4.10.∗Tolerance Limits for Distributions 315- For each such samplej:
(a)Label the sample of sizen 1 byx∗and label the sample of sizen 2 byy∗.
(b)Calculatevj∗=y∗−x∗.- The estimatedp-value isp̂∗=#{vj∗≥y−x}/M.
(a)Suppose we have two samples each of size 3 which result in the realizations:
x′=(10, 15 ,21) andy′=(20, 25 ,30). Determine the test statistic and the
permutation test described above along with thep-value.(b)If we ignore distinct samples, then we can approximate the permutation test
by using the bootstrap algorithm with resampling performed at random and
without replacement. Modify the bootstrap programboottesttwo.sto do
this and obtain this approximate permutation test based on 3000 resamples
for the data of Example 4.9.2.(c)In general, what is the probability of having distinct samples in the approx-
imate permutation test described in the last part? Assume that the original
data are distinct values.4.9.11.Letz∗be drawn at random from the discrete distribution that has mass
n−^1 at each pointzi=xi−x+μ 0 ,where (x 1 ,x 2 ,...,xn) is the realization of a
random sample. DetermineE(z∗)andV(z∗).4.9.12.For the situation described in Example 4.9.3, show that the value of the
one-samplet-test ist=0.84 and its associatedp-value is 0.20.
4.9.13. For the situation described in Example 4.9.3, obtain the bootstrap test
based on medians. Use the same hypotheses; i.e.,
H 0 :μ=90versusH 1 : μ> 90.4.9.14.Consider the Darwin’s experiment onZea maysdiscussed in Examples 4.5.1
and 4.5.5.(a)Obtain a bootstrap test for this experimental data. Keep in mind that the
data are recorded in pairs. Hence your resampling procedure must keep this
dependence intact and still be underH 0.(b)Write an R program that executes your bootstrap test and compare itsp-value
with that found in Example 4.5.5.4.10∗ToleranceLimitsforDistributions...................
We propose now to investigate a problem that has something of the same flavor
as that treated in Section 4.4. Specifically, can we compute the probability that a
certain random interval includes (orcovers) a preassigned percentage of the prob-
ability of the distribution under consideration? And, by appropriate selection of