500 Optimal Tests of HypothesesSample − 0. 389 − 2. 177 0.813 − 0. 001
1 − 0. 110 − 0. 709 0. 456 0. 135
Sample 0. 763 − 0. 570 − 2. 565 − 1. 733
1 0. 403 0. 778 − 0. 115
Sample − 1. 067 − 0. 577 0. 361 − 0. 680
2 − 0. 634 − 0. 996 − 0. 181 0. 239
Sample − 0. 775 − 1. 421 − 0. 818 0. 328
2 0. 213 1. 425 − 0. 165(a)Obtaincomparison boxplotsof these two samples. Comparison boxplots con-
sist of boxplots of both samples drawn on the same scale. Based on these
plots, in particular the interquartile ranges, what do you conclude aboutH 0?(b)Obtain theF-test (for a one-sided hypothesis) as discussed in Remark 8.3.2
at levelα=0.10. What is your conclusion?(c)The test in part (b) is not exact. Why?8.3.18.For the skewed contaminated normal random variableXof Example 8.3.4,
derive the cdf, pdf, mean, and variance ofX.
8.3.19. For Table 8.3.1 of Example 8.3.4, show that the half-width of the 95%
confidence interval for a binomial proportion as given in Chapter 4 is 0.004 at the
nominal value of 0.05.
8.3.20.If computational facilities are available, perform a Monte Carlo study of
the two-sidedt-test for the skewed contaminated normal situation of Example 8.3.4.
The R functionrscn.Rgenerates variates from the distribution ofX.
8.3.21.SupposeX 1 ,...,Xnis a random sample onXwhich has aN(μ, σ^20 ) distri-
bution, whereσ 02 is known. Consider the two-sided hypotheses
H 0 :μ=0versusH 1 : μ =0.Show that the test based on the critical regionC ={|X|>√
σ^20 /nzα/ 2 }is an
unbiased levelαtest.8.3.22.Assume the same situation as in the last exercise but consider the test
with critical regionC∗={X>√
σ 02 /nzα}. Show that the test based onC∗has
significance levelαbut that it is not an unbiased test.
8.4 ∗The Sequential Probability Ratio Test
Theorem 8.1.1 provides us with a method for determining a best critical region
for testing a simple hypothesis against an alternative simple hypothesis. Recall its
statement: LetX 1 ,X 2 ,...,Xnbe a random sample with fixed sample sizenfrom
a distribution that has pdf or pmff(x;θ), whereθ={θ:θ=θ′,θ′′}andθ′andθ′′