Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.5. Multiparameter Case: Testing 401

in its denominator with a consistent estimate. Recall that̂pi→pi,i=1,2, in

probability. Thus underH 0 ,thestatistic

Z∗=

̂p 1 −p̂ 2

√

pb 1 (1−bp 1 )

n 1 +

pb 2 (1−bp 2 )

n 2

(6.5.27)

has an approximateN(0,1) distribution. Hence an approximate levelαtest is
to rejectH 0 if|z∗|≥zα/ 2. Another consistent estimator of the denominator is
discussed in Exercise 6.5.14.

EXERCISES

6.5.1.On page 80 of their test, Hollander and Wolfe (1999) present measurements
of the ratio of the earth’s mass to that of its moon that were made by 7 different
spacecraft (5 of the Mariner type and 2 of the Pioneer type). These measurements
are presented below (also in the fileearthmoon.rda). Based on earlier Ranger
voyages, scientists had set this ratio at 81.3035. Assuming a normal distribution,
test the hypothesesH 0 :μ=81.3035 versusH 1 :μ =81.3035, whereμis the
true mean ratio of these later voyages. Using thep-value, conclude in terms of the
problem at the nominalα-level of 0.05.

EarthtoMoonMassRatios

81.3001 81.3015 81.3006 81.3011 81.2997 81.3005 81.3021

6.5.2.Obtain the boxplot of the data in Exercise 6.5.1. Mark the value 81.3035 on

the plot. Compute the 95% confidence interval forμ, (4.2.3), and mark its endpoints

on the plot. Comment.

6.5.3.Consider the survey of citizens discussed in Exercise 6.4.1. Suppose that the
hypotheses of interest areH 0 :p 1 =p 2 versusH 1 :p 1
=p 2. Note that computation
can be carried out using the R functionp2pair.R, which can be downloaded at the
site mentioned in the Preface.

(a)Test these hypotheses at levelα=0.05 using the test (6.5.20). Conclude in

terms of the problem.

(b)Obtain the 95% confidence interval, (6.5.21), forp 1 −p 2. What does the

confidence interval mean in terms of the problem?

6.5.4.LetX 1 ,X 2 ,...,Xnbe a random sample from the distributionN(θ 1 ,θ 2 ).

Show that the likelihood ratio principle for testingH 0 :θ 2 =θ′ 2 specified, andθ 1

unspecified against∑ H 1 :θ 2 = θ′ 2 ,θ 1 unspecified, leads to a test that rejects when

n

1 (xi−x)

(^2) ≤c 1 or∑n
1 (xi−x)
(^2) ≥c 2 ,wherec 1 <c 2 are selected appropriately.
6.5.5.LetX 1 ,...,XnandY 1 ,...,Ymbe independent random samples from the
distributionsN(θ 1 ,θ 3 )andN(θ 2 ,θ 4 ), respectively.