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Preface
This solution manual was prepared as an aid for instrctors who wil benefit by
having solutions available. In addition to providing detailed answers to most of theproblems in the book, this manual can help the instrctor determne which of theproblems are most appropriate for the class.
The vast majority of the problems have been solved with the help of availablecomputer software (SAS, S~Plus, Minitab). A few of the problems have been solved withhand calculators. The reader should keep in mind that round-off errors can occur-parcularly in those problems involving long chains of arthmetic calculations.
We would like to take this opportnity to acknowledge the contrbution of manystudents, whose homework formd the basis for many of the solutions. In paricular, wewould like to thank Jorge Achcar, Sebastiao Amorim, W. K. Cheang, S. S. Cho, S. G.Chow, Charles Fleming, Stu Janis, Richard Jones, Tim Kramer, Dennis Murphy, RichRaubertas, David Steinberg, T. J. Tien, Steve Verril, Paul Whitney and Mike Wincek.Dianne Hall compiled most of the material needed to make this current solutions manualconsistent with the sixth edition of the book.
The solutions are numbered in the same manner as the exercises in the book.Thus, for example, 9.6 refers to the 6th exercise of chapter 9.
We hope this manual is a useful aid for adopters of our Applied MultivariateStatistical Analysis, 6th edition, text. The authors have taken a litte more active role inthe preparation of the current solutions manual. However, it is inevitable that an error ortwo has slipped through so please bring remaining errors to our attention. Also,comments and suggestions are always welcome.
Richard A. JohnsonDean W. Wichern
1.1
1.2 a)
Xl =" 4.29
51i = 4.20
.
Chapter 1
X2 = 15.29
522 = 3.56 S12 = 3.70
Scatter Plot and Marginal Dot Plots
.
. . . .. . .
. .
. .
. .
17.5.
.
.
15.0.
.
.
12~5.
.
I')C
10.0.
.
7.5 . ..
.
5.0 . .
0 2 4 6 8 10 12xl
b) SlZ is negative
c)Xi =5.20 x2 = 12.48 sii = 3.09 S22 = 5.27
SI2 = -15.94 'i2 = -.98Large Xl occurs with small Xz and vice versa.
d)
(5.20 )x = 12.48
( 3.09S -n - -15.94
-15.94)5.27
R =( 1 -.98)-.98 1
1.3
UJSnJ6 : -~::J
L (synetric) 2 .x =-
1.4 a) There isa positive correlation between Xl and Xi. Since sample size issmall, hard to be definitive about nature of marginal distributions.However, marginal distribution of Xi appears to be skewed to the right. .The marginal distribution of Xi seems reasonably symmetrc.
.....'....._.,..'....,...,..'.":
SCtter.PJot andMarginaldt:~llt!;
. . . . . . . . . .
25 . .
20.
.
I' . . . ;..)C . .
15.
.
.. . .
10 ...
.
50 100 150 200 250 300xl
b)Xi = 155.60 x2 = 14.70 sii = 82.03 S22 = 4.85
SI2 = 273.26 'i2 = .69Large profits (X2) tend to be associated with large sales (Xi); small profitswith small sales.
R =
(1 .577. (synet:; c )
2
-. 40~. 3~OJ
31.5 a) There is negative correlation between X2 and X3 and negative correlationbetween Xl and X3. The marginal distribution of Xi appears to be skewed tothe right. The marginal distribution of X2 seems reasonably symmetric.The marginal distribution of X3 also appears to be skewed to the right.
Sttr'Plotl'(i'Marginal.DotPi_:i..~sxli.
. .
. . . . . . . .
1600.
.
1200 . .. .
. .
M.800 . .)C . .
400. . . .
. . . .
010 15 20 25
x2
. '-'. .Scatir;Pltnd:Marginal.al.'lli:ltfjt.I...
. . . . . . .. . .
1600.
.
1200 . ..
.
. .
M 800 . .)C . .
400. . . . .
. .
050 100 150 200 250 300
xl
1.5 b)
(155.60Jx = 14.70
710.91
R = ( ~69-.85
1.6 a) Hi stograms
HIDDLE OFINTERVAL5.6.
7.8.9.
10.
HIDDLE OFINTERVAL
30.40.50.60.70.80.90.
100.110.
HIDIILE OFINTERVAL
2.J.4.5.6.7.
HIDDLE OFINTERVAL
1...
.. .
:3 .4.s.
( 82.03Sn = 273.26
- 32018.36-.85)-.42
1
-32018.36)-948.45
461.90
273.264.85
- 948.45
.691
-.42
Xi
4
Xs
HIDIILE OF NUMBR. OFNUMBER OF INTERVAL 08SERVATIONSOBSERVATioNS co 2 **oJ .
5 ***** 6. J U*S ******** 7. S *****7 u***** a. s *****
11 *********** 9. 6 ******5 un* 10. 4 ****6 ****** 11. 4 u**
12. . S Uu.*X2 13. 4 ****14. 1 *
15. 0NUHBER OF 16. 1 *OBSERVATIONS 17. 0
1 * LS. 1 *J n* 19. 02 ** 20. 03 *** 21. 1 *
10 **********12 ************
a ******** Xl2 n1 *
I' I DOLE .OFINTE"RVAL
X32.4.6.a.
10.12.14.16.18.20.22.24.26.
NUHBER OFOBSERVA T I'ONS1 *:s *****19 *******************9 *********.3 U*S u***
X4
NUMBER OF.OBSERVA T IONS1 J ***$*********15 ***************a ********5 ui**1 *
HIDDLE OFINTERVAL
2.J.4.s.
NUMBER OFOBSERVATIONS7 *******25 *************************9 *********1 *
NUltEiER OFOI4SERVATIOllSJ ***4 ****7. *******7 *******B ********5 n***2 **2 u1 *oo2 **1 *
X7
51.6 b) 7.5 2.440 -2 . 714 -.369-.452 - . 571 -2.1 79 .,.67
73 .857 293.360 3.816 -1 .3546.602 30.058 :609
4.548 1.486 .6582.260 2.755 .138
- 2. 191 S =1 . 154 1 . 062 -.7-91 .172
x = n
1 0 . 04811. 093 3.052 1 .019
9.40530.241 .580
3.095 (syrtric) .467
The pair x3' x4 exhibits a small to moderate positive correlation and so does the
pair x3' xs' Most of the entries are small.
1.7
ill b) 3x2
4 . 4 . .
Xl 2
2 .
.
2 4Scatter.p1'Ot
(vari ab 1 e space)~ ~tem space.)
1
-6
1.8 Using (1-12) d(P,Q) = 1(-1-1 )2+(_1_0)2; = /5 = 2.236
Using (1-20) d(P.Q)' /~H-1 )'+2(l)(-1-1 )(-1-0) '2t(-~0);' =j~~ = 1.38S
Using (1-20) the locus of points a c~nstant squared distance 1 from Q = (1,0)
is given by the expression t(xi-n2+ ~ (x1-1 )x2 + 2t x~ = 1. To sketch thelocus of points defined by this equation, we first obtain the coordinates of
some points sati sfyi ng the equation:(-1,1.5), (0,-1.5), (0,3), (1,-2.6), (1,2.6), (2,-3), (2,1.5), (3,-1.5)
The resulting ellipse is:
X1
1.9 a) sl1 = 20.48 s 22 = 6. 19s 12 = 9.09
X2
5 .
.
.
. .
0 5 10xi
.
.
.
-"5
71.10 a) This equation is of the fonn (1-19) with aii = 1, a12 = ~. and aZ2 = 4.
Therefore this is a distance for correlated variables if it is non-negative
for all values of xl' xz' But this follows easily if we write
2. 2. 1 1 15 2.xl + 4xZ + x1x2 = (xl + r'2) + T x2 ,?o.
b) In order for this expression to be a distance it has to be non-negative for2. :
all values xl' xz' Since, for (xl ,x2) = (0,1) we have xl-2xZ = -Z, we
conclude that this is not a validdistan~e function.
1.11d(P,Q) = 14(X,-Yi)4 + Z(-l )(x1-Yl )(x2-YZ) + (x2-Y2):'
= 14(Y1-xi): + 2(-i)(yi-x,)(yZ-x2) + (xz-Yz):' = d(Q,P)
Next, 4(x,-yi)2. - 2(xi-y,)(x2-y2) + (x2-YZ): =
=,(x1-Yfx2+Y2):1 + 3(Xi-Yi):1,?0 so d(P,Q) ~O.
The scond term is zero in this last ex.pr.essi'on only if xl = Y1 andthen the first is zero only if x.2 = YZ.
81.12 a) If P = (-3,4) then d(Q,P) =max (1-31,141) = 4
b) The locus of points whosesquar~d distance from (n,O) is , is
.1
1
..
-1 17
-1
X2
x,
c) The generalization to p-dimensions is given by d(Q,P) = max(lx,I,lx21,...,lxpl)'
1.13 Place the faci'ity at C-3.
1.14 a)
360.+ )(4
320.+
280.+
240.+
200.+
9
.
.
.
..
.
.
.
....
. .
. .
. ..
I:
I:
.
.. .
..
.
*
160.+ +______+_____+-------------+------~.. )(2130. 1:5:5. 180. 20:5.' 230. 2:5:5.
Strong positive correlation. No obvious "unusual" observations.
b) Mul tipl e-scl eros; s group.
x =
42 . 07
179.6412.31
236.6213.16
116.91 61 .78 -20.10 61 . 1 3 -27 . 65812.72 -218.35 865.32 90.48
Sn = 3 as . 94 221 '. 93 286.601146.38 82.53
(synetric) 337.80
10
1 .200 -. H)6 .167 -.1391 .438 .896 .173
R = 1 .375 .8921 .133
( synetrit: ) 1
Non multiple-sclerosis group.
37 . 99147.21
i = 1 .561 95.57
1.62
273.61 95.0811 0.13
5.281.841.78
1 01 . 67
1 03 .28
2.22183 . 04 .
s =n
(syietric)
1 .548 .239 .454 .1271 .132 .727 .134
R = 1 .123 .244
1 .114
(symmetric) 1
3.2u2.15
.492.352.32
1.15 a) Scatterplot of x2 and x3.
11
., ..".. ... . . . ., .. .0 .. . . . . . + . . . . l . . . . + . . . . + . . . . . . . . . +. . . . . . . . . + . . .. .... + . . . . . . . . . + . .. . . . . . . . . .
.l -.
I.
.
1.
.
3. it .
J.
..
- .
..1.2~
1 t 1 1 I.
-
1. , . -.
1.
t.
"'. ..
. .
1 1 1.
~
I I t : 1.
--
-.
.
E 1 1.
E 2. cl III -.- '_ 1 I 1 +..
~
.
2 I 1 I 1.X:i I I.
. ..
.z.o .1 1 +3 I 1 t
..
. .
.
I.
1 1 I.
. \1 \1 1. .I t
.
i 1 1 1.
.i
.J .2 I 1 t 1 l.
.
1 J.
1I 2 1 1.
I.
.80 .1
.
. .
. . .. .. . . . . . . .. . .. . .. . .. '. ... .. . . .. ... ...- .. . ".--. . .
. 75f) 1.251.88 t.~~ 1.75 t. ~ e 2.25 Z "~A 2.7'5 3.253.P,1) 3.S11 3.75 G.251I.llflACTIVITY X%
b)3.541.812.14~.212.581.27
x =
1.15 4.61 ..92 .58 .27 1.~6 .15.61 .11 .12 .39 -.02
.57 .09 .34 .11Sn = .11 .21 .02
.85 -.01;. (synetric) .85
12
1 .551 .362 .386 .537 . 077 '1 .187 .455 .535 -.035
1 .346 .496 .156R = 1 .704 .071
1 -. 01 0
(syretric) 1
The largest correlation is between appetite and amount of food eaten.
Both activity and appetite have moderate positive correlations with
symptoms. A1 so, appetite and activity have a moderate positive
correl a tion.
13
1.16There are signficant positive correlations among al variable. The lowest correlation is
. 0.4420 between Dominant humeru and Ulna, and the highest corr.eation is 0.89365 bewteenDominant hemero and Hemeru.
x -
0.84380.81831. 79271. 73480.70440.69380.01248150.00996330.02145tiO0.01928220.00875590.0076395
Sn -
1.00000 0.85181 0.69146 0.66826 0.74369 0.677890.85181 1.00000 0.61192 0.74909 0.74218 0.809800.69146 0.61192 1.00000 -0.89365 0.55222 0.4420
, R = 0.66826 0.74909 0.89365 1.00000 0.ti2555 0.61882
0.74369 0.74218 0.55222 0.62555 1.00000 0.728890.67789 0.80980 0.44020 0.61882 0.72889 1.00000
0.0099633 0.0214560 0.0192822 0.0087559 0.00763950.0109612 0.0177938 0.0202555 0.0081886 0.00855220.0177938 0.0771429 0.0641052 0.0161635 0.01233320.0202555 0.0641052 0.0667051 0.0170261 0.01612190.0081886 0.0161635 0.0170261 0.0111057 0.00774830.0085522 0.0123332 0.0161219 0.0077483 0.0101752
1.17There are large positive correlations among all variables. Paricularly largecorrelations occur between running events that are "similar", for example,the 1 OOm and 200m dashes, and the 1500m and 3000m runs.
11.3623.1251.99
x = 2.024.199.08
153.62
So=
.152
.338
.875.027.082.230
4.254
.338 .875
.847 2.152
2.152 6.621
.065 .178
.199 .500
.544 1.400
10.193 28.368
.027.065.178.007.021.060
1.197
.082
.199
.500.021.073.212
3.474
.230.544
1.400. .060.212.652
10.508
1.000 .941.871 .809 .782 .728 .669
.941 1.000 .909 .820 .801 .732 .680
.871 .909 1.000 .806 .720 .674 .677
R = .809 .820 .806 1.000 .905 .867 .854.782 .801 .720. .905 1.000 .973 .791
.728 .732 .674 .867 .973 1.000 .799
.669 .680 .677 .854 .791 .799 1.000
4.25410.19328.368
1.1973.474
10.508265.265
1.18
14
There are positive correlations among all variables. Notice the correlationsdecrease as the distances between pairs of running events increase (see the firstcolumn of the correlation matrx R). The correlation matrix for running eventsmeasured in meters per second is very similar to the correlation matrix for therunning event times given in Exercise 1.17.
8.818.667.71
x = 6.60
5.995.544.62
.091 .096 .097 .065 .082 .092 .081
.096 .115 .114 .075 .096 .105 .093
.097 .114 .138 .081 .095 .108 .102
Sn = .065 .075 .081 .074 .086 .100 .094
.082 .096 .095 .086 .124 .144 .118
.092 .105 .108 .100 .144 .177 .147
.081 .093 .102 .094 .118 .147 .167
1.000 .938 .866 .797 .776 .729 .660
.938 1.000 .906 .816 .806 .741 .675
.866 .906 1.000 .804 .731 .694 .672
R = .797 .816 .804 1.000 .906 .875 .852
.776 .806 .731 .906 1.000 .972 .824
.729 .741 .694 .875 .972 1.000 .854
.660 .675 .672 .852 .824 .854 1.000
15
1.19 (a)
ULNA ILULNA tlUME~US LHUI.ERUSRADIUS o _R A 0 IUS
I
c c c- .. ,... .. ..
o' '0
~
..
o' :
.' '.
"
" .'
.. CI -
.. - -
.. .QI ..
C C C-
... ,.z .. ..
.,. t-.' .0:; .. ".00o. : 00... : ..
.~.,..
.. ..-
- '" QI CI..
.. '"
II I: I:- .. ,... ..
..
,
0'
~
,
i::,
0,o'. '
- -..
.. ....
.. .. ..
II I: C- .. ~
"z .. ..
o'
~
"
,.
.0.-
"
. ..
o',.
"00
- - ..
.. .. ..
.. ....
I: C C- .. ,.
.. ....
: " t- . ..:.. " ,,'co co CI
UI .. -..
- ..
I: C C- ..
,.
..co ..
.~ ., .. 00 .'.
".
.0,
".
.
.. ....
.. ..CD
QI '" -
oi::;:oc:en
::;:-0c:en
oi=c:c..
::c:in
=c:c..
:;c:en
QIc:i-z:;:
c:i-z:;:
16
1.19 (b)
~. .... " . ... A..c
~l ., ..'lt.., .,..... .~.
-Ii . P. _. .-, . .~. . .,. .. ... .. . . , ". . . .
. . .
. . . .
. . ";t':",o;,
i: l .i ~ ' tl,' !t1" .~._... ~.,.to ... .... ...
... -. ..~
. . . .
. .
. . ,
l. t .' \. :.-i-. ..(. .!,l :i .~~-l \; ..0' :,. .. . .1, . .- ,-..' . . . . . . .. - .
.
. .
"-:f'~,:~
.
.1.;.\ I! ~ .'\:~ .8.,., . . ~.... 'L . ..I.
t.:. " " ...".., . . . .... .. . . . . ..
. . . .
.'
\.~
~; .~ . ~c,.. . . .. .. -it .
.t:l . . t- . .-~ ... - , . . ..... . . . .
. .
.
.~,.. . .
~..:.~~ . .~.
. ,.
ll . .. \........ ..
..
. .. .l. .
. .
.
\. \. .~. . .1f .1. i 0 ~: i: '.~:. .;;- -\1: ii . .~~
:,. I.l " . . . . .. . : ... f
(b) ,,' A L_-l_ XL _ _ (,"" i. \ l'.l l~'T '\ .\ . . ....
\. . . ." r
. '"... ,"..
. .
1.20
(a) Xl.
.. . .
.. ...
... '.
.
. ..... .. . .. .
... .
~
x3.
x1.
.
.
.
.
~'~t.
.
.
\,\~
.
X1
.
.
.
.
(a1 The plot looks like a cigar shape, but bent. Some observations. in the lower left handpart could be outliers. From the highlighted plot in (b) (actually non-bankrupt groupnot highlighted), there is one outlier in the nonbankruptgroup, which is apparentlylocated in the bankrupt group, besides the strung out pattern to the right.
(ll) The dotted line in the plot would be an orientation for the classific.tion.
17
X3
...
. ... .
...... .
. . .. .
. . .... ..
.
.
1.21
(a)
.
..
.
.
.
.
.
.
. .
.
. .
.
.
.
.
oOutlier
.
.
.
X1.
X3.
(b)
.
tfe',e
~~0'"
~~G.
.
. .
18
oOutlier
X1
.
.
.
Outlier Q
.
.
.
.
.
. .. .
...... .
. ...- - .l..-- ~...
.
.
.
(a) There are two outliers in the upper right and lower right corners of the plot.(b) Only the points in the gasoline group are highlighted. The observation in the upper
right is the outlier. As indiCated in the plot, there is an orientation to classify into twogroups.
1.22 possible outliers are indicated.
GOutlier
X1
. .
. ...
. .
.
.. .
. ..
. .
fi ..
..
...
..
.
19
~~~
~.
t I ~~\e/ \, X1
Xz/
.
..
x,
..
.
.
.. ..
.
. ..
.
.
..
. .
.
. .
.
Xz
.."
..
.. /. ./
././
./;I
./ .
.
..
.
.
.
. ,.. ..
fi.
..
. .
.
.
.. .
.
Xz
)l"
./
~,etot)~
.. .~
. .
x,
.
.
.
. .
.
~~e
),. il~\.e
./ ~e~
..~
.
. .
..s..Outliers
VI 20Q..Q
-0 VICI ...VI s.C .. U
s. u0 I s.U c: faV) ois. =II s. V)C' Cd
.:: oi s.V) :: ciV) Co
=V)
i.s.ci~
VI:: IIi- ai ciu VI u
~ ..
:: a:z:4; ~ci~ ..
CJ Cd . ..I) s. ::.,. ~ a.c:
.
~ C"a ~i- s.ci ci s..Q ~ a.
VI ~c: :: ci VI toci i- .. ::~ U - i- .-- -i Co Cden -uVI ci- Q.CIVIciuCd..
..Iac:s.ci.cu c
c: VIoi N s. a:c: ci ..- s. ~VI ci c ra= ~ .. ci
VI ;I .cen :: - ::c: i- ra.. u a:s.ci~
VI -::i-Ui-Cd en:: i-VI -- ..
~ s. uci i- ::
c( +J n: "'VI +J e:: 0
.. - l- a..G uMN
.
'"
1.24
20
14
18
22
Cl uster 1
13C1 uster 2
10
9
19
3
C1 uster 3
.s
21
4
1
85
2
Clust~r 4
16
Cl uster 5
21
Cluster 6
12
C1 uster 7
7. '5
22
11
17
We have cluster~d these faces
in. the same manner as those inExample 1.12. Note, however,
other groupings are~qually .plausible. for instance, utilities9 and 18 l1ight be swit.ched from
Cluster 2 toC1 uster 3 and so
forth.
1.25 We illustrate one cluster of "stars.l. Theshown) can be gr~uped in 3 or 4 additional
r.emai ni ngcl usters.
stars
4 10
'.
/ ....1."-,,. -.
- ... ..l ".......;-
.... .'." ..~.
.....: l. fi .... ... -,'-1
'/. ...0: .. .":
",. -.
. I....~ " :
~
20 13
23
(not
....
'-a.-
1.26 Bull data
(a) XBAR
4. 38161742.4342
50.5224995.947470.88166.31580.1967
4. 12631555.2895
24
RBreed SalePr YrHgt FtFrBody PrctFFB Frame BkFat SaleHt SaleWt1.000 -0.224 0.525 0.409 0.472 0.434 -0.~15 0.487 0.116-0.224 1.000 0.423 0.102 -0.113 0.479 0.277 0.390 0.317o . 525 0 .423 1 .000 O. 624 0 . 523 0 . 940 -0.344 0 . 860 0 . 3680.409 0.102 0..624 1.000 0.691 0.605 -0.168 0.699 0.50.472 -0.113 0.523 0.691 1.000 0.482 -0.488 0.521 0.1980.434 0.479 0.940 0.605 0.482 1.000 -0.260 0.801 0.368-0.615 0.277 -0.344 -0.168 -0.488 -0.260 1.QOO ~0.282 0.2080.487 0.390 0.860 0.699 0.521 0.801 -0.282 1.00 0.~660.116 0.317 0.368 0.555 0.198 0.368 0.208 0.566 1.000
SnBreed SalePr YrHgt FtFrBody PrctFFB Frame BkFat SaleHt 'SaleWt9.55 -429.02 2.79 116.28 4;73 1.23 -0.17 3.00 46.32
-429 ..02 383026.64 450.47 5813.09 -226.46 272.78 15.24 480.56 25308 . 442.79 450. 47 2.96 98.81 2.92 1.49 -0.05 2.94 81. 72
116.28 5813.09 98.81 8481. 26 206 . 75 51. 27 -1.38 128.23 6592.414.73 -226.46 2.92 206. 75 10.55 1.44 -0.14 3.37 82.821.23 272.78 1.49 51.27 1.44 0.85 -0.02 1.47 43.74
-0. 17 15.24 -0.05 -1. 38 -0.14 -0.02 0.01 -0.05 2.383.00 480 . 56 2.94 128.23 3.37 1.47 -0.05 3.97 145.35
46.32 25308.44 81.72 6592.41 82.82 43.74 2.38 145 . 35 16628.94
. . . . . .
Breed. . . . . . . .
. . . . . . . . . . .
. ...
. . .
. . . . . . . . .
Frame. . . . . . . . . . .
,
. . . . . . . .
cci
a..
:i
~
5.0 6.0 7.0 8.0 90 1100 t30.,
CD CD
..
Breed
'" N
~
8 :- .-
!
.
.
I.
.
. ... .I-
. '..,.,;......~'.l . .
. .., .1, .. . .
. - . -:-
FtFrB
oCD~
. .
. . . .
. . . . .
. . . . . BkFat
. . . . .
. . . . . . .
. . . . . . .
. . .
2 4 6 8
. . .I . -. .l ..,... ....
. \..i'..l: -:,: .-.-'..
. ,.
CD..
~ CDon
'"d SaleHt :;'"d ~
gdO. t 0.2 0.3 0.4 0.5 50 52 54 1i 58 602 4 6 8
25
1.27
(a) Correlation r = .173
Scatterplot of Size Y5 viSitors
2500
2000
1500
iIi
.1000
500
0
0
.
.
.
.
.
..
..
.
..
. .
1 2 3 4 5 6Visitors
Gct '5lio\~"' .
7 8 9
(b) Great Smoky is unusual park. Correlation with this park removed is r = .391.This single point has reasonably large effect on correlation reducing the positivecorrelation by more than half when added to the national park data set.
(c) The correlation coefficient is a dimensionless measure of association. Thecorrelation in (b) would not change if size were measured in square milesinstead of acres.
2.1
26
Chapter 2
a)I,
; i
I...
i : ; ,,
----. . i I
: i . Ii
, ~
: :
. , IoJ /l r ,
, .
... ------I!
~
I.
, ,,
, I . ,
-A : t- ~ ..1'-, ':, . - ,i :=-: -j.1 3 d~j =,:': ~ ~, ,,, . .,;' .,.. .: _ __ ~.; " ,1 '
..,:/ ,1.. . . . /" 1-.
~~ /. -~'"'-.A "'7' .! . _ ... i I .: 'JO " : i i ; , , ,
../ :
7../
: '7K (-~ '~ =, -ii;;;;;.I~l-' :
I ,; I . ; ;I I
i i :~. .
, I.
. : ,
! i Ii
I,
: iI ; ,
I i ,I I , I
, , :
i !! I i : I.
; '.,;! i/'i : g'-' ,ILl / i-tI ! i i ,, , 1
! I
I
. J :
.
I ,t
i I;! : i I i
: :, 'i; " I I./i ,,/
_.
== i i Ib) i) Lx = RX = I = 5.9l'i
il) cas(e) x.y 1 .051= .. = =LxLy 19.621- -
e = arc cos ( .051 ) ;, 870
; 1 i) proJection of L on x ; s lt~i x =i
x
(1 1 31'is1is 3 = 7~35'35
c)':
: i
I . ! I , :
. i, I
= i i : l : i i
02-2: i.!- .
i 'i, i
I i
:i :3~
~
.1
I
;I
.;-- I 1I i. I
::ti I
i'.1i,
:i:-'.i.1 ..
"T
~-:~~-';'-i-' .~._~-:.i" 1 .' :.. :. -"-- --_..-- ---
27
( -~
15)r-6
-q2.2 a) SA = b) SA = - ~20 1 a -6
(-1 :
-9
-: )c) AIBI = d) C'B = (12, -7)
-1
e) No.
. (~
1). A i2.3 a) AI so (A I) = A' = A3 .
b) C'.(: :l (C'f"l~ J)1 a 10i2 4 J
(C' J' 'l- 1~il). (t''-'
-1 -a l,c = 3 1 i
i -i ' 10 -T
c)
(1:7)'
U8
': )(AB) ,= =
4 11 11
B'A' = (i n (~ ~)-
(~ ':)= (AB) i
11
d) AB has (i ~j )th entryk
a,. = a"b1, + a'2b2' +...+ a,,,b,,, = i aitb1j1 J 1 J 1 J 1 J R.=1Consequently, (AB) i has (' ,)th .1 ~J' entry
kc , , = I ajR,b1i ,Jl 1=1
Next I has .th row (b, i ,b2i ~'" IbkiJ and A' lias. jth2
28
column (aji,aj2"",ajk)1 so SIAl has ~i~j)th entryk
bliaji+b2ibj2+...+bk1~jk = t~l ajtb1i = cji
51 nce i and j were arbi trary choices ~ (AB) i = B i A I .
2.4 a) I = II and AA-l = I = A-1A.and 1= (A-1A)' = A1(A-l)l.of Al or (AI r' = (A-l)'.
bl (S-lA-l)AS _ B-1 (f1A)B - B-1S' I so AS has inverse (AS)-1
I
B-1 A- i. It was suff1 ci ent to check for a 1 eft inverse but we may
also verify AB(B-1A-l) =.A(~Bi~)A-i = AA-l = I ,
Thus I i = I = (AA - ~ ) I = (A-l)' A,IConsequently, (A-l)1 is the inverse
2,6
s 12l r _121 r
1 :J .l:~1
IT IT IT 13 = 1 69 = QIQ ,QQI =
-12 5 12 513 i3 IT i3 a 169
a) 5i nce A = AI, A' is symetric.
b) Since the quadratic form
x' Ax . (xi ,x2J ( 9-:)(::1-
9xi - 4x1 X2 + 6X2
- - .. -2
2.5
~ (2Xi.x2)2 + 5(x;+xi) ~ 0 for tX,lx2) -l (O~O)
we conclude that A is positive definite.
2.7 a) Eigenvalues: l = 10, 2 = 5 .Nonnalized eigenvectors: ':1 = (2/15~ -1/15)= (,894~ -,447)
~2 = (1/15, 2/15) = (.447, .894)
-
29
b) A' V-2 ) . 1 fIlS r2/1S. -1//5 + 5 (1/1S1 (1/IS, 2//5
-2 9-1/~ 2/~c) -1 1 (:
2) . (012 0041A = 9(6)-( -2)( -2) 9 ,04 .18
d) Eigenval ues: ll = ,2, l2 = ,1
Normal;z~ eigenvectors: ;1 = (1/;~ 2/15J
;z =: (2/15~, -1I/5J
2.8 Ei genva1 ues: l1 = 2 ~ l2 = -3
Norma 1; zed e; genvectors: ;~ = (2/15 ~ l/~ J
A (:
=~ = (1/15. -2/15 J
2) = 2 (2//5) (2/15, 1/15J _ 3( 1/1S)(1//s' -2/151 '
-2 1/15 -2/~2.9 ) -1 1 (-2 -2) =i1131 11a A = 1(-2)-2(2) -2 1 - --3 6
b) Eigenvalues: l1 = 1/2~ l2 = -1/3
Nonna1iz.ed eigenvectors: ;1 = (2/, l/I5J
cJ A-l =(t
;z = (i/~ -2/I5J
11 = 1 (2/15) (2/15, . 1//5J _ir 1/15) (1//5, -2/1-1 2 1/15 3L-21 5
30
2.10B-1 _ 1 r 4.002001 -44,0011
- 4(4,D02001 )-(4,OOl)~ ~4,OOl .
( 4,OZOCl= 333,333
-4 , 001
~.0011
1 ( 4.002-1A = 4(4,002)~(4,OOl)~ -4,001
-: 00011
. ( 4.002= -1,000,000
-4 , 001
-: 00011
Thus A-1 ~ (_3)B-1
2.11 With p=l~ laii\ = aii- and with p=2,
aaiia
= a11a2Z - 0(0) = aiia22a22
Proceeding by induction~we assume the result holds for any
(p-i)x(p-l) diagonal matrix Aii' Then writing
aii a aA =
a. Aii(pxp) ..a
we expand IAI according to Definition 2A.24 to findIAI = aii I
Aii I + 0 + ,.. + o. S~nce IAnl =, a2Za33 ... ~pp
by the induction hypothesis~ IAI = al'(a2Za33.... app) =al1a22a33 ,.. app'
31
2.12 By (2-20), A = PApl with ppi = pip = 1. From Result 2A.l1(e)IAI = pi IAI Ipil = AI. Since A is a diagonal matrix wlth
diagonal elements i,2~..., , we can apply Exercise 2.11 top pget I A I = I A I = n , .
'1 11=
2.14 Let be ,an eigenvalue of A, Thus a = tA-U I. If Q ,isorthogona 1, QQ i = I and I Q II Q i I = 1 by Exerci se 2.13. . Us; ngResult 2A.11(e) we can then write
a = I Q I I A-U I I Q i I = I QAQ i -I I
and it follows that is also an eigenvalue of QAQ' if Q isorthogona 1 .
2.16 (A i A) i = A i (A i ) I = A i AYl
Y = Y 2 = Ax.
show; ng A i A ; s symetric.
Then a s Y12+y22+ ,.. + y2 = yay = x'A1Axp _.. .. ..yp
2.18
and AlA is non-negative definite by definition.
Write c2 = xlAx with A = r 4 -n1. Theeigenvalue..nonnalized- - tl2 3
eigenvector pairs for A are:
i = 2 ~'=1 = (.577 ~ ,816)
2 = 5,':2 = (.81 6, -, 577)
'For c2 = 1, the hal f 1 engths of the major and minor axes of the
elllpse of constant distance are
~ = -i = ,707 and ~ =.. = .447~1 12 ~ ~
respectively, These axes 1 ie in the directions of the vectors ~1
and =2 r~spectively,
32
For c2 = 4~ th,e hal f lengths of the major and mlnor axes arec 2 '- = - = 1.414 and:, .f
c _ 2 _-- - -- - .894 .:2 ' IS
As c2 increases the lengths of, the major and mi~or axes ; ncrease.
2.20 Using matrx A in Exercise 2.3, we determne
i = , ,382, :1 = (,8507, - .5257) i2 = 3.6'8~ :2 = (.5257., .8507)1
We know
__(' .376A '/2 = Ifl :1:1 + 1r2 :2:2
,325
,325)1. 701
A-1/2 = -i e el + -- e el _ ( ,7608If, -1 -1 Ir -2 _2 ~ -,1453
- .1453 J
.6155
We check
Al/ A-1/2 =(:~) . A-l/2 Al/2
2,21 (a)
33
A' A = r 1 _2 2 J r ~ -~ J = r 9 1 Jl1 22 l2 2 l190= IA'A-A I I = (9-A)2- 1 = (lu- A)(8-A) , so Ai = 10 and A2 = 8.Next,
(b)
U;J ::J i ~ J :~ J
10 :~ J gives- (W2J- ei - . 1/.;
8 :~J gives 1/.; J
- e2 = -1/.;
AA'= ~-n U -; n = n ~J
12-A 0 4o = /AA' - AI I - .1 0 8 - 04 0 8-A
= (2 - A)(8 - A)2 - 42(8 - A) = (8 - A)(A -lO)A so Ai = 10, A2 = 8, andA3 = O.
(~ ~ ~ J ~ J - 10 (~J
.gves 4e3 - 8ei8e2 - lOe2
so ei= ~(~J
~
0
~ J :: J8 (~J8 -
0
gives4e3 - Gei
so e,= (!J4ei - U
Also, e3 = 1-2/V5,O, 1/V5 J'
\C)
34
u -~ J - Vi ( l, J ( J" J, 1 + VB (! J (to, - J, I
2,22 (a)
AA' = r 4 8 8 Jl 3 6 -9r : ~ J = r 144 -12 J
l8 -9 L -12 126o = IAA' - I I = (144 - )(126 - ) - (12)2 = (150 - )(120 - ) , soi = 150 and 2 =' 120. Next,
r 144 -12) r ei J = .150 r ei JL -12 126 L e2 le2 . r 2/.; )gives ei = L -1/.; .and 2 = 120 gives e2 = f1/v512/.;)'.
(b)
AI A = r: ~ J
l8 -9r438 8Jl 6-9
- r ~~ i~~ i~ J
l 5 10 145
25 - 5050= IA'A - I 1= 50 100 - 10 = (150 - A)(A - 120)A
5 10 145 - so Ai = 150, A2 = 120, and Ag = 0, Next,
25 50 5 J50 100 105 10 145
r ei J' r ei Jl :: = 150 l::
gives -120ei + 60e2 0 1 ( JO or ei = 'W0521-25ei + 5eg VU
( ~ i~~ i~ J5 lD 145 ( :~ J = 120 (:~ Jeg e2
35
gives -l~~~ ~ -2:~: ~ or., = ~ ( j J
Also, ea = (2/J5, -l/J5, 0)'.(c)
(4 8 8J3 6 -9
= 150 ( _~ J (J. vk j, J + 120 ( ~ J (to ~ - to J
2.24
( 1
a
n
1 = 4,=l=('~O,OJ';-1 = ~
a) 1 b) 2 = 9 ~=2 = (0,1,0)''9
a 3 = 1,=3 = (0,0,1)'
c) ~-lFor + : 1 = 1/4,2 = 1 /.9, = 1,3
':1 = (1 ,O,~) i
':2 = (0 ~ 1 ,0) ,el = (OlO~l)1-3
36
2.25
Vl/2 "(:
a OJ ( 1 -1/5 4fl5Jil
-.2 .26~
a) 2 o 'if.= ,-1/5 1 1/6= - 2 1 .1'67" ~:i'67
a 3 4/15 1/6 1 ' 1 67 i
b) V 1/2 .e v 1/2 =
(:0 0Jt 1 -1/5 4flJ (5 OJ i5 -1 4/3) (5 a
:J2 a -1/5, 1 1/6 0 2 = -2/5 2 1/3 a 2
a 3 4/15 1/6 1 a a 3 4/5 1/2 3 a 0
= (~:-241 n =f
2.26 a)1/2 i /2
P13 = 13/11 22, = 4/13 q = 4l15 = ,27
b) Write Xl = 1 'Xl + O'X2 + O-X3 = ~~~. with ~~ = (1 ~O~O)
1 1 i , i 1 12 x2 + 2 x3 = ~2 ~ W1 th ~2 = (0 i 2' 2" J
Then Var(Xi) =al1 = 25. By (2-43), ~1X 1X ,+ 1 2 1 .19
Var(2" 2 +2" 3) =':2 + ~2 =4 a22 + 4 a23 + '4 33 = 1 + 2+ 4
15= T = 3.75
By (2-45) ~ (see al so hi nt to Exerc,ise 2.28),
1 1 i 1 1Cov(X, ~ 2Xi + 2 Xi) = ~l r ~2 = "'0'12 +"2 13 = -1 + 2 = 1
~o
2.27
37
1 1Corr(X1 ~ '2 Xl + '2 X2) =
, 1COy(X" "2X, + '2X2) 1~r(Xi) har(~ Xl + ~ X2) =Sl3 := .103
a) iii - 2iiZ ~ aii + 4a22 - 4012
b) -lll + 3iZ ~ aii + 9a22 - 6a12
c) iii + \12 + \13' aii + a22 + a3i + 2a12 + 2a13 +2a23
d) ii, +~2\12 -. \13, aii' +~a22 + a33 + 402 - 2a,.3 - 4023
e) 3i1 - 4iiZ' 9a11 + 16022 since a12 = a .
2,31 (a)
38
EX(l)J = ,(l) = :i (b) A,(l) = 1 -'1 1 ~ J = 1
(c)
COV(X(l) ) = Eii = ~ ~ J
(d)
COV(AX(l) ) = AEiiA' = i -1 i ~ n -iJ = 4
(e)
E(X(2)J = ,,2) = n tf) B,(2) (~ -iJ n = n
(g)
COV(X(2) ) = E22 = -; -: J
(h)
COV(BX(2)) = BE22B' = ~ -~ J (-; -: J (-~ ~ J - (~: -~ J
0)
COV(X(l), X(2)) = ~ ~ J
(j)
COV(AX(1),BX(2))=AE12B'=(1 -1) ~ ~J _~ n=(O 21
2,32 ~a)
39
EIX(l)j = ILll) = ~ J (b) AIL(l) = ~ -~ J ~ J = -~ J
(c)
Co(X(l) ) = En = l-i -~ J
td)
COV(AX(l)) = AEnA' = ~ - J -i -~ J L ~~ ~ J - i ~ J
(e)
E(Xl2)j = IL(;) = ( -~ J (f) BIL(2) = ~ ; -~ J ( -~ i = -; J
(g)
( 6 1 -~1 iCOV(X(2) ) = ~22 = 1 4
-1 0
(h)COV(BX(2) ) = BE22B' ,
= U i -~ J (j ~ -~ J U -n - 12 9 J9 240)
CoV(X(1),X(2)) = l ::J ~ J
(j)COV(AX(l) i BX(2)) = AE12B'
2,33 (a)
40
- U j J H =l n ( j J - l ~ ~ J
E(X(l)j = Li(l) = ( _~ J (b) Ati(l) = L -~ ~ J ( _~ J - ~ J
(c)
( 4 i 6-i~JCov(X(l ) = Eii = - ~ - ~
(d)COV(AX(l) ) = Ai:iiA' ,
= ( -~ ~) (- -~!J (-~ n -234)
4 63
(e)
E(X(2)J = ti(2) = ~ ) (f) Bti(2) = ~ - J ~ J = I ; )
(g)
Co( X(2) ) = E" = n
(h)
CoV(BX,2) ) = BE"B' = U - ) L ~ J D - ~ J - I 1~ ~ J
(i)
41
( _1 0 JCOv(X(1),X(2))= -1 0
1 -1
)COV(AX(l), BX~2)) = A:E12B1
= 2 -1 0 J (=!O J1 1 3 i 01 -1 ~ - ~ J = -4,~ 4,~ J
42
2.34 bib = 4 + 1 + 16 + a = 21, did = 15 and bid = -2-3-8+0 = -13- -
(I~)Z = 169 ~ 21 (15) = 315
2.35 bid = -4 + 3 = -1- -
biBb = (-4, 3) L: -:J (-~ J= (-:14 23)
( -~ J 125- -
( 5/6
2/6 ) il )
d I B-1 d = (1~1) 2/6 = 11/62/6 1
so 1 = (bld)Z s 125 (11/6)" = 229.17- - '
2.36 4x~ + 4x~ + 6xix, = x'Ax wher A = (: ~).(4 - ).)2 - 32 = 0 gives ).1 = 7,).2 = 1. Hence the maximum is 7 and the minimum is 1.
2.37 From (2~51), maxx'x=l- -
X i Ax = max~fQ
~ 'A!~13 = 1
where 1 is the largest eigenvalue of A. For A given inExercise 2.6, we have from Exercise
2.7 ~ i = 10 and
el . (.894, -,447), Therefore max xlAx = 10 andth2s-1 x I x Flmaximum is attained for : = ~1.
2.38Using computer, ).1 = 18, ).2 = 9, ).3 = 9, Hence the maximum is 18 and the minimum is 9,
2.41 (8) E(AX) = AE(X) = APX = m
(b) Cav(AX) = ACov(X)A' = ALXA' = (~
43
o OJ18 0o 36
(c) All pairs of linear combinations have zero covarances.
2.42 (8) E(AX) = AE(X)= Apx =(i
(b) Cov(AX) = ACov(X)A' = ALxA' = ( ~o OJ12 0o 24
(c) All pairs of linear combinations have zero covariances.
3.2
44
Chapter 3
3.1
a) ~ = (:) b) ~, = ~, - i,! = (4 tOt -4) i':2 = ~z - x2! = (-1 t '. 0) I
c) L = m.e t..1
L = 12:2
Let e be the angl e between .:, and :2' then os ~e) ~
-4//32 x 2 = -.5
Therefore n s" = L2 or $" = 32/3; n S = i2 or S22 = 2/3;:, 22 ~2n 5'2 = ~i':2 or slZ = -4/3. Also, riZ = cos (e) = -.S. Conse-
(32/3 -413) "( 1" -.5)quently S = and R =n -4/3 2/3 -.5 1
a) g = (;J b) :1 = II - xl! = (-', 2, -11'
~2 = l2 - xz! = (3, -3, 0)'
c) L =/6; L =11':1 ~2Let e be the angle between ':1 and ~2' then eOs (e) =
-9/16 x 18 = - .866 .
Therefore n 31, = L!l or s" = 6/3 = 2; n 522 ~ l~ or szi =
= 18/3 = "6; n i.2 = :~ -:2 or :5'2 = -9/3 = -3. Also, r1Z =
~"s (e) = _ .8:6'6. Consequently So =( Z -3) and R= ( 1 - .8661-3 '6 . -.86'6 1 J
-
45
3.3II = (1, 4, 4)';
xl ! = (3,3. 3); II - ii ! = (-2, 1, 1 J'Thus
1 3 _2J= 4 = 3 + 1 - ii ! + (ll - xl l)li . -
4 3 1
l'.(~ 5:) ; (:
5
:Ja) - l3.5 X 1&
3 i
_.) - ')'=e
0
-:)E~; 1. ( 32 -:J
2S=(X-xl CX-xl.. ..
1 o -4
so S .l6~2J
-2
, (3
6
:J ~
b) 1: =4 -2
and l sIc: l2
i l' (; 41 ;J
-1 32
-:J e -9)~ -3 = -9 18-3-1 0(-31
2 S = (X - 1 x')' ( X-I ?) =". ,. .. ..
so S =.. ( 3 -9/2 J and Isl = 2.7/4-9/2 9
3.6 a) X'- 1 x' = r -~ ~ -~ J. Thus d'i = (-3, 0, -3),- N 3 -1 2 N
!2 = to, 1, -1) and ~/3 = (-3, 1,2) .Since,i = .92 = 23' the matrx of deviations is not offull ra.
3.7
46
b)(X - 1 X')' ( X-I xl) = ( ~ ~
.. "i w --15
15J-1l4
2 S =
-32
-1
So
( 9 -3/2.S = -3/2 1
15/2 -1/2
1'5/2)-1/2
7
Isl = 0 (Verify). The 3 deviation vectors lie in a 2-dimensionalsubspace. - The 3-dimensional volume ,enclosed by ~he deviation. .vectors 1 s zero.
c) Total sample varia-nce = 9 + 1 + 7 = 17 .
All e11 ipses are 'centered at
i) For S = (: : J '
-
x .-
S";1 (~ -:~: -4/9J519
Eigenvalue-normalized eigenv~ctor pairs for 5-1 are:
1 = 1. ;1 = (.707, -.707)
2 = 1/9, !~ = (.707, .7n7)
Half lengths of axes of ellipse (x-x)'S-l(X-X) S 1.. - ..-
are l/Ir = 1 and l/~ = 3 respectively. The major axis
of ell ipse 1 ies in the direction of ~2; the minor axis
1 ies in the direction of :1.
if)( 5 -4) -1
s= , s =-4 5 . (5/94/9
4/9)5/9
5-1 are:
For
Eigenvalue-normal ized eigenvectors for
i = 1. :~ = (.707. .707)i2 = 1/9, ~2 = (.7~7, -.77)
47
Half l~ngths ~faxes of ell ipse (x - x)'S-l (x - x) ~ 1 are,.... ...
again. l/lr = 1 and 1/1. = 3. The major axes of the
ellipse li.es in the direction of ':2; the minor axis lieS'
in the directi~n of =1. Note that ~2 here is =1 in"part (i) above and =1 here is =2 in part (i) above.
iii) For S = (3 0),. S-l = (1/3 OJo 3 0 l/3Eigenvalue-normalized eigenvector pairs for 5-1 are:
Half lengths of axes of ellipse
).1 = 1 13; ~i = (1, 0)).2 = 1/3, !~ = (0. lJ
(x - x)' 5-1 (x - x) s 1.... _..
are
equal and given by l/ir = l/lr = 13. Major and minor
axes of ellipse can be taken to lie in the directions of
" the
coordinate axes. Here, the salid ellipse.is, fn fact, a solidsphere.
Notice for aii three cases 1s1 = 9.
3.8 a) Total sample variance in both cases is 3 .
S. G0
~J.b) For 1 Isl = 1
0
For S =(-1~2-1/2
-1/21
- l/2
-1/2J-1/2 ,
1
Isl = 0
48
3.9 (8) Vve calculate = (16,18,34 l and
-4 -1 -52 2 4
Xc= -2 -2 -44 0 40 1 1
and we notice coh( Xc)+ coh( Xc) = cOli( Xc)
so a = fl, 1, -1 J' gives Xca = O.
(b)
( J I I 10(2.5)(18.5) + 39(15.5) + 39(15.5)S = 1~ 2.~ 5~~ so S = -(13)2(2.5) _ 9(18.5) -55(5.5) = 013 5.5 18.5 "As above in a)
Sa = ( 3 ~ ;53 -= 5~~ J - o~ J13 + 5.5 - 18.5
( c) Check.
3.10 (a) VVe calculate = (5,2,3 J' and
-2 -1 -31 2 3
Xc= -1 0 -12 -2 00 1 1
and we notice coh( Xc)+ ~012( Xc) = cOli( Xc)
so a = iI, 1, -1 J' giv.es Xca = O.
~b)
( 2.5 .0 2.5 JS =. 0 2.5 2.5
2.52.5 5I S I - 5(2.5)2 + 0 + 0so - -(2.5)3 _ 0 - (2.5)3 = i)
Using the save coeffient vector a as in Part a) Sa = O.
49
(c:) Setting Xa = 0,3ai + a2 = 07ai + 3ag = 0 so5ai + 3a2 + 4ag = 0
ai -5ai
g-"jag
3(3ai) + 4ag = 0
so we must have ai = as = 0 but then, by the first equation in the fil"tset, a2 = O. The columns of the data matrix are linearly independent.
3.11
i14808
1 4213 J
S = Con~equently14213 15538
( 09:70
09:70) ;01/2 = (121 ~6881
o )R =
124 .6515
and 0-1/2 =(" 0:82 00:0 J
The relationships R = 0-1/2 S 0-1/2 and S = 0'12 R 01/2
can now be verifi ed by direct matrix multiplication.
50
3.14. a) From fi rst pri nciples we hav.e
f ~l (2 3) (~J' 21-
Similarly ' ~2 = 19 and ' ~3 = 8 so
sample mean =2l+19+8 = 16
3
sampl~ vari ance = (21_16)1+(19-16)2+(8-16)2 = 492
Also :' ~1 (-1 2) (~J = -7;
so
IC X = 1_ -2 and :' ~3 = 3
sampl e mean = -1
sampl~ variance = 28
Finally sample covariance = (21-16)(-7+1)+(19-16)(1+1)+(8-16)(3+1) =
-2.8.
) -I (b ~ = 5 2) and
.
S ( ~: -12 J
Using (3-36)
3.15
sample mean of b' X =~' ~. (2 3) (:1 = 16
sample mean of :' ~ = (-1 2) (:1.-1
sample variance of b' X ~' S~ (2 3) e: -121(: 1 = 49
sample variance of C' X = :' S:.' (-1 21 C: -121 (";1 .28
sample covariance of b' X and c' X.. .. .. -
l6 -2J (-11:b'Sc=(23) " . =-28- -, -2 1 2
Resul ts same as those ; n part (a).
E (;1. (13S = -2.5
1.5
1.SJ-1.5
3
-2.51
-1.5
sampl e mean of b. X= 12- -
sample mean of c. X = -1- -
samp1 e variance of b' X = l2- -
sampl e vari ance of c' X = 43sample covariance of b' X and c' X = -3
51
'"
52
3.16 S 1 nee tv =E(~ -~V)(~ -~V)'
I , I I)= E(~ - ~V - ~V~ +~VJ:V
, 'E(V' )" ,;: E(~ ) - E(~)!:V - ~V _ +~V!:V, , , ,
:: E(~ ) - !:VJ:V -: !:V~V + ~V!:V
= E(~') - !:V!;V. '
we have E(VV') = * + !;V!;~ .
3.18 (a) Let y = Xi+X2+X3+X4 be the total energy consumption. Then
y=(1 1 1 l)x=1.873,
s~ =(1 1 1 I)S(1 1 1 1) =3.913
(b) Let y = Xl -X2 be the excess of petroleum consumption over natural gasconsumption. Then
y=(i -1 0 0)x=.258,
s~ =(1 -1 0 O)S(1 -1 0 0) =.154
S3
Chapter 4
4.1 (a) We are given p = 2 iI E I = .72 and
i=(;J E= 2 -.8 x V2i J-.8 x J2 50
( i V2)E-1 = ~:
4: :7i (i 1 2 2V2 2 )I(:i) = V: exp -- ( -(Xi - 1) + -(Xl - i)(X2 - 3) + -(X2 - 3)2)(27l) .72 2 .72.9 .72
-(b) 1 ( )2 2V2( 2 2- Xl - 1 + - Xl - 1)(x2 - 3) + -(X2 - 3).72.9 .72
4.2 ta) We are given p = 2 , I' = (n E =( 21V2
-4~) so I E I = 3/2
and2v'
. V2 2-T "J
I( ) i ( 1 (2 2 2V2 4()2 ) ):i = (27l)'3/2 exp -2" 3Xi - 3Xl(X2 - 2) + 3 X2 - 2
L-l =
(b) 2 2 2V2 4 2-Xi - -Xi~X2 - 2) + -(X2 - 2 )3 3 3
~c) c2 = x~(.5) = 1.39. Ellpse centered at (0,2)' with the major ax liav-in, haif-length . c = \12.366\11.39 = 1.81. The major ax liesin the direction e = I.SSg, .460)'. The minor axis lies in the directione =i-A-O , .B81' and has half-length ' c = \I;34v'1.S9 = .94.
54
Constant density contour that contains50% of the probability
oc?
I.~
C' 0x ~
I...
o..
-3 -2 -1 o 1 2 3
x1
4.3 We apply Result 4.5 that relates zero covariance to statisti~a1 in-
dependence
a) No, 012 1 0
b) Yes, 023 = 0
c) Yes, 013 = 023 = 0
d) Yes, by Result 4.3, (X1+XZ)/Z and X3 are jointly normal
their covariance is 10 +1a. = 02 1 3 2 3 . (0 ,e) No, by Result 4.3 with A = _~ 1
to see that the covari anc.e i~ 10 and not o.
and
_ ~ ), form A * A i
4.4 a) 3Xi - 2X2 + X3 is N03,9)
b) Require Cov (X2,X2-aiXi-a3X3)
~ i = tai ,a3J of the fonn ~ i
requirement. As an example,
a) Xi/x2 is N(l'(XZ-2),~)
b) X2/xi ,x3 is N(-2xi-5, 1)
c) x3lxi ,x2 is N(x1+X2+3) ,!)
4.5
ss
= : - a, - 2a3 = O. Thus any
= (3-2a3,.a3J wi 11 meet the
a' = (1,1).-
4.6 (a) Xl and X2 are independent since they have a bivariate normal distributionwith covariance 0"12 = O.
(b) Xl and X3 are dependent since they have nonzero covariancea13 = - i.~c) X2 and X3 are independent sin-ce they have a bivariate normal distribution
with covariance 0"23 = O.
(d) Xl, X3 and X2 are independent since they have a trivariate normal distri-bution where al2 = and a32 = o.
te) Xl and Xl + 2X2 - 3X3 are dependent since they have nonzero covariance
au + 20"12 - 3a13 = 4 + 2(0) - 3( -1) = 7
4.7 (a) XilX3 is N(l + "&(X3 - 2) , 3.5)
.(b) Xilx2,X3 is N(l + .5(xa - 2) ,3.5) . Since)(2 is independent of Xi, condi-tioning further on X2 does not change the answer from Part a).
S6
4.16 (a) By Result 4.8, with Cl = C3 = 1/4, C2 = C4 - -1/4 and tLj = /- for. j = 1, ...,4 we have Ej=1 CjtLj = a and ( E1=1 c; ) E = iE. Consequently,
VI is N(O, lL). Similarly, setting b1 = b2 = 1/4 and b3 = b4 = -1/4, wefind that V2 is N(a, iL).
(b) A.gain by Result 4.8, we know that Viand V 2 are jointly multivariatenormal with covariance4 (1 1 -1 1 1 -1 -1 -1 )( L bjcj ) L = -( -) + -( - ) + -( -) + -( - ) E = 0j=1 4 4 4 4 4 4 4 4That is,
( ~: J is distributed N,p (0, (l; l~ J )so the joint density of the 2p variables is
I( v, v,) = (21l)pf lE I exp ( - ~(v;, v; J (l; l~ r (:: J )
1 . (1 i -1 i -l ) )= (27l)pl lE I exp - s( VI E Vl + V2 E V2
4.17 By Result 4.8, with Cl = C2 = C3 = C4 = Cs = 1/5 and /-j - tL for j = 1, ...,5 wefind that V 1 has mean EJ=1 Cj tLj = tL and covariance matrix ( E;=1 cJ ) .L =
lL.Similarly, setting bi = b3 = bs = 1/5 and b2 = b4 = -1/5 we fid that V2 hasmean :;=i bj/-j = l/- and covariance matrix ( :J=1 b; ) L = fE.Again by Result 4.8, we know that Vi and V2 have covariance4 (1 1 -1 1 1 1 -1 1 1 1) 1(~b'c.)L= -( -) +- (-) .J -( -) +-( - )+-( -) ~=-E.;; J 1 "5 5 5. 5 l 5 5 5"5 '5 5 25
57
4.18 By Result 4.11 we know that the maximum 1 He1 i hood estimat.es of II
and t are x = (4,6) i and
n1 L - -)'_ (x.-x)(x.-xn j=l -J - -J- = t tmH~J)(m-mHm-(~J)((:1-(m'
.(GJ-m)(~H~J) '.((~J-m)mimn .
= t tc~J Gi aj.~o -i).m(i j) .(~JfP 1)1
b) From (4-23), ~ - N~(~,io t). Then ~-~ - N~(~,io t) and
finally I2 (~-~) - N(~,t)
c) From (4-23), 195 has a Wishart distribution with 19 d.f.
4.20 8(195)B' is a 2x2 matrix distributed as W19('1 BtBt) with 19 d.f.where
a) BtB i has 1 1 1(1,1) entry =011 + ~22 + tf33 - 012 - G13 + Z'23
1 1 1 1 1'1 l'(1 ,2) entry = -r14 of :t.24 +tf34 -'ZOlS +:tZ5 +-r35 +?'l - za26 - f13'1 1 i(2,2) entry = 06 + :t55 + tf44 - 46 - S6 + zc45
b)
=l 11stB'
31
131 .
G33J
S8
4.21 (a) X is distributed N4(J.1 n-l~ )(b) Xl - J- is distributed N4\OI L ) so ( Xl - J. )'L-1( Xl - J. ) is distributed
as chi-square with p degrees of freedom.
(c) Using Part a)i( X - J. )'( n-1L )-l( X - J. ) = n( X - Jl )'~-l( X - J. )
is distributed as chi-square with p degrees of freedom.
(d) Approximately distributed as chi-square with p degrees of freedom. Sincethe sample size is 1 arge i L can be replaced by S.
59
4.22' a) We see that n = 75 is a sufficiently lar"ge sample (compared
with p)and apply R,esult 4.13 to get I(~-!:) is approximately
Hp(~,t) and that ~ is approximately Np(~'~ t).
c i -1(- )By (4-28) we ~onclude that n(X-~) S ~-~ is approximatelyb)
X2p.
4.23 (a) The Q-Q plot shown below is not particularly straight, but the samplesize n = 10 is small. Diffcult to determine if data are normally distributedfrom the plot.
Q-Q Plot for Dow Jones Data30
. ..
.
20.
10-C)C .
0 .
..
-10
.
-20-2 -1 0 1 2
q(i)
(b) TQ = .95 and n = 10. Since TQ = .95 ~ .9351 (see Table 4.2), cannot rejecthypothesis of normality at the 10% leveL.
4.24 (a) Q-Q plots for sales and profits are given below. Plots not particularlystraight, although Q-Q plot for profits appears to be "straighter" thanplot for sales. Difficult to assess normality from plots with such a smallsample size (n = 10).
Q-Q Plot for Sales300
250
200a..
'I~
150 .
100.
50-2 -1 o
q(i)
.. Q"4 P1t for l)rofits
.
lS
10
-2 ~1 oq(i)
1
.
.
1
2
2
60
(b) The critical point for n = i 0 when a = . i 0 is .935 i. For sales, TQ = .940 and for
profits, TQ = .968. Since the values for both of these correlations are greaterthan .9351, we cannot reject normality in either case.
61
4.25 The chi-square plot for the world's largest companies data is shown below. Theplot is reasonably straight and it would be difficult to reject multivariate normalitygiven the small sample size of n = i O. Information leading to the construction ofthis plot is also displayed.
5
4
1iis 3g'"
l!u 2'!o
1
o
o 1 2 3 4 5ChiSqQuantii.
6 7 8
(155.6Jx = 14.7
710.9 ( 7476.5
S = 303.6-35576
303.6 -35576 J26.2 -1053.8
-l053.8 237054
Ordered SqDist.3142
1.28941.40731.64182.01953.04113.18914.35204.83654.9091
Chi-square Ouantiles.3518.7978
1.21251.64162.10952.64303.28314.10835.31707.8147
x=( 5.20J s=( 10.6222 -17.7102 J s-I =(2.1898 1.2569 J4.26 (a) 12.48 ' -17.7102 30.8544' 1.2569 .7539
Thus dJ = 1.8753, 2.0203, 2.9009, .7353, .3105, .0176, 3.7329, .8165,1.3753, 4.2153
(b) Since xi(.5) = 1.39, 5 observations (50%) are within the 50% contour.
(c) The chi-square plot is shown below.
C1i~squre plt for
.
.
. .
.
. 2
(d) Given the results in pars (b) and (c) and the small number of observations(n = 10), it is diffcult to reject bivarate normality.
62
4.27 q-~ plot is shown below.63
**
100. ** 2
2 2
so.**2*
*3**"..
:;3 32*
2 *60.'
*
*40. ** *
:\
20.\
-2. SI
-0.5 ii.5 \'a(i)%.5-1.S 0.5
Since r-q = .970 -i .973 (See Table 4..2 for n = 40 and .a = ..05) twe would rejet the hypothesis of normality at the 5% leveL.
64
4.29
(a).
x = (~~4~:~~:~)' s = (11.363531 3~:~~:~~~).
Generalized distances are as follows;
o .4607 o . 6592 2.3771 1 . 6283 0.4135 o . 47 1 1. 184910.6392 0.1388 0.8162 1.3566 o .6228 5.6494 0.31590.4135 0.1225 o . 8988 4. 7647 3.0089 o . 6592 2.77411.0360 o . 7874 3.4438 6.1489 1 .0360 O. 1388 0.8856O. 1380 2.2489 0.1901 o .4607 1.1472 7 . 0857 1 .4584O. 1225 1 .8985 2 .7783 8.4731 0.6370 o . 7032 1. 80 14
(b). The number of observations whose generalized distances are less than X2\O.ti) = 1.39 is26. So the proportion is 26/42=0.6190.
(c). CHI-SQUARE PLOT FOR (X1 X2)
8
w 8a:c
~4
~
2
0
0 2 4 6 8 10
~saUARE
4.30 (a) ~ = 0.5 but ~ = 1 (i.e. no transformation) not ruled out by data. For
~ = 1, TQ = .981 ~.9351 the critical point for testing normality withn = 10 and a = .10. We cannot reject the hypothesis of normality atthe 10% level (and, consequently, not at the 5% level).
(b) ~ = 1 (i.e. no transformation). For ~ = 1, TQ = .971 ~.9351 the critical
point for testing normality with n = 10 and a = .1 O. We cannot reject thehypothesis of normality at the 10% level (and, consequently, not atthe 5% level).
(c) The likelihood function 1~" --) is fairly flat in the region of , = 1, -- = 1so these values are not ruled out by the data. These results are consistent withthose in parts (a) and (b).
n-n niot~ follow
65
4.31The non-multiple-scle"rosis group:
Xi X2 X3 X4 Xs
rQ 0.94482- 0.96133- 0.95585- 0.91574- 0.94446-Transformation X-o.s Xi3.S (X3 + 0.005).4 X3.4 (Xs + 0.'(05).321
*: significant at 5 % level (the critical point = 0.9826 for n=69).The multiple-sclerosis group:
Xl X2 X3 X4 Xs
rQ 0.91137 0.97209 0.79523- 0.978-69 0.84135-Transformation - - (X3 + 0.005).26
- (X5 + 0.005).21
*: significant at 5 % level (the critical point = 0.9640 for n=29).Transformations of X3 and X4 do not improve the approximaii-on to normality V~l"y muchbecause there are too many zeros.
4.32Xl X2 X3 X4 Xs X6
rQ 0.98464 - 0.94526- 0.9970 0.98098- 0.99057 0.92779-Transformation (Xl + 0.005)-0.59 x.0.49
- XO.2S - (Xs + 0.5)0.Sl4
*: significant at 5 % level (the critical point = O.USO for n=98).
4.33Marginal Normality:
Xl X2 X3 X4rQ 0.95986* 0.95039- 0.96341 0.98079
*: significant at 5 % level (the ci"itical point = '.9652 for n=30).Bivariate Normality: the X2 plots are
given in the next page. Those for (Xh X2), (Xh X3),(X31 X4) appear reasonably straight.
66
CHI-SQUARE PLOT FOR (X1,X2) CHI-SQUARE PLOT FOR (X1,X3)
88
68
~
wi:
~
c
" ~"
.
is is2 2
0 0
0 2 " 6 8 10 0 2" 6 8 10
e-SOUARE e-SORE
CHI-SQUARE PLOT FOR tX1 ,X4) CHI-SQUARE PLOT FOR (X2,X3)
88
6 6w
w
i: a:c c~
~
~
""
.:f isCJ
2 2
0 0
0 2 " 8 8 10 12 0 2" 8 8 10 12
e-SOARE e-SOARE
CHI-SQUARE PLOT FOR (X2,X4) CHI-SQUARE PLOT FOR (X3,X4)
8 8
8 6w
w
i: a:
~
c
" ~
"
:f :fCJ
(.2
2
00
0 5 10 15 02 " 6 a
e-SOUARE e-SORE
4.34Mar,ginal Normality:
67-
Xl X2 X3 X4 Xs X6rQ. 0.95162- 0.97209 0.98421 0.99011 0.98124 0.99404
Bivariate Normalitv: Omitted.
*: significant at 5 % level (the critical point == 0.9591 for n==25).
4.35 Marginal normality:Xl (Density)
rQ I .897*& (MachDir) X;i ,(CrossDir)
.991 .924*
* significant at the 5% level; critical point = .974 for n = 41
From the chi-square plot (see below), it is obvious that observation #25 is amultivariate outlier. If this observation is removed, the chi-square plot isconsiderably more "straight line like" and it is difficult to reject a hypothesis ofmultivariate normality. Moreover, rQ increases to .979 for density, it is virtuallyunchanged (.992) for machine direction and cross direction (.926).
Chi-square Plot3S
:l
25
:!
15
10
6 10
Chi-square Plot without observation 25
12
2 4 6 B 10 12
68
4.36 Marginal normality:
100m 200m 400m 800mrQ I .983 .976* .969* .952*
1500m 3000m Marathon.909* .866* .859*
* significant at the 5% level; critical point = .978 for n = 54
Notice how the values of rQ decrease with increasing distance. As the distanceincreases, the distribution of times becomes increasingly skewed to the right.
The chi-square plot is not consistent with multivariate normality. There areseveral multivariate outliers.
4.37 Marginal normality:
100m 200m 400m 800mrQ I .989 .985 .984 .968*
1500m 3000m Marathon.947* .929* .921*
* significant at the 5% level; critical point = .978 for n = S4
As measured by rQ, times measured in meters/second for the various distancesare more nearly marginally normal than times measured in seconds or minutes(see Exercise 4.36). Notice the values of rQ decrease with increasing distance. Inthis case, as the distance increases the distribution of times becomes increasinglyskewed to the left.
The chi-square plot is not consistent with multivariate normality. There areseveral multivariate outliers.
..r = 0.9916 normal 0 r = 0.9631 not nonnalI/ 0
C'..
C\ =-_I/
"8 001:i ai 0;; 0 ~ ::I/ u.
II 00.. 01
-2 .1 0 1 2-2 -1 0 1 2
Quantiles of Standard Norml Quantlles of Standard Nonnal
I/0 r = 0.9847 nonnal c: r = 0.9376 not nonnalII..
c:ai I/u. i-
~~u.~
oX 0a. ai0i- C\
.0
lt ... ...co d
-2 -1 0 1 2 .2 -1 0 1 2Quantiles of Standard Nonnal Quantiles of Standard Nonnal
0 0co 0
r = 0.9956 normal 01 r = 0.9934 normal..IIlt0
co 0_I/ _ i-:i s: ..CI Gl ;g 0en 0I/C\ ..I/
0 0I/ 0C'..
-2 -1 0 1 2-2 -1 0 1 2
Quantiles of Standard Nonnal Ouantiles of Standard Norml
4.38. Marginal and multivariate normality of bull data
Normaliy of Bull DataA chi-square plot of the ordered distances
oC\
r:l/. ..'CCI~ 0- ..o
.... . .
..'
~'".
lt
2 4 6 8 10 12 14 16 18qchisq
69
70
XBAR SYrHgt FtFrBody PrctFFB BkFat SaleHt SaleWt
5-0.5224 2 . 9980 100.1305 2 . 9600 -0.0534 2.9831 82.8108995.9474 100 . 1305 8594.3439 209.5044 -1.3982 129.9401 6680 . 308870.881-6 2 . 9600 209.5044 10.6917 -0.1430 3.4142 83 .92540.1967 -0 .0534 -1. 3982 -0.1430 o .0080 -0.0506 2.4130
54. 1263 2.9831 129.9401 3.4142 -0.0506 4.0180 147.28961555.2895 82.8108 6680. 3088 83.9254 2.4130 147.2896 16850.6618
Ordered Ordered Ordereddsq qchisq dsq qchisq dsq qchisq
1 1 . 3396 0.7470 26 3.8618 4 .0902 51 6 . 6693 6 . 84392 1. 7751 1.1286 27 3 . 8667 4.1875 52 6.6748 6 .98363 1 . 7762 1.3793 28 3 .9078 4.2851 53 6 .6751 7 . 12764 2.2021 1 .5808 29 4.0413 4 .3830 54 6.8168 7 . 27635 2.3870 1.7551 30 4.1213 4.4812 55 6 . 9863 7 .430 16 2.5512 1 . 9118 31 4. 1445 4.5801 56 7. 1405 7 .58967 2.5743 2.0560 32 4 . 2244 4.6795 57 7 . 1763 7 . 75548 2.5906 2.1911 33 4.2522 4 . 7797 58 7.4577 7 .92819 2. 7604 2.3189 34 4.2828 4 . 8806 59 7.5816 8.1085
10 3.0189 2.4411 35 4.4599 4.9826 60 7 .6287 8.297511 3 . 0495 2.5587 36 4. 7603 5 . 0855 61 8 . 0873 8 . 496312 3 . 2679 2 .6725 37 4. 8587 5. 1896 62 8 .6430 8 .706213 3.2766 2.7832 38 5. 1129 5 . 2949 63 8 . 7748 8 .928614 3.3115 2.8912 39 5 . 1876 5 .4017 64 8.7940 9. 165715 3.3470 2.9971 40 5.2891 5 .5099 65 9.3973 9.419716 3 . 3669 3.1011 41 5 . 3004 5.6197 66 9 . 3989 9.693717 3.3721 3 . 2036 42 5.3518 5 . 7313 67 9 .6524 9.991718 3.4141 3 . 3048 43 5 .4024 5 .8449 68 10.6254 10.319119 3 . 5279 3.4049 44 5.5938 5 .9605 69 10.6958 10.682920 3.5453 3 . 5041 45 5 .6060 6.0783 70 10.8037 11. 093621 3 . 6097 3 .6027 46 5.6333 6.1986 71 10.9273 11.566522 3.6485 3 . 7007 47 5 .7754 6.3215 72 11.3006 12.126323 3.6681 3. 7983 48 6.2524 6.4472 73 11.321$ 12.816024 3 . 7236 3. 8957 49 6 . 3264 6.57'60 74 12.4744 13.722525 3.7395 3.9929 50 6.6491 6.7081 75 17.6149 15.0677
76 21.5751 17.8649
From Table 4.2, with a = 0.05 and n = 76, the critical point f.or the Q - Q plot corre-lation coeffcient test for normality is 0.9839. We reject the hypothesis of multivariatenormality at a = 0.05, because some marginals are not normaL.
4.39 (a) Marginal normality:independence
rQ I .991support benevolence
.993 .997conformity leadership
.997 .984*
* significant at the 5% level; critical point = .990 for n = 130
(b) The chi-square plot is shown below. Plot is straight with the exception ofobservation #60. Certainly if this observation is deleted would be hardto argue against multivariate normality.
Chi-square plot for indep, supp, benev, conform, leader
15~&..
. .
....
10..
"..du)A2
5
o..
o 2 4 6 8 10 12 14 16 18q(u-.5)/130)
71
(c) Using the rQ statistic, normality is rejected at the 5% level for leadership. Ifleadership is transformed by taking the square root (i.e. 1 = 0.5), rQ = .998 and
we cannot reject normality at the 5% leveL.
4.40 (a) Scatterplot is shown below. Great Smoky park is an outlier.
500G..u;t .;..01/.0':o
.'.o -...
cO " 5 6Visitors
7 8 9
(b) The power transformation -l = 0.5 (i.e. square root) makes the sizeobservations more nearly normaL. rQ = .904 before transformation andrQ = .975 after transformation. The 5% critical point with n = 15 for thehypothesis of normality is .9389. The Q-Q plot for the transformdobservations is given below.
10
-1 1
(c) The power transformation ~ = 0 (i.e. logarithm) makes the visitorobservations more nearly normaL. rQ = .837 before transformation andrQ == .960 after transformation. The 5% critical point with n = 1"5 for the
hypothesis of normality is .9389. The Q-Q plot for the transformedobservations is .given next.
72
(d) A chi-square plot for the transformed observations is shown below. Giventhe small sample size (n = i 5), the plot is reasonably straight and it would behard to reject bivarate normality.
.. _.,..,.'...... ....,.- .. ",-,-,-, ...........'..,.... -, ".. .. .......:.,.
Chi-square plot for transformed nat
. .
. .
o
1 2 3 4 5Chi~square quantiles
6 7
73
4.41 (a) Scatterplot is shown below. There do not appear to be any outliers with thepossible exception of observation #21.
(b) The power transformation ~ = 0 (i.e. logarithm) makes the durationobservations more nearly normaL. TQ = .958 before transformation andTQ = .989 after transformation. The 5% critical point with n = 25 for thehypothesis of normality is .9591. The Q-Q plot for the transformedobservations is given below.
Q-QPlatfotNatural Log Dut60n3.0
2.5
(I5' 2.0o~
1.5
1.0-2 -1 o
q(i)1 2
74
7S
(c) The power transformation t = -0.5 (i.e. reciprocal of square root) makes theman/machine time observations more nearly normaL. rQ = .939 beforetransformation and rQ = .991 after transformation. The 5% critical point withn = 25 for the hypothesis of normality is .9591. The Q-Q plot for thetransformed observations is given next.
,QeQ',plot for Re:iprocal of Square Root of ManlMachinl! Time
. .
. .
. .
. .
. .
.2-1 o
q(i)1 2
(d) A chi-square plot for the transformed observations is shown below. The plot isstraight and it would be difficult to reject bivariate normality.
Ci., ",:,-,..,::'_',::-...','__d"":,'d,.'.' _ ',' "',, , ' ,,:':::;_:,":':'"::-_'..,.,..,'.:.;c_',:::,.,:;,',"',""" ._,_, _ "".:;..:',_'"':::',, -,-," 'J;'g'~
Chi-square plot for transformed,snow,rernovat data ...0 0'010
. .
. . .
8
6
o
.. .
.
.....
......
o 2 3 4 5Chi-squa.~ quanti
6 7 8
7-6
Chapter 5
5.1 .) ~ "
(i60) ; s" (-i-:/3-i:13).
f2 = 1 SO/ll = 13.64
b) T2 is 3ri,2 (see (5-5))
c) HO :~. ~ (7,11)
a =- .05 so F2,2(.05~ - 19.00
Since T2 _ 13.64 (;' 3FZ,2(.05) = 3(19) =57; do not reject H1l at
the a - .05 1 eve1
5.3 a) TZ ;.
n
(n-1)!.J.g1 (:j-~O)(:j-~O)'!n ,- - (n-1) = 3(~4) - 3 = 13:64
! r (x.-i)(xJ.-i)'! .j=l -J - - -
b) li - (I Ji (~j-~)(~j-~) 'I 'r =- I j~i (~r~~H~j-~o)' i/
Wil ks i 1 ambda '" A2/n = A1/Z = '.0325 - .1803
(44)2244 =.0325
5.5 HO:~' = (.'5'5,;60); TZ = 1.17
a -.05; FZ ,40( .05) ;. 3.23
Since TZ '" 1.17 (; 2~~) F2,40( .05) =- 2.05(3.23) = 6.62,we do not reject HO at the a" .05 level. The r,esult is ~onsistentwith the 9Si confidence ellipse
for ~ pi~tured in Figure 5.1 since
\11 = (.'55,.60) is inside the ellipse.-
77
5.8 -1(- )el CI S X-\1 .:- - -0 f227.273 -181.8181
t18L818 212.121 J((.Sti4 J -( .5'5 J )
.603 .60
= (2.636 J-1.909
tZ = n(~'(~-~O))Z =a' SA- -
(.014) 242(~.,'3L. -1.9a'"J . )
.003
r2.636 -1 9091 .(.0144 .01 i71f2.63611. ':J .0117 .0146jL-i.909j
= 1.31 = TZ
5.9 a) Large sample 95% T simultaeous confdence intervals:
Weight: (69.56, 121.48) Girt: (83.49, 103.29)Body lengt: (152.17, 176.59) Head leng: (16.55, 19.41)Neck: (49.61, 61.77) Head width: (29.04, 33.22)
b) 95% confidence region determined by all Pi,P4 such that
L,002799 - .006927J(9S.52 -Pi)(95.52 - ,up93.39 -,u4 ~ 12.59/61 = .2064- .006927 .019248 93.39 - P4
Beginng at the center x' = (95.52,93.39), the axes of the 95%
confidence ellpsoid are:
major axis :t .J3695.52.Ji 2.5 9(' 939).343
:t .J45.92.J12.59(- .343).939
(See confidence ellpsoid in par d.)c) Bonferroni 95% simultaneous confidence intervals (m = 6):
160 (.025 / 6) = 2.728 (Alternative multiplier is z(.025/6) = 2.638)
. ,minor axis
Weight: . (75.56, 115.48) Gii1h: (86.27, 100.51)Body lengt: (155.00, 173.76) Head length: (16.~, 19.0g)Neck: (51.01, 60.37) Head width: (29.52, 32.74)
d) Because ofthe high positive correlation between weight (Xi) and girt~X4),the 95% confidence ellpse is smaller, more informative, than the 95%Bonferroni rectangle.
5.9 ,Continued)
0..
..
LO0..
00..
"' LO)( C'
0C'
LOCD
0CD
78
Large sample 95% confidence regions.
large sample simultaneousBonferroni
-- --- -~ -------- - - - - ---- - ----
- - - - - - - - - - - - - - - - -.
II .. . . . .. .. . . . . .'. .. . . . .. . ... . . . . . . . . . -' . . . . . . . . . . . . "l . . . . . . ~I I :: i :, : :i ,
60 70 80 100 110 120 13090
x1
e) Bonferroni 95% simultaneous confidence interval for difference betweenmean head width and mean head lengt (,u6 - tls ) follows.(m = 7 to allow for new statement and statements about individual means):t60 (.025/7) = 2.783 (Alternative multiplier is z(.025/7) = 2.690)
- (0036) S66 - 2sS6 + sss = (31.13 -17.98) +_ 2.78~~2i.26 -2(13.88) + 9.95x6 -xs :tt60 . Jn 61or 12.49:: tl6 -,us:: 13.81
79
5.10 a) 95% T simultanous confidence intervals:Lngt: (13D.65, 155.93) Lngt4: (160.33, 185.95)Lngt3: (127.00, 191.58) Lngt5: (155.37, 198.91)
b) 95% T- simultaneous intervals for change in lengt (ALngt):
~Lngth2-3: (-21.24, 53.24)~Lngt-4: (-22.70, 50.42)~Lngth4-5: (-20.69, 28.69)
c) 95% confidenceregon determined by all tl2-3,tl4-S such that
. ( i.Oll024 .009386J(16 - ,u2-3)16-tl2_3,4-tl4_s . ~72.96/7=10.42.009386 .025135 4 - ,u4-S
where ,u2-3 is the mean increase in length from year 2 to 3, and tl4-S isthe mean increase in length from year 4 to 5.
Beginnng at the center x' = (16,4), the axes of the 95% confidence
ellpsoid are:
maior axis.~~.895)
:tv157.8 72.96.- .447
( 447):t .J33.53.J72.96 .
.895(See confidence ellpsoid in par e.)
mior axis
d) Bonferroni 95% simultaneous confdence intervals (m = 7):Lngt: (137.37, 149.21)Lngth3: (144.18, 174.40)
..6Lngth2-3: (-1.43, 33.43)i1Lngt3-4: (-3.25, 30.97)
Lngt4: (167.14, 179.14)Lngth5: (166.95, 187.33)
i1Lngth4-5: (-7.55, 15.55)
-5.10 (Continued)
'80
e) The Bonferroni 95% confidence rectangle is much smaller and moreinformative than the 95% confidence ellpse.
0C\
0., ,.
Iv::
0
0,.
I
0C\
I
95% confidence regions.
o"l
oC" ...... ..........
....................................................
simultaneous T"2Bonferroni
IIIIIIIIII. I
--~--------- ----------------~; I I: I I: I I; I I; I I; I I.. . , . .~. . ... .. , ... , , . , . . . J. . . . . .. . .. , . . . , , . . . . , . . , , , . , .. . . .: i: I: I
-20 o 20
J.2-3
40
81
5.11 a) E' =- (5.1856, 16.0700)
S = (176.0042 . 287.2412J;287.2412 527.8493
-1 ( .0508S =-.0276
~ .0276 J
.0169
Eigenvalues and eigenvectors of S:
,t = 688.759A
.42 = 15.094
,
1 = (.49,.87),
i. = (.87,-.49)i 16Fp,n_p(.10) =: 7 F2.7(.10) = T (3.26) = 7.45~--
Confidence Region
45
40
~L.
V)'ON)(
15 I 20 25 30 35 40 45x1 ( C r )
35
-10 -,! '
I -10 J
b) 90% T intervals for the full data set:Cr: (-6.88, 17.25) Sr: (-4.83, 36.97)
(.30, 1 OJ' is a plausible value for i..r-
825.11 (Continued)
c) Q-Q pJotsfor the margial distributions of both varables
oi.
30
020
10
.o. ......-l. -UL .0.5 0.0 os 1.0 1.5
nomscor
Since r = 0.627 we rejec the hypothesis of normty for ths varable at a = 0.01
80
7Ieo
50
u; 40
30
20
10
0 .
-1.
.
. .
. .
.
. .
-1.0 .0.5 0.0 0.5 1.0 1.5nomsrSr
Since r= 0.818 we rejec the hypothesis of normty for this varable at a = 0.01
d) With data point (40.53, 73.68) removed,
ii = (.7675, 8.8688); r .3786S =b .03031.0303 J
69.8598.
-1 (2.7518S - .0406
-.0406 J. 0149
1. F (.10)= 7(62t F" 6(.10) '" 164 (3.4'6) ~. 8.07-T p1n-p 'I90% r intervals: Cr: (.15, 1.9) Sc: (.47, 17.27)
83
5.12 Initial estimates are
( 4 i - - (0.5 0.0 0.5 i' - 6, ~ - 2.0 0.0 .2 1.5
The first revised estimates are
( 4.0833 i -( 0.6042 0.1667 0.8125 i' = 6.0000 , E = 2.500 0.0.2.2500 1.9375
5.13 The X2distribution with 3 degrees of freeom.5.14 Length of one-at-a time t-interval / Length of
Bonferroni interval = tn_i(a/2)/tn_i(a/2m).
n 215 0.854625 0.8632-50 0.8691100 0.871800 0.8745
m4
0.74890.7644D.77490.7799"0.7847
100.64490.66780.68360.69110.6983
5.15
(0).E(Xij) = (l)Pi + (0)(1 - Pi) = Pi.
Var(Xij) = (1 - pi)2pi +(0 - p)2(1 - Pi) = Pi(1 - Pi)(b). COV(Xij, Xkj)
= E(XijXik) - E(Xij)E(Xkj) = 0 - PiPIi =-PiPk.5.16
(6). Using Pj:: vx3.(0.05)VPj(1 - pj)ln, the 95 % confidence intervals for Pi, P2, 11, P4, Psare
(0.221, 0.370),(0.258, 0.412), (0.098, 0.217), (0.029, 0.112),\0.084, .a.198) respectively.(b). Using Pi - i :l Vx3.(0.05)V(pi(1 - i) + i(1 - i) - 2iPi) In, the 95 % confdenceinterval for Pi - P2 is (-0.118, 0.0394), There is no significant difference in two proportions.
5.17i = 0.585, i = 0.310, P3 = 0.105. Using Pj:l vx'5(O.-D5)VPj(1 - Pi)fn, the 95 %.confidenceintervals for Pi, P2, 11 are "(0.488, 0.682), (0.219, 0.401), ('0.044, 0.l6), respectively.
84
5.18\lo). Hotellng's T2 = 223.31. The critical point for the statistic (0: = 0.05) is 8.33. We rejectHo : fl = (500,50,30)'. That is, The group of students represented by scores are significantlydifferent from average college students.(b). The lengths of three axes are 23.730,2.473, 1.183. And directions of corresponding ax..are
( 0.994 )
0.103 .,0.038
.(c). Data look fairly normaL.
( -0.104 )
0.995 ,0.006 ( -0.037 )
-0.010 .0.999
.
. 35700 70
~
-
.
.
.I- I" -30 .I' 60 r .60 I' -i .I~
.Lf M-
;C 50 J x 25 -/ .500 i -.40 J -..."1
i 20 .io.
-
-
400-
30 15
-2 -1 0 1 2 -2 -1 0 1 2 .2 -1 0 1 2
NORMA SCORE NORMA SCORE NORMAL SCORE
700
600. I... . ....
..- .'1...e : .._
-, ... ..~....
. .a. .I . ...0
. . . ...- . , .
;c50' .
400
30 40 50 60 70
X2
700
.
.
.
70 0 . . .0I .
. ..
. . 01 10
. . .
60 :. .. 00.
.
.
.
.! I.oi..
N . 0 050 . o.x . .- .. ..
40 i . ..
.
.
30
15 20 2S 30 35
X3
60.
.. .a.
. ..t : .
. . ... . . .
. . .. . . .. .. . ...
. i .: I... .. . .-.
.. . ..
. .
. .-.: :.o ..
x500
400.
o
15 20 2S 30 35
X3
5.19 a) The summary statistics are:
-x __ (180. 50Jn = 30,~354 .13
and s = (124055.17361621 .03
361"621 .031
348"6330.9'0 J
85
wher~ S has e i g~nva 1 ues and e; g~nv ect~rs
1 = 3407292
2 = 82748
e~ = (.105740, .994394)_1
!2 = (.994394,-.1 0574~)
Then, since 1 p(;:~) .Fp n_p(a) = 3~ 2~i) F2 2St .tl5) = .2306,, n, 'a 95% confidence region for ~
-
is given by the set of \1. -
(124055.17 3'61~21.03J' ~1(1860.5tl-~lJ(1860.'50-\11' 8354.13-~2) . .361621 .03 348633tl. 90 83~4 .13-~2
. ~ .2306
The half lengths of the axes of this ellipse are 1.2300 Ir = 886.4 and
l. 2306 .~ = 138 ~ 1. Th~refore the ell ipse has the form-------_. --_...__.. . ...... --_..- .._._.. ".-----------_.- ------_._-
-~"
'12,000 , :
-
,
; /,. 10000 ; ,,
: j f,
'.
:, i
: ,. ; I; ;
i,
!,
; i /' . "Ji-- 'v.. i
,
, ,
:,
: , ,
,
!
; , ; : ! ;~ I
: i :: ,~~.So . I ,I i : ! ! ! :-,
'-,
- - ~5'4.13 i. i
:
- : iI
;
! I,
l .1 :; . I , i I l~ ... J I: : : ; i
,
; , , 11: :
,
,i : ! !
. i J iI
, ,
ii
.1: ;
. 1:
:
';fJ"w: ;
,,
,
i : : :i
~E.
:--
,
I i , I1 ,'. . IOQ" 2,.ao. ' . 3OO ' '. l.o.ft Xl
86
b) Since ~O = (2000, 10000)' does not fall within the 9Siconfidence
ellipse, we would reJect the hypothesis HO:~ = ~O at the 5% level.
Thus, the data analyz~d are not consistent with these values.
c) The Q-Q plots for both stiffness and bending strength (see below)
show that the marginal normal ity is not seri ously viol ated. Ai so;
the correlation coefficients for the test of normal ity are .989 and
.990 respectively so that we fail to reject even at the ii signifi-
cance level. Finally, the scatter diagram (see below) does not indi-
cate departure from bivariate normality. So, the bivariate normal
distribution is a plausible probability model for these data.
Q-Q Plot-Bend i n9 StrengthX2
12000. . *I.!
* * *
10000. *******
. ._-*
...._-----
---- -.- --
8000 .**
..2--..' ..*****
*- _.. .,-------***" ._-"- . -._--_._--
* * *
..._--- -_._-
6000.- ...... * ._--.._.._....... _.
------_.._---
4000. :i,
-2.0 l0.0 2.0-1.'0 1.~ 3.0
t:rr.e 1 at; on .989
Xi2800.
2400 .
::ooo.
1600 . ***** *
*
1200. **
800. .._----------- ._-- ..---"..
-2.0_ ____, ._-_ _=J.!.9..._
. _.._ . ..Correlation .. -.990 .
Q-Q Plot-Stiffness
* *
***
*
..
***2*2
****
* *
*
87
."__0"" ._____. .._--_ -_._---"
I
0.0.....-._.. ---~------I--:.
2.0_......-_.. ._.__....__.- ~.-
,
.
1.0 ._ _ _ _.._~ .9.___
2400 ;. .
2000.
1600.
1200. . *
800.I
4000 .
Sea tter 01 agram
*
* .
*
* *
* *
* *
* * *** *
. '.._-.- . ........_... .. . ...
*
*
***
*
**
*
'88
-_.. . -...._....~.. ..-
- -------- - ...-
* .
--- ----_..-
*
**.....-_.._-- -_..__...
*
-- ---.- -._----
..._-_..__....- _.. ......._. .. ,...__. ---------
I
80O.. - - 10000.
_._-~- .. .-:-6000.._-.. -- ---- . _._........--.__.,.. .1---0- -~r.12000. X2
.. . i 4000 .
89
5.20 (6). Yes, they are plausible since the hypothesized vector eo (denoted as . in theplot) is inside the "95% confidence region. .
96li S1mullJeouB Cooldence Region for Wean Veclor
(11).
i i I
ii.ii.i .~ii i
ii."
i.oi ..i"i' .
i, i
Uii' .
... 110 II .
Bonferroni C. i.:
Simultaneous C. i.:
'ii ,., ... 1'. ,.. "7 ...iiu.
LOWER UPPER189 .822 197 . 423274.782 284.774
189.422 197 .823274. 25S 285.299
Simultaneous confidence intervals are larger than Bonferroni's confidence intervals. Simul-taneous confidence intervals wil touch the simultaneous confidence region from outside.(c). Q~Q plots suggests non-normality of (Xii X2). Could try tra.nsforming XI.
Q-Q PLOT FOR X1
210-
.
-
-
.
.
200 ...
.
)( ...
190 ---
.
180.
..
.2 -1 0 2
NORMAL SCORE
Q-Q PLOT FOR X2
310.
..
300 ..
.
290 .JrN 280x
./270.
.
260 .-..
250
-2 -1 0 2
NOMA SC-QRE
310. .
.
300
29. .
. : - . ..280 ..N .. . .x270 . '.
260
250
ISO 20X1
90
5.21
HOTELLING T SQUARE - 9 .~218P-VALUE 0.3616
T2 INTERVAL BONFERRONIN MEAN STDEV TO TOxl 2S 0.84380 0.11402 .742 .946 .778
.909x2 25 0.81832 0.10685 .723 .914 .757
.880x3 25 1.79268 0.28347 1.540 2.046 1. 629 1. 95"6x4 25 1.73484 0.26360 1. 499 1. 970 1. 583 1. 887x5 25 0.70440 0.10756 .608 .800 .642 .766x6 25 0.69384 0.10295 .602 .786 .635 .753
The Bonferroni intervals use t ( .00417 ) - 2.88 andthe T2 intevals use the constant 4.465.
5.22
la). After eliminating outliers, the approximation to normality is improved.91
a-a PLOT FOR X1 a-a PL-DT FOR X2 a-a PLOT FOR X3
30 182S 15 111
.
.'. 14 ..20 .
.. 12.,.
~
10 ..X .. x 10,.'--15 .' ...C/ ,. - 8
....10 _.. 5 - 6 0a: . . .' .W 5 . .. 4...
-2 -I 0 2 -2 .1 0 2 -2 -1 0 2l-:: NOMA SCRE NOMA SCORE NORMA SCORE0::l-~ 18 18
15 18 16. 14 . . 14.
12'. 12 . .~
10 .. . . M10 I. .X 10 , .. x .
.l o .. .' 8 . o . 8 ...5 .. 8 6.. .. 4 4
5 10 is 20 25 30 5 10 15 20 25 30 S 10 1SXi X1 X2
a-a PLOT FOR X1 a-a PLOT FOR X2 a-a PLOT FOR X3
14 . . 1818 16CJ 14 . . 12 14 .... ..o. 10a: 12 .. 12 ...
X..
~8 . '" ..W 10 ..- .. x 10 .""
- 6 _.. .... 8
.
- 8 .... . 4 .l- . 66 . ..:: . 2 4 .4a
-2 -1 0 2 .2 .1 0 2-2 -I 0 2l- NORMA SCRE NOM4 SCORE NOMA SCRE::a
::l-~
iI ,.14111 ILL12
14. . 141012 .. . 12
~8 . '"
:. .'" I... X 10 . . x 10 . .
. .
II . . . 8 . ... . .4 . 6 II2 . . . 4 4
4 II 8 10 14 4 8 8 10 14 2 4 6 8 10 14XI X1 X2
l. Outliers remov.edi~
Bonferroni c. i.:
Simul taneous C. i.:
92
LOWER UPPER9.63 12.875.24 9.678.82 12.34
9.25 13.244.72 10.198.41 12.76
Simultaneous confidence intervals are larger than onferroni's confidence intervals.
(b) Full data set:
Bonferroni C. I.:
Simultaneous C. I.:
Lower9.795.788.65
Upper15.3310.5512.44
9.165.238.21
15.9611.0912.87
93
5.23 a) The data appear to be multivanate normal as shown by the "straightness" ofthe Q-Q plts and chi-square plot below.
.
.140 -
.
. 140.
.
..
..
.-
..c
. or.
't. 0)
.CD 130 - . :i.x
. Ul.t' lU 130 .:2 . ID
..
.
.
.
.
120 -..
120-1 i . I i-2
-1 0 1 2-2
-1 0 2NScMB NScBHi- = .97'6
~= .994
... .
.
.
55 -.110 -
..
.
.
.
-s:. .c
.
-
. C)C). :i 50 -
.
-i. UlUl 100 - . lU
.
lU. Zm
..
.
..
..
.
45 -90 -.
.
-T .1 I I ,'. I ! i
-2 -1 0 1 2-2
-1 0 2NScBL NScNHr;= .995i. = .992
.10 -.
.
.
..
d) .5 _. . ...
..
.....
..
....
..
o - . ...
I i "U 5 10
.fe,4l.( -.5)/30)
5.23 (Continued)
94
b) Bonferroni 95% simultaneous confidence intervals (m = p = 4):t29 (.05/8) = 2.663
MaxBrt:BasHgth:BasLngth:NasHgt:
(128.87, 133.87)(131.42, 135.78)(96.32, 102.02)(49.17, 51.89)
95% T simultaneous confidence intervals:4(29) F (.05) = 3.49626 4.26
MaxBrt:BasHgt:BasLngt:NasHgth:
(128.08, 134.66)(130.73, 136.47)(95.43, 102.91)(48.75, 52.31)
The Bonferroni intervals are slightly shorter than the T intervals.
9S
5.24 Individual X charts for the Madison, Wisconsin, Police Department data
xbar s LeL UCLLegalOT 3557.8 06.5 1738. 1 5377.4ExtraOT 1478.4 1182.8 -2.070.0 5026.9 use L-CL = 0
Holdover 2676.9 1207.7-946 . 2 6300 . 0 use LCL = 0
COA 13563.6 1303.2 9654.0 17473.2MeetOT 800.0 474.0 -622. 1 2222 . 1 use LCL=O
00ai .0:: CDii;: 0ii 00:: ("
"C":;
'e.5 0
.00
...
00ai 0:: ,.ii ..;:ii .00:: a"C ("":;
..
'e.5 000Q)
The XBAR chart for x3 = holdover hours
. . .
..
. ....a................y........................;..........;..........................__......................;................................
. . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
2 4 6 8 10 12 14 16
Observation Number
The XBAR chart for x4 = COA hours
... . .
. .
............................................................--;.........................................................................................
..
.
. .
2 4 6 8 10 1412 16
Observation Number
Both holdover and COA hours are stable and in control.
96
5.25 Quality ellpse and T2 chart for the holdover and COA overtime hours.All points ar.e in control. The quality control 95% ellpse is
1.37x 10-6(X3 - 2677)2 + 1.18 x 10-6(X4 - 13564)2+1.80 X 1O-6(x3 - 2677)(X4 - 13564) =5.99.
The quality control 95% ellipse forholdover hours and COA hours
000r-..
000co..
.00.0It..
..
0.in 0
:i 0 .'I0 ..J:. .+c(
.0 0u 00t'
...
.
0 .00C\..
000T-T-
-1000 0 1000 3000 5000
Holdover Hours
a:
r-
The 95% Tsq chart for holdover hours and COA hours
UCL = 5.991ci .................. '''..n..... ............ ........ ..... ...... .._...... ..... ..............._.._...........__.ini:t! 'It'C\
o
97
5.26 T2 chart using the data on Xl = legal appearances overtime hours, X2 - extraordinaryevent overtime hours, and X3 = holdover overtime hours. All points are in control.
The 99% Tsq chart based on x1, x2 and x3
o..
................................................................................................................................................
.
CD
C'~
co
v
N
o
5.27 The 95% prediction ellpse for X3 = holdover hours and X4 = COA hours is1.37x 10-6(x3 - 2677)2 + 1.18 x 1O-6(x4 - 13564)2+1.80x 1O-6(x3 - 2677)(X4 - 13564) = 8.51.
The 95% control ellpse for future holdover hoursand COA hours
oooo..
000co..
.
.
..
!!0
.0j 0 .0 v:z ..
. .+c(.0()
000N..
-1000 0 1000 3000 5000
Holdover Hours
98
5.28 (a)
x=
-.506-.207"-.062-.032
.698-.065
s=
.0626 .0616
.0616 .0924
.0474 .0268
.0083 -.0008
.0197 .0228
.0031 .0155
.0474 .0083 .0197 .0031
.0268 -.0008 .0228 .0155
.1446 .0078 .0211 -.0049
.0078 .1086 .0221 .0066
.0211 .0221 .3428 .0146
-.0049 .0066 .0146 .0366
The fl char follows.
(b) Multivariate observations 20, 33,36,39 and 40 exceed the upper control limit.
The individual variables that contribute significantly to the out of control datapoints are indicated in the table below.
Point Variable P-ValueGrea ter Than UCL 20 Xl O. 0000
X2 0.00.01X3 0.0000X4 0.0105X5 0.0210X6 0.0032
33 X4 .0.0088X6 O. 0000
36 Xl o . 0000X2 \) .0000X3 \). OO.QOX4 0.0343
39 X2 0.0198X4 0.0001X5 0.0054X6 o . 000'0
40 XL 0.0000X2 O. 0088X3 0.0114X4 0.0-013
99
2 472' 2 29(6) .5.29 T = 12. . Since T = 12.472 c: -- F6,24 (.05) = 7.25(2.51) = 18.2 , we do not
reject H 0 : . = 0 at the 5% leveL.
5.30 (a) Large sample 95% Bonferroni intervals for the indicated means follow.Multiplier is t49 (.05/2(6)):: z(.0042) = 2.635
Petroleum: .766:t 2.635(.9251,J) = .766:t .345 -7 (.421, 1.111)
Natural Gas: .508:t 2.635(.753/.J) = .508:t .282 -7 (.226, .790)
Coal: .438:t2.635(.4141.J) = .438:t.155 -7 (.283, .593)Nuclear: .161:t 2.635(.207/.J) = .161 :t.076 -7 (.085, .237)Total: 1.873:t 2.635(1.978/.J) = 1.873 :t.738 -7 (1.135, 2.611)
Petroleum - Natural Gas: .258:t2.635(.392/.J) = .258:t.146 -- (.112, .404)
(b) Large sample 95% simultaneous r intervals for the indicated means follow.Multiplier is ~%;(.05) = .J9.49 = 3.081
Petroleum: .766:t3.081(.9251.J) = .766:t.404 -- (.362, 1.170)
Natural Gas: .508:t3.081(.753/.J) = .508:t.330 -- (.178, .838)
Coal: .438:t3.081(.414/.J) = .438:t.182 -- (.256, .620)Nuclear: .161:t3.081(.207/.J) =.161:t.089 -- (.072, .250)Total: 1.873:t 3.081(1.978/.J) = 1.873:t .863 -- (1.010, 2.736)
Petroleum - Natural Gas: .258:t 3.081(.392/.J) = .258:t .171-- (.087, .429)
Since the multiplier, 3.081, for the 95% simultaneous r intervals is larger thanthe multiplier, 2.635, for the Bonferroni intervals and everything else for a giveninterval is the same, the r intervals wil be wider than the Bonferroni intervals.
100
5.31 (a) The power transformation ~ = 0 (i.e. logarthm) makes the durationobservations more nearly normaL. The power transformation t = -0.5
(i.e. reciprocal of square root) makes the man/machine time observationsmore nearly normaL. (See Exercise 4.41.) For the transformed observations,say Yi = In Xi' Y2 = 1/' where Xl is duration and X2 is man/machine time,
- = p.171JY l .240 s = r .1513 -.0058J
l- .0058 .0018S-i - r 7.524 23.905J
l23.905 624.527
The eigenvalues for S are . = .15153, . = .00160 with corresponding, ,eigenvectors ei = (.99925 - .03866), e2 = (.03866 .99925l Beginning atcenter y, the axes of the 95% confidence ellpsoid are
maior axis: IT 2(24):! v . F2 23 (.05) ei = :t.208el25(23) .
. .mInor axis: r: 2(24):tv. F223(.OS)e2 =:t.021e225(23) .
The ratio of the lengths of the major and minor axes, .416/.042 = 9.9, indicatesthe confidence ellpse is elongated in the ei direction.
(b) t24 (.05/2(2)) = 2.391, so the 95% confidence intervals for the two componentmeans (of the transformed observations) are:
Yi :tt24(.0125); = 2.171:t2.391.J.1513 = 2.171:t.930 ~ (1.241, 3.101)
Y2 :tt24 (.0125)' =.240:t2.391.J.0018 =.240:t.101 ~ (.139, .341)
Chapter 6WI
ii.1Ei9~nvalues andei9~nvectnrs of Sd are:
"1 = 449.778,
"2 = 168.082,
!1 = (.333, .943)
~~ = (.943, -.333)
Ellipse cent~rl!d at r = (-9.36,.13.27). Half length of major axis is
20.57 units. Half length of minor axis is 12.58 units. Major and minor
axes lie in :1 and !2 d;r~ctions, respetively.
Yes, the t.est answers the question: Is = 0 ins1tfe the 95i confi-
dence e 11 ipse 1
6.2 Using a critical value tn_i(cr/2p) = tio(O.0l25) = 2.6338,
Bonferroni ~. I.:UlWER
-20 . 57-2.97
-22 . 45-5.70
Simul taneous 'C. I.:
UPPER1.85
29.523.73
32.25
Simultaneous confidence intervals are larger than Bonferroni's confidence intervals.6.3 The 95% Bonferroni intervals are
LOWER UPPER-21.92 -2.08-3.31) 20.56
-23.70 -~ . 30-5 .50 22.70
Bonferroni 'C. i.:
Simultaneous C.I. :
Since the hypothesize vector '6 = 0 (denoted as * in the plot) is outside the joint confidenr.egion, we reject Ho : '6 = O. Bonferroni C.!. are consistent with this result. After theelimination of the outlier, the difference between pairs became significant.
95% Simultaneous Conidence Region (or Della Vector
.3 0
M 20U12
10MU22 0
- 10-.3 0
-20 -10MU11-MU21
Problem 6.3
6.4(a). HoteHing's T2 _Ho : = o.
.. ...
(b).
102
o
10.215. Since the critical point with cr
Lower-1.09-0.04
Bonferroni C. I.:
T Simultaneous C. 1.:-1.18-0.10
- 0.05 is 9.4'59, we reject
Uoner-0.02
0.64
0.070.69
..1..
95% Cofidence Slips Ab the Me Vecor'.0...
...
..S
...
0.'0.'...
-0.'-0.2
-0.4
(o~O)
..
ld ...."'1 -it
-0."-1.1i -1.5 -1.4 -1.3 -I.R -1.1 ....0 -o.S -0.. -0.7 _0.. -0.& -0.4 -0.8 -0.2 -e., 0.0 0.' .... o.a
Figure 1: 95% Confidence Ellpse and 'Simultaneous T2 Interv for the Mea
Diffence
(c) The Q-Q plots for In(DiffBOD) and In(DiffSS) are shown below. Marginalnormality cannot be rejected for either variable. The.%2 plot is not straight(with at least one apparent lJivariate outlier) and, although the sample size(n =11) is small, it is diffcult to "argue for bivariate normality.
o _o.s
i5.... -I..
a-a Ploto.S
....
. .--~/
. //-/..,,~
.////'/////
-.,/../".
~/-~......,/-'
./.-......-/
////-_1.5 . r"''//-/
..../
1.25
1.00
0.76..
CI~
0.60
is..
..0.25
_2.0
_0.25
_0.50
//_1.5 o 0.&
lr' Qlt.. 1_
Q-Q Plots
~/..///"///
,../-
.. /////"/.. //~'_.//
.....
.-///-../
....
../....//
o ..5..1~...1_
Chi -squa-e Aot d th OderedDistcnced.
0-11,
3,
4-'-
103
.i7
104
6.5 a) H: Cii = 0 wher.e C = ('0. 0 - -
-1
1 ). ~. = (~1'~2'~3) -~
i:x = (-11.2), CSt' = (55.5- 6.9 -32.6
-32.6)66.4
T2 = n(Ci)' (ese' )-1 (ei) = 90.4; n = 40; q = 3- -
((~~~li)l) Fq_l.n_a+i(.05) = (3~~2 (3.25) = 6.67
Since T2 = 90.4 ~ 6.67 reject H :Cii = 0o - -b) 951 simultaneous confidence intervals:
111 - 1-2: (46.1 - 57.3) :! -/6.67 J5~5 = -11.2 :! 3.01-2 - 1-3: ti.9 :! 3.3
111 - 1-3: -4.3 :! 3.3
The means are all different from one another.
IOS
6.'6 a) Tr"eatment 2: Sampl e mean vector (:l sampl e covariance matrix(-3;2
-3~2 J
Trea tment 3: Sampl e mean vector
(:) ; sampl covariance matrix
r~13-4/3 J4/3
Spoled =
( 1.6-1.4
b) TZ (2-3, 4-2)((1 + 1) (1.6
;1.~
r (:~J= 3.-88=
-1.4
("1 +n2-2)p _ (5)2 _-(" +n p 1) Fp n +n p 1 ( .01) - 4 (18) - 451 2- - '1 2--
Since TZ = 3.88 ~ 45 do not reject HO=l2 -!!3 = ~ at the ci = .011 evel .
c). 99%simul taneous confi-dence intervals:
1121 - l1:n: (2-3) :! I4 ~+l)l.,- = -1 :16.5
1122