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Statistics
Date
Apr 3, 2024
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(?)r-.rr=Jmarks) h=Yt+tTrt=@tK='@
X ecotogist wishes to investigate the levels of mercury pollution in four laKes. ne citches 4 fish from lake 1,
7 fish from lake 2, 5 fish from lake 3, and 5 fish from lake 4.
The following table gives a summary of measurements (in parts per million) for each of four samples.
Lake (treatment)
Sample data
1
2
-l
4
tli
4
7
5
5
n.
t' 1.,.. - T.
? ,'ll I
./ =l
52
69
7t
6t
n.
I
z
.i=1
2
1' . .
753
798
t248
9t2
Assume that concentr.tions of mercury in all fish in a lake constitute the population of values.
Assume that all three populations are normal and have common variance.
a. Construct the ANOVA table for the sample data.
use the rest of page for the calculations required for ANovA table.
b. Test at 5yo significance level to determine if the data provide sufficient evidence to indicate a
difference in larr'-population means. lf required, use Tukey' method to determine which means differ'
c. construct an individ ual g|%o confidence interva! for the difference between mean concentrations of
mercury in the fish from lake # 2 and lake # 1.
Write the solutions for questions b. and c. on the following page'
OVA T
Source of variation
Degrees of freedom
Sums of squares
Mean squares
F ratio
Treatment
(-l=3
60,
t,
gf
LO,
l6t
0. 16 q
Error
h* f. -- l1
60t. 1 rt
3S. Y 33
Tota I
h-t=20
5 6 Z-. 9SZ
Ffr =-- ?'5 Er+';g, "Y-s;,;:@
+ a'a
d7r r{h;,-: _---
ss; =rtd-+'z-3?rr E ry=@
ntJ- , ? ^-
tlc,
ssrr=E# #={.q*rF-q
9sE = sST- SSI-r = aqz,tg+
o
b. Test at 5% significance level to determine if the data provide sufficient evidence to
indicate a difference in four population means. lf required, use Tukey' method to
determine which means differ.
a. State null and alternative hypotheses. (0.5 mark)
F{o: F,= lrr = ft = 1, ,
I{,: 6+ ArAryI' *ru r"tun iE il1'tr1*t'* W W Yl4'trtt
b. State the calculated value of test statistic and criticalvalue of test. (1 mark)
.f*,. = 0. S59
f-.- !-o-o.sr3f l? =3.zt
c. Decide whether null hypothesis is true and make the conclusions.
lf required, use Tukey's procedure. (0.5 mark)
Lrrrq l*. z ?, we da vuv+ W lfin. a+ sL = O'os-
\t1,g ti,'"w[[iu.uN+ W wi)a'uu l4^/* ('u/+L a'&d''[4t'w
PPru"'** ttt^'artq
T,r4"" W te vw+ w,ywiva4.
c. Construct an individ ual gil%confidence interval for the difference between.mean
concentrationsofmercuryinthefishfromlake#1andIake#z.Lz,niin>".I-
-f
[,. = * = * = B t tr . =+" =+ = s-sb i F-[r.= 3.[Y,
t_
to, zsllr
MtE[*,*frJ = 2,uD
@e4,s7-
=l,ll0
":r
fi.= [l.ht, ) Tr.=[st)6= t06t T.z.=-!9s)y=YY0, Il.=(F0)6=?a0
T,, =fl.nTz.*T3. : s 0o t yyo r 3 og :@
) 5 SIr cnn ^$(. f,ut ,,i i" 1ry0 wu.atf,.
@ ssrr- t#,- F_- -h*-#,H- F= #-#.y-w:-
5\t+09.8 t- 5qt806 - 9Y,?0t,8 = luo.za
i',.=F=W=F2.3^
- Lake (treatment)
sa m p re o ari'--------------
1
2
a
J
tti
6
8
6
i.
51
55
50
s.
I
5.4
4.9
5.2
Assume that concentrations of mercury in allfish in a lake constitute the population of values.
Assume that all three populations are normal and have common variance.
a. Construct the ANOVA table for the sample data.
Use the rest of page for the calculations required for ANOVA table.
b. Test at 5%o significance level to determine if the data provide sufficient evidence to indicate a
difference in three population means. lf required, use Tukey' method to determine which means differ.
c. Construct an individ ual95% confidence interval for the difference between mean concentrations of
mercury in the fish from lake # 2 and lake # 1.
Write the solutions for questions b. and c. on the following page.
ANOVA TABLE
Source of variation
Degrees of freedom
Sums of squares
Mean squares
F ratio
Treatment
lc-t = /
t0 0, Lo
F0, t0
4.{0
Error
h-Kal+
o,{r. o}
26, 4z
Total
[1-] = l!
s\s, ?\"
SLSS-52.il2 + 6LST-sZ.3l2 = lo0,L
q000 _
\Z .
tJl
= 15,605 +-Zyrlsotl
o s sTr =Z,n;L[;-8,
S 9l'r = 6 Ll t- SZ. 3)z +
. SsE = Jt,.(n;-f)
t=l
"= S;,{1(s) * t{.gt(?) r ;.ZL(,s) =Vyg.O+ @
b. Test at 5% significance level to determine if the data provide sufficient evidence to
indicate a difference in three population means. lf required, use Tukey' method to
determine which means differ.
' -" State null and alternative hypotheses. (0.5 mark)
F[: dr{- at W V+a,k uul
'*kf,[+al WWyY
State the calculated value of test statistic andtiiticalvalrie of test. (1 mark)
f*= to.og,zrtL=3,50'
Ff,: tlU= lh.=Vt
' *ltu"dr
.fut,
= 4, g0
Decide whether null hypothesis is true and make the conclusions.
lf required, use Tukey's procedurs. iron
'5 mark)
t Jto.oqrr
2b.\rc(lrt)
@_Jg: t*.,we Aav'olyw( ltro a**= o,oy
d,vd w,t/),",rq il^rr* l4/'ptL r9 w d4[41^r/^/L &ful(/)^
la4puw@^/L wtA,t^^'
Inr,&r,t' L Wdilrt,ls no4 wry.,rc/.
c.Constructan,no,,',,ffinceinterva!forthedifferencebetweenmean
concentrations of mercury in the fish from lake # 2 and lake # 1'
tt- tl b L,ll
_,1"g6 , ltr^
!
I
*rhr*,)
L6 ,tro (f +f) . l, - !, , ss-sl t z .,r
, $tnf -1n&r{ A[r|$17A , M ril/vi{1/^^k4 d-vian.
6)t, +3+2=(,r;;;"" --"T*'['=63," vt= 5+t{ +r=11' 0
\#order to study the differences in operation cost of different types of machines, an engineer examined
the operation costs for three different types of machines. Random samples of operation costs (in $ per
hour) were obtained for each type of machine. The data obtained by the engineer are represented in the
table below. one observation is missed, not all sample sizes are the same.
1,
2
3
4
5
[,
Type I machine
L0.4
9.9
10.6
8.9
1,1,.0
50"8
Type ll machine
8.8
1,1,.6
L0.9
1,1,.4
Yz,a
Type lll machine
L1,.2
13.4
11,.9
13.3
13.1
62.5
Itb.Y
h
9u= t0, ft
ff2, = 10.6t
k ,= lz'58
$$rl:1 77462
7,, '"
SgT=
9SIr =
9$E=
l4gh =
IV'I9E =
l6 r0?,1,' a
6. oll
if ti-+=
r3t 6:l v
*t-t,=
Er h; h
L
= [t. 3g g.
Ir
a. Construct ANOVA table. Leave 3 digits after decimal point for the values of SSTr, SSE, SST, MSTr, MSE, and
2 digits for the value of F ratio. Use the rest of page for the calculations required for ANOVA table.
b. Test at 5% significance level to determine if differences exist in mean operation costs of 3 types of machines.
lf required, use Tukey's method to determine which means differ. Use a = 0.05.
c. Construct 90 % individual confidence interval for the difference in mean operation costs for Type I machine
and Type 2 machine.
Write the solutions for question b. - c. on the following two pages.
ANOVA TABLE
Source of variation
Degrees of freedom
Sums of squares
Mean squares
F ratio
Treatment
K-l:t
t6,ozt
f ,ou
+,+\
Error
h-tl- =11
,1. e ,E
1,03r
Total
1a-1 = l3
z+. t{09
lT+Y" 6L -
lE6. Y I
-.
l4
= L+, rlo 3
5!-C*\z.lz r hr* 156.Y-t = lb, r,zt.
#lr{
sgr -
5e.
f(-l
sE_
h -tu
ssrr = L+.rurl-!'itliu = rt ,388
=. l.0et
(0
uww\ s!.tnl4oyu c*vl4 rrtl- tuul *anvw W dll (W 4
W4ltt4dta
b.
' il\WT"iti"to determine if the data provide sufficient evidence to
Test at 5% y$ni
in operation costs of 3 types of machines. State null and alternative
hypoth
(1 mark)
r{,: *the* e\t- vyuwieu w,r+ $ d'4[t+rr+/^rf fu fua1 olfuS-S'
State the calculated value of test statistic and critical value of test. (1 mark)
f*.= {e,asrzrn = 3.98'
.foor.= +r ?t{
Decide whether nun hypothesis is true Jt'm;.[]l"r.,rsions.
lf required, use Tukey's procedure. Draw the diagram for corresponding population
mean and underline the means that do not differ significantly. (3 marks)
W-"fu y @ vNn-c is w; il^*'*n
Mek@r.FT,r,,-,w%%
T*. = {*,r, vr-,e
ry,f r.
,6,
r.
(f,
I (a-
,Ta
<T
trr
lr
ri
>li
>l
L<
r, E le.g 8 - [0, 16 =2.Y2,
L, 2 lz,gt - 10:68 z,t.g0
Ir =,0.68- 10. 15= 0,S
h
,{
I
*
[i
Ir
F
3.
,2t
6,.Conftru.t 6O % individual confidence interval for the difference in mean operation costs for Type
: 0, 6tL
0,5L- {.T96 (
o.sl - 4" L\
t,+q6 r0,6gY)
Z9
o.1r t f, - tn
r. = [0. [6 t
)=16s6 i+;, {EsT-'
ind icate a
I
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