4.2. Confidence Intervals 2454.2.4.In Example 4.2.4, for the baseball data, we found a confidence interval for
the mean difference in heights between the pitchers and hitters. In this exercise,
find the pooledt95% confidence interval for the mean difference in weights between
the pitchers and hitters.
4.2.5.In the baseball data set discussed in the last exercise, it was found that out
of the 59 baseball players, 15 were left-handed. Is this odd, since the proportion of
left-handed males in America is about 11%? Answer by using (4.2.7) to construct a
95% approximate confidence interval forp, the proportion of left-handed professional
baseball players.
4.2.6.LetXbe the mean of a random sample of sizenfrom a distribution that is
N(μ,9). Findnsuch thatP(X− 1 <μ<X+1)=0.90, approximately.
4.2.7.Let a random sample of size 17 from the normal distributionN(μ, σ^2 ) yield
x=4.7ands^2 =5.76. Determine a 90% confidence interval forμ.4.2.8.LetXdenote the mean of a random sample of sizenfrom a distribution that
has meanμand varianceσ^2 = 10. Findnso that the probability is approximately
0.954 that the random interval (X−^12 ,X+^12 ) includesμ.
4.2.9.LetX 1 ,X 2 ,...,X 9 be a random sample of size 9 from a distribution that is
N(μ, σ^2 ).
(a)Ifσis known, find the length of a 95% confidence interval forμif this interval
is based on the random variable√
9(X−μ)/σ.(b)Ifσis unknown, find the expected value of the length of a 95% confidence
interval forμif this interval is based on the random variable√
9(X−μ)/S.
Hint: WriteE(S)=(σ/√
n−1)E[((n−1)S^2 /σ^2 )^1 /^2 ].(c)Compare these two answers.4.2.10.LetX 1 ,X 2 ,...,Xn,Xn+1be a random sample of sizen+1,n>1, from a
distribution that isN(μ, σ^2 ). LetX=
∑n
1 Xi/nandS(^2) =∑n
1 (Xi−X)
(^2) /(n−1).
Find the constantcso that the statisticc(X−Xn+1)/Shas at-distribution. If
n= 8, determineksuch thatP(X−kS < X 9 <X+kS)=0.80. The observed
interval (x−ks,x+ks) is often called an 80%prediction intervalforX 9.
4.2.11.LetX 1 ,...,Xnbe a random sample from aN(0,1) distribution. Then the
probability that the random intervalX±tα/ 2 ,n− 1 (s/
√
n)trapsμ=0is(1−α). To
verify this empirically, in this exercise, we simulatemsuch intervals and calculate
the proportion that trap 0, which should be “close” to (1−α).
(a)Setn=10andm= 50. Run the R codemat=matrix(rnorm(m*n),ncol=n)
which generatesmsamples of sizenfrom theN(0,1) distribution. Each row
of the matrixmatcontains a sample. For this matrix of samples, the function
below computes the (1−α)100% confidence intervals, returning them in a
m×2 matrix. Run this function on your generated matrixmat. What is the
proportion of successful confidence intervals?