Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

252 Some Elementary Statistical Inferences

poissonci(124,25,4,4.4,.95); $solution 4.287836
Since ppois(125, 25 ∗ 5 .5) = 0.1528 and ppois(125, 25 ∗ 6 .0) = 0.0204, for the second
equation, 5.5 and 6.0 bracket the solution. Hence, the computation of the lower
bound of the confidence interval is:
poissonci(125,25,5.5,6,.05); $solution 5.800575
So the confidence interval is (4. 287 , 5 .8), with confidence at least 90%. Note that
the confidence interval is right-skewed, similar to the Poisson distribution.

A brief sketch of the theory behind this confidence interval follows. Consider
the general setup in the first paragraph of this section, whereTis an estimator of
the unknown parameterθandFT(t;θ)isthecdfofT. Define

θ =sup{θ:FT(T;θ)≥α 1 } (4.3.8)

θ =inf{θ:FT(T−;θ)≤ 1 −α 2 }. (4.3.9)

Hence, we have

θ>θ ⇒ FT(T;θ)≤α 1

θ<θ ⇒ FT(T−;θ)≥ 1 −α 2.

These implications lead to

P[θ<θ<θ]=1−P[{θ<θ}∪{θ>θ}]

=1−P[θ<θ]−P[θ>θ]

≥ 1 −P[FT(T−;θ)≥ 1 −α 2 ]−P[FT(T;θ)≤α 1 ]

≥ 1 −α 1 −α 2 ,

where the last inequality is evident from equations (4.3.8) and (4.3.9). A rigorous

proof can be based on Exercise 4.8.13; see page 425 of Shao (1998) for details.

EXERCISES

4.3.1.Recall For the baseball data (bb.rda), 15 out of 59 ballplayers are left-

handed. Letpbe the probability that a professional baseball player is left-handed.

Determine an exact 90% confidence interval forp. Show first that the equations to

be solved are:

∑ 14

j=0

(n

j

)

θj(1−θ)n−j=0.95 and

∑ 15

j=0

(n

j

)

θ

j

(1−θ)n−j=0. 05.

Then do the following steps to obtain the confidence interval.

(a)Show that 0.10 and 0.17 bracket the solution to the first equation.

(b)Show that 0.34 and 0.38 bracket the solution to the second equation.

(c)Then use the R functionbinomci.rto solve the equations.

4.3.2.In Example 4.3.2, verify the result for the asymptotic confidence interval for

θ.