522 Inferences About Normal Linear Models

(d)Note that one assumption for theF-test is that the random errorseijin Model

(9.2.1) are normally distributed. An estimate ofeijisxij−x·j.Theseare

calledresiduals, i.e., what is left after the full model fit. Compute these

residuals and then obtain a histogram, a boxplot, and a normalq−qplot of

them. Comment on the normality assumption. Use the code:

resd <- lm(quailmat[,2]~factor(quailmat[,1]))$resid

par(mfrow=c(2,2));hist(resd); boxplot(resd); qqnorm(resd)

9.2.7.Letμ 1 ,μ 2 ,μ 3 be, respectively, the means of three normal distributions with
a common but unknown varianceσ^2. In order to test, at theα= 5% significance
level, the hypothesisH 0 :μ 1 =μ 2 =μ 3 against all possible alternative hypotheses,
we take an independent random sample of size 4 from each of these distributions.
Determine whether we accept or rejectH 0 if the observed values from these three
distributions are, respectively,

X 1 :5968

X 2 :11131012

X 3 :10699

9.2.8.The driver of a diesel-powered automobile decided to test the quality of three

types of diesel fuel sold in the area based on mpg. Test the null hypothesis that the

three means are equal using the following data. Make the usual assumptions and

takeα=0.05.

Brand A: 38.7 39.2 40.1 38.9

Brand B: 41.9 42.3 41.3

Brand C: 40.8 41.2 39.5 38.9 40.3

9.3 Noncentralχ^2 andF-Distributions...................

LetX 1 ,X 2 ,...,Xndenote independent random variables that areN(μi,σ^2 ),i=
1 , 2 ,...,n, and consider the quadratic formY=

∑n

1 X

2

i/σ

(^2) .Ifeachμiis zero, we
know thatYisχ^2 (n). We shall now investigate the distribution ofYwhen eachμi
is not zero. The mgf ofYis given by
M(t)=E
[
exp
(
t
∑n
i=1
X^2 i
σ^2
)]

∏n
i=1
E
[
exp
(
t
Xi^2
σ^2
)]
.
Consider
E
[
exp
(
tXi^2
σ^2
)]

∫∞
−∞
1
σ
√
2 π
exp
[
tx^2 i
σ^2
−
(xi−μi)^2
2 σ^2
]
dxi.