4.2. Confidence Intervals 2474.2.17.It is known that a random variableX has a Poisson distribution with
parameterμ. A sample of 200 observations from this distribution has a mean equal
to 3.4. Construct an approximate 90% confidence interval forμ.
4.2.18.LetX 1 ,X 2 ,...,Xnbe a random sample fromN(μ, σ^2 ), where both param-
etersμandσ^2 are unknown. Aconfidence intervalforσ^2 can be found as follows.
We know that (n−1)S^2 /σ^2 is a random variable with aχ^2 (n−1) distribution. Thus
we can find constantsaandbso thatP((n−1)S^2 /σ^2 <b)=0.975 andP(a<
(n−1)S^2 /σ^2 <b)=0.95. In R, b = qchisq(0.975,n-1), while a = qchisq(0.025,n-1).
(a)Show that this second probability statement can be written asP((n−1)S^2 /b < σ^2 <(n−1)S^2 /a)=0. 95.(b)Ifn=9ands^2 =7.93, find a 95% confidence interval forσ^2.(c)Ifμis known, how would you modify the preceding procedure for finding a
confidence interval forσ^2?4.2.19.LetX 1 ,X 2 ,...,Xnbe a random sample from a gamma distribution with
known parameterα= 3 and unknownβ>0. In Exercise 4.2.14, we obtained an
approximate confidence interval forβbased on the Central Limit Theorem. In this
exercise obtain an exact confidence interval by first obtaining the distribution of
2
∑n
1 Xi/β.
Hint: Follow the procedure outlined in Exercise 4.2.18.
4.2.20.When 100 tacks were thrown on a table, 60 of them landed point up. Obtain
a 95% confidence interval for the probability that a tack of this type lands point
up. Assume independence.
4.2.21.Let two independent random samples, each of size 10, from two normal
distributionsN(μ 1 ,σ^2 )andN(μ 2 ,σ^2 ) yieldx=4. 8 ,s^21 =8. 64 ,y=5. 6 ,s^22 =7.88.
Find a 95% confidence interval forμ 1 −μ 2.
4.2.22.Let two independent random variables,Y 1 andY 2 , with binomial distribu-
tions that have parametersn 1 =n 2 = 100,p 1 ,andp 2 , respectively, be observed
to be equal toy 1 =50andy 2 = 40. Determine an approximate 90% confidence
interval forp 1 −p 2.
4.2.23.Discuss the problem of finding a confidence interval for the differenceμ 1 −μ 2
between the two means of two normal distributions if the variancesσ^21 andσ^22 are
known but not necessarily equal.
4.2.24.Discuss Exercise 4.2.23 when it is assumed that the variances are unknown
and unequal. This is a very difficult problem, and the discussion should point out
exactly where the difficulty lies. If, however, the variances are unknown but their
ratioσ^21 /σ 22 is a known constantk, then a statistic that is aTrandom variable can
again be used. Why?