520 Inferences About Normal Linear Models
oneway.test(elasticmod~ind,var.equal=T)
F = 12.565, num df = 2, denom df = 19, p-value = 0.0003336
With such a lowp-value, the null hypothesis would be rejected and we would con-
clude that the casting method does have an effect on the elastic modulus.
In this example, the experimenter would also be interested in the pairwise com-
parisons of the casting methods. We consider this in Section 9.4.
EXERCISES
9.2.1.Consider theT-statistic that was derived through a likelihood ratio for test-
ing the equality of the means of two normal distributions having common variance
in Example 8.3.1. Show thatT^2 is exactly theF-statistic of expression (9.2.11).
9.2.2.Under Model (9.2.1), show that the linear functionsXij−X.jandX.j−X..
are uncorrelated.
Hint:Recall the definition ofX.jandX..and, without loss of generality, we can
letE(Xij) = 0 for alli, j.
9.2.3. The following are observations associated with independent random sam-
ples from three normal distributions having equal variances and respective means
μ 1 ,μ 2 ,μ 3.
I II III
0.5 2.1 3.0
1.3 3.3 5.1
− 1 .00.01.9
1.8 2.3 2.4
2.5 4.2
4.1
Using R or another statistical package, compute theF-statistic that is used to test
H 0 :μ 1 =μ 2 =μ 3.
9.2.4.LetX 1 ,X 2 ,...,Xnbe a random sample from a normal distributionN(μ, σ^2 ).
Show that
∑n
i=1
(Xi−X)^2 =
∑n
i=2
(Xi−X
′
)^2 +
n− 1
n
(X 1 −X
′
)^2 ,
whereX=
∑n
i=1Xi/nandX
′
=
∑n
i=2Xi/(n−1).
Hint: ReplaceXi−Xby (Xi−X
′
)−(X 1 −X
′
)/n. Show that
∑n
i=2(Xi−X
′
)^2 /σ^2
has a chi-square distribution withn−2 degrees of freedom. Prove that the two
terms in the right-hand member are independent. What then is the distribution of
[(n−1)/n](X 1 −X
′
)^2
σ^2
?