Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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

?