Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

248 Some Elementary Statistical Inferences

4.2.25.To illustrate Exercise 4.2.24, letX 1 ,X 2 ,...,X 9 andY 1 ,Y 2 ,...,Y 12 rep-
resent two independent random samples from the respective normal distributions
N(μ 1 ,σ^21 )andN(μ 2 ,σ^22 ). It is given thatσ^21 =3σ 22 , butσ^22 is unknown. Define a
random variable that has at-distribution that can be used to find a 95% confidence
interval forμ 1 −μ 2.

4.2.26.LetXandYbe the means of two independent random samples, each of size
n, from the respective distributionsN(μ 1 ,σ^2 )andN(μ 2 ,σ^2 ), where the common
variance is known. Findnsuch that

P(X−Y−σ/ 5 <μ 1 −μ 2 <X−Y+σ/5) = 0. 90.

4.2.27.LetX 1 ,X 2 ,...,XnandY 1 ,Y 2 ,...,Ymbe two independent random samples
from the respective normal distributionsN(μ 1 ,σ^21 )andN(μ 2 ,σ^22 ), where the four
parameters are unknown. To construct aconfidence interval for the ratio,σ 12 /σ^22 ,of
the variances, form the quotient of the two independentχ^2 variables, each divided
by its degrees of freedom, namely,

F=

(m−1)S 22

σ 22 /(m−1)

(n−1)S^21

σ^21 /(n−1)

=

S 22 /σ^22

S 12 /σ^21

,

whereS 12 andS 22 are the respective sample variances.

(a)What kind of distribution doesFhave?

(b)Critical valuesaandbcan be found so thatP(F<b)=0.975 andP(a<

F<b)=0.95. In R, b = qf(0.975,m-1,n-1), while a = qf(0.025,m-1,n-1).

(c)Rewrite the second probability statement as

P

[

a

S 12

S 22

<

σ^21

σ^22

<b

S^21

S^22

]

=0. 95.

The observed values,s^21 ands^22 , can be inserted in these inequalities to provide

a 95% confidence interval forσ 12 /σ^22.

We caution the reader on the use of this confidence interval. This interval does

depend on the normality of the distributions. If the distributions ofX andY

are not normal then the true confidence coefficient may be far from the nominal

confidence coefficient; see, for example, page 142 of Hettmansperger and McKean

(2011) for discussion.

4.3 ∗Confidence Intervals for Parameters of Dis-

crete Distributions

In this section, we outline a procedure that can be used to obtain exact confidence

intervals for the parameters of discrete random variables. LetX 1 ,X 2 ,...,Xnbe a