8.3. Likelihood Ratio Tests 497EXERCISES8.3.1.Verzani (2014) discusses a data set on healthy individuals, including their
temperatures by gender. The data are in the filetempbygender.rdaand the vari-
ables of interest aremaletempandfemaletemp. Download this file from the site
listed in the Preface.(a)Obtain comparison boxplots. Comment on the plots. Which, if any, gen-
der seems to have lower temperatures? Based on the width of the boxplots,
comment on the assumption of equal variances.(b)As discussed in Example 8.3.3, compute the two-sample, two-sidedt-test that
there is no difference in the true mean temperatures between genders. Obtain
thep-value of the test and conclude in terms of the problem at the nominal
α-level of 0.05.(c)Obtain a 95% confidence interval for the difference in means. What does it
mean in terms of the problem?8.3.2.Verify Equations (8.3.2) of Example 8.3.1 of this section.
8.3.3.Verify Equations (8.3.3) of Example 8.3.1 of this section.
8.3.4.LetX 1 ,...,XnandY 1 ,...,Ymfollow the location model
Xi = θ 1 +Zi,i=1,...,n
Yi = θ 2 +Zn+i,i=1,...,m,whereZ 1 ,...,Zn+mare iid random variables with common pdff(z). Assume that
E(Zi)=0andVar(Zi)=θ 3 <∞.
(a)Show thatE(Xi)=θ 1 ,E(Yi)=θ 2 ,andVar(Xi)=Var(Yi)=θ 3.(b)Consider the hypotheses of Example 8.3.1, i.e.,H 0 :θ 1 =θ 2 versusH 1 : θ 1
=θ 2.Show that underH 0 , the test statisticT given in expression (8.3.4) has a
limitingN(0,1) distribution.(c)Using part (b), determine the corresponding large sample test (decision rule)
ofH 0 versusH 1. (This shows that the test in Example 8.3.1 is asymptotically
correct.)8.3.5.In Example 8.3.2, the power function for the one-samplet-test is discussed.(a)Plot the power function for the following setup:Xhas aN(μ, σ^2 ) distribution;
H 0 :μ=50versusH 1 :μ = 50;α=0.05;n= 25; andσ= 10.(b)Overlay the power curve in (a) with that forα=0.01. Comment.(c)Overlay the power curve in (a) with that forn= 35. Comment.