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normal distribution (Exercise 14). This can be used to derive a large-sample
approximation for the value c in interval (14.7). The result is
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776 =~ cuarrer 14 Alternative Approaches to Inference
¢ wt a jam n+) a ) (14.8)
As with the signed-rank interval, the rank-sum interval (14.7) is quite effi-
cient with respect to the f interval; in large samples, (14.7) will tend to be only a bit
longer than the f interval when the underlying populations are normal and may be
considerably shorter than the ¢ interval if the underlying populations have heavier
tails than do normal populations. And once again, the actual confidence level for
the t interval may be quite different from the nominal level in the presence of
substantial nonnormality.
Exercises | Section 14.3 (17-22)
17. The article “The Lead Content and Acidity of 20. The following observations are amounts of hydro-
Christchurch Precipitation” (New Zeal. J. Sci., carbon emissions resulting from road wear of bias-
1980: 311-312) reports the accompanying data belted tires under a 522-kg load inflated at
on lead concentration (jzg/L) in samples gathered 228 kPa and driven at 64 km/h for 6 h (“Charac-
during eight different summer rainfalls: 17.0, terization of Tire Emissions Using an Indoor Test
21.4, 30.6, 5.0, 12.2, 11.8, 17.3, and 18.8. Assum- Facility,” Rubber Chem. Tech., 1978: 7-25): .045,
ing that the lead-content distribution is symmetric, .117, .062, and .072. What confidence levels are
use the Wilcoxon signed-rank interval to obtain a achievable for this sample size using the signed-
95% CI for ju. rank interval? Select an appropriate confidence
18. Compute the 99% signed-rank interval for true level andicomputesthe: interval.
average pH j (assuming symmetry) using the 21. Compute the 90% rank-sum Cl for jz, — jp using
data in Exercise 3. [Hint: Try to compute only the data in Exercise 9.
those pairwise averages having relatively small 95° Compute a 99% CI for jay — js using the data in
or large values (rather than all 105 averages).] wore
Exercise 10.
19. Compute a CI for jp of Example 14.2 using the
data given there; your confidence level should be
roughly 95%.
Bayesian Methods
Consider making an inference about some parameter 6. The “frequentist” or
“classical” approach, which we have followed until now in this book, is to regard
the value of @ as fixed but unknown, observe data from a joint pmf or pdf
Ff (x1,-++,%n;), and use the observations to draw appropriate conclusions. The
Bayesian or “subjective” paradigm is different. Again the value of 0 is unknown,
but Bayesians say that all available information about it—intuition, data from past
experiments, expert opinions, etc. —can be incorporated into a prior distribution,
usually a prior pdf g(@) since there will typically be a continuum of possible values
of the parameter rather than just a discrete set. If there is substantial knowledge
about 0, the prior will be quite peaked and highly concentrated about some central
value, whereas a lack of information is shown by a relatively flat “uninformative”
prior. These possibilities are illustrated in Figure 14.3.
In essence we are now thinking of the actual value of 0 as the observed value
of a random variable ©, although unfortunately we ourselves don’t get to observe
the value. The (prior) distribution of this random variable is g(0). Now, just as in
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14.4 Bayesian Methods 777
Prior pdf
1.0
08
f\
| Narrow
0.6 | \ ae
04
I |
\ :
0.2 oo ee eg get
0.0 b=" —Sae)
i) 2 4 6 8 10
Figure 14.3 A narrow concentrated prior and a wider less informative prior
the frequentist scenario, an experiment is performed to obtain data. The joint pmf or
pdf of the data given the value of @ is p(x1,...,xXn|@) or f(x1,...,%n|0). We use a
vertical line segment here rather than the earlier semicolon to emphasize that we are
conditioning on the value of a random variable.
At this point, an appropriate version of Bayes’ theorem is used to obtain
A(O\xy,....Xn), the posterior distribution of the parameter. In the Bayesian world,
this posterior distribution contains all current information about 0. In particular, the
mean of this posterior distribution gives a point estimate of the parameter.
An interval [a, b] having posterior probability .95 gives a 95% credibility interval,
the Bayesian analogue of a 95% confidence interval (but the interpretation is
different). After presenting the necessary version of Bayes’ Theorem, we illustrate
the Bayesian approach with two examples.
Bayes’ theorem here needs to be a bit more general than in Section 2.4 to
allow for the possibility of continuous distributions. This version gives the posterior
distribution A(@ | x1, x2, ..., X») as a product of the prior pdf times the conditional
pdf, with a denominator to assure that the total posterior probability is 1:
f (x1, 49, + %nl)8 (8)
W(Pbnszaisest8) FoF x2, Xn]0) @(0)d0
Suppose we want to make an inference about a population proportion p. Since the
value of this parameter must be between 0 and 1, and the family of standard beta