the hitters and pitchers,
respectively). Then obtain 95% confidence intervals for the median weights of the
hitters and pitchers (use the R functiononesampsgn). Comment.