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UNIVERSITI TUN HUSSEIN ONN MALAYSIA
FINAL EXAMINATION
SEMESTER
I
SESSTON 20r0t2011
COURSE NAME
ENGINEERING STATISTICS
COURSE CODE
DSM 2932
PROGRAMME
2DFTI3DFT/ 3DDT/
3 DET/3DEE
EXAMINATION DATE
NOVEMBER/DECEMBER 20 I O
DURATION
2II2
INSTRUCTIONS
ANSWER ALL QUESTIONS IN
PART A AND THREE (3)
QUESTIONS IN PART B
HOURS
THIS QUESTION PAPER CONSISTS OF NINE (9) PAGES
DSM2932
PART A
Ql
Refer to the data in Table 1:
Table
x
v
(a)
Find
I
xi,Z
60
74
54
76
52
66
/i,2 xi2, 2 xi !i,
42
70
48
76
arrrdl y?
I
36
68
34
62
28
54
26
64
l8
54
.
(5 marks)
(b) Find,S",, S- and ^!o.
(3 marks)
(c)
calculate the sample correlation coefficient and interpret the result.
(3 marks)
(d)
By using the least square method, estimate the regression line.
(6 marks)
(e)
Estimate x if
y:72.
(3 marks)
Q2 (a) A manufacturer
of car batteries claims that the lifetime of his batteries is approximately
normally distributed with a standard deviation o : L2 years. A random sample of 15 of
these batteries has an average lifetime 2.9 yearc and a standard deviation ol t.g yeaxs.
Use a level of significance d = 0.02.Test the hypothesis about the average lifetime of
batteries with
Hs : p:2.0
fi:
P#2.0
(9 marks)
(b) A teacher wants to determine whether the new method of teaching that was recently
being introduced is better than the traditional method. One test was given to both
groups from the same school by the end of week 6. Each group was taught using
different method (one group with the new method, another group using the tradilional
method). The group which has the higher test score is assumed to have the better
method. The data is in Table 2:
Table 2
New Method
\
Traditional Method
= 69.2
7z = 67.5
sl = 49.3
nr=50
si = 64.5
n, =80
Test the hypothesis for the above statement
witha = 0.05 for
DSM2932
Ho:Fr-ltz=0
Ht:ltt-lJr>0.
(11 marks)
PART B
Q3 (a) A discrete random variableXis given by
'r+1. x=0. 1.2
10
"*r=
9
-2x
x=J)
{ l0 '
+
(i) Show that p(x) is a pdf.
(ii) Find P(0 < X <3).
(iii) Find the cumulative distribution, F(x).
(10 marks)
(b) A continuous
random variable xhas a probability density function
r, \ l*'*bx+1, 0<x<l
otherwise
L 0'
Jlx)=1
where a and D are constants.
(D
(ii)
Given that E(g:
?,
Show that Var(X)
=!
J
firrd the value of
a
arrd, b.
.
45
(10 marks)
Q4 (a) A supervisor from a company found out that 7 o/o of theproduct is defective and cannot
be exported
.If
20 of the product were selected, find the probability that
(i) none is defected,
(ii) between 2 and 5 are defected,
(iii) at most 3 are defected.
(7 marks)
(b) If the average of summons given per week is 10, find the probability that
(i)
(ii)
exactly 4 summons were given per week,
at least 2 summon were given per week.
(6 marks)
DSM 2932
(c)
Marks for Statistics test have a mean 73 and standard deviation 12. Find the probability
that one student is selected at random, the marks is
(i)
(ii)
between 65 and 78,
more than 75.
(7 marks)
Qs
(a)
ll,
l,
Given a population of 7 numbers which are I 12, 13,
14,9 and 7.
sample of 5 can be drawn from that population with replacement, find
population mean and standard deviation,
sample mean and standard deviation,
the probability that the sample mean is greater than 10.
rf arandom
(i)
(ii)
(iii)
(8 marks)
(b)
An electronics company manufactures two types of resistors, type X and type 1z The
resistors of type Xhave a mean resistance of 100 Cl and a standard deviation of l0 O.
The resistors of type )'have a mean resistance of 99 O and a standard deviation of 15
C). A random sample of 25 for each one is selected from normal population.
(i)
(ii)
(iii)
State the distribution of X and V .
Find the probability that sample mean of type
X
greater than sample mean
of
Wpe Y.
Find the probability that p(l .0 <N
-
F< t.s;.
(12 marks)
Q6 (a) Given a random sample of size, n = 50 from a normal population with variance
o'=225
and mean
population mean.
i=64.3.
Construct the 95
% confidence interval for
the
(8 marks )
(b)
The wall thickness of 25 glass 2-liter bottles was measured by a quality control
engineer. The sample mean was x :4.058 mm and the sample standard deviation
w&s 11 :0.081 mm. Another sample of wall thickness 17 mm with the sample mean
y :3.912 mm and the sample standard deviation s2 :0.075 mm was also selected.
Assume that the populations are approximately normal with equal variances.
(i)
(ii)
State the point estimate for the difference means of both wall thicknesses.
Construct the 99Yo confidence interval for the difference means of both wall
thicknesses.
(12 marks )
DSM2932
BAHAGIAN A
51
Berpandu kepada data di Jadual
(a)
l:
Dapatkan D xi,E yi,Z xiz, D xi yi, dan} yi2
(5 markah)
(b) Dapatkan S,y, So and qo.
(3 markah)
(c)
Dapatkan korelasi koefisien sampel dan terangkan keputusan.
(3 markah)
(d)
Tentukan persamaan garis regresi dengan menggunakan
kaedah kuasa dua terkecil.
(6 markah)
(e)
Anggarkan
rjika y :
72.
(3 markah)
s2
(a)
Suatu kilang yang membuat bateri kereta mendakwa bahawa jangka
hayat bagi bateri
yang dihasilkan mempunyai taburan hampir normal dengan
riritr- piawai o = 1.2
tahun. Satu sampel rawak 15 bateri y*g dipitih mempunyai
min ha*t 2.9 tahun dan
sisihan piawai 1.8 tahun. Guna aras keertiaan a= 0.02.
Uji hipotesis terhadap min jangka hayat bateri, iaitu
Hs:
p:2.0
H: P*2'o
(9 markah)
(b)
leorang guru ingin tahu sama ada kaedah mengajar yang baru diperkenalkan lebih baik
daripada kaedah tradisi. Satu ujian diberikan tepaaa'aua kumpulan
pelajar dari sekolah
yang sama pada akhir minggu ke-6. Kumpulan masing-masing
diajar dengan kaedah
berlainan (satu kumpulan dengan kaedah yang baru, rito m-luan
lagi JZng- kaedah
yang lama). Kumpulan yang menghasilkan skor ujian yang
tinggi dianigap teknik
mengajar kumpulan itu lebih baik. Data yang terhasil di Jadual2:
Jadual
2
Kaedah Baru
Kaedah Lama
4 = 69.2
7z = 67.5
sl = 49.3
sl = 64.5
\=50
nz
=80
DSM2932
Lakukan ujian hipotesis untuk menguji pemyataan di atas pada aras keertian a 0.05
=
iaitu
Ho:lJt-Pz=0
Hr:pt-ltr>0
(l I markah)
BAHAGIAN B
53 (a)
Suatu pembolehubah rawak
diskritXdiberi oleh
'r+1. x=0. 1.2
l0
,*,=
9
{
-2x
l0'
x=3,
4
(i) Tunjukkan p(r) adalah pdf.
(ii) Dapatkan P(0 < X <3).
(iii) Dapatkan fungsi kumulatif, F(x).
(10 markah)
(b) Pembolehubah rawak selanjarXmempunyai
fungsi ketumpatan kebarangkalian
t, \ l*'*bx+|, 0<x<l
selainnya
L 0,
J\x)=1
dengan
a
dan
b adalahpemalar.
(i)
Diberi
(ii)
Tunjukkan bahawa Var(/)
E(X): ? ,
J "*nilai
bagi
a
dan b.
: +.
45
(10 markah)
S4 (a) Seorang penyelia daripada sebuah syarikat mendapati 7 % danpada pengeluaran adalah
rosak dan tidak dapat di ekspot. Jika2} daripada pengeluaran ini dipilih, dapatkan
kebarangkalian
(i)
(ii)
(iiD
tiada yang rosak,
di antara 2 dan 5 rosak,
selebih-lebihnya 3 rosak.
(7 markah)
(b) Jika purata saman yang diberikan dalam seminggu ialah 10, dapatkan
(i)
tepat4 saman dalam seminggu,
kebarangkalian
DSM 2932
(ii)
sekurang-kurangnya2dalam seminggu.
(6 markah)
(c)
Markah ujian Statistik mempunyai min 73 dan sisihan piawai 12.Dapatkan
kebarangkalian jika seorang pelajar dipilih secara rawak, markah adalah
(i)
(ii)
di antara 65 dan78,
lebih dari 75.
(7 markah)
55 (a) Diberi satu populasi terdiri daripada 7 nombor iaitu I l, 12, 13, lI,14,9 dan 7. Jika
sampel rawak sebanyak 5 diambil daripada populasi itu dengan pengembalian,
dapatkan
(i)
(ii)
(iii)
min dan sisihan piawai populasi,
min dan sisihan piawai sampel,
kebarangkalian min sampel melebihi
10.
(8 markah)
(b)
Suatu kilang elektronik menghasilkan 2 jenis perintang, jenis X dan jenis ). perintang
jenis Xmempunyai nilai purata 100 Q dan sisihan piawai l0 f). Manakala, perintang
jenis mempunyai nilai purata 99 Q dan sisihan piawai 15 C). Satu sampel rawak
bersaiz 25 bagi setiap jenis perintang telah dipilih masing-masing dari populasi
normal.
I
(i)
(ii)
(iiD
Q6 (a)
Nyatakan taburan bagi X dan 7.
Cari kebarangkalian min sampel bagi jenisXmelebihi min sampel jenis
Dapatkan kebarangkalian p(l .0 < N P< t.S).
(12 markah)
I.
-
Diberi satu sampel rawak bersaiz, n = 50 daripada satu populasi normal yang
mempunyai varians o' =225 dan min i =64,3. Dapatkan 95 % selang keyakinan
bagi min populasi.
(8 marks )
(b)
Ketebalan dinding bagi 25 botol kaca 2-liter telah diukur oleh seorang jurutera
kawalan mutu. Didapati bahawa min sampel
4.058 mm dan sisihan piawai
sampel sl
0.081 mm. Satu sampel ketebalan dinding yang lain bersaiz 17 mm
dengan min sampel y :3.912 mm dan sisihan piawai sampel s2:0.075 mm telah
dipilih. Anggap kedua-dua populasi tersebut sebagai taburan hampir normal dengan
varians yang sama.
:
(i)
(ii)
f :
Nyatakan titik anggaran bagi bezaantanmin kedua-dua ketebalan dinding.
Bina 99Yo selang keyakinan bagi beza arfiara min kedua-dua ketebalan
dinding.
(12 markah )
DSM2932
F'INAL EXAMINATION
SEMESTER / SESSION: SEM I I 201012011
COURSE NAME : ENGINEERING
: 2DFTI3DFTI
3DDT/3DET/3DEE
: DSM2932
PROGRAMME
STATISTICS
COURSE CODE
Formulae
srr:Z*,y,-Z*'?'',
,sB,
lilx,)=r,
i=-o
P(x)=(:),'
=I *' -E !')'
n
:/-FtI,
E(X)=lx7.:x), f..F{*)
Vr
dx
=1, E(X)= f_xp(x) dx, Var(X) = E(X' ) - [E(X))',
p)'-' r = 0, l, ..., fl , P(X - r) = e-,
.(l -
x - N(/,, o'), z-
,sr:l!,'- Er,)'
,p,
r =0,
rl.
N(0, r) and
t
=
+,
N
- *(r,+), z =
1, ...
-+#-
,
o,
N(0, r),
r =-ffi,
o?,o|\
- - -,(
Xr-Xr-Nlpr-pz,---t,
ur nz)
\
(7,
-7r)- zat2 -14-,.4. p, - Fz < (7t -ir)*
Vq
(i,
(7,
-ir)-
tr":
zo12 l:+-
n2
1r,
t
n2
. p, - trz < (rr -rr)+ r,,r,lt*t
-ir)- r",r,EE
n2
n2
l\
torz,, ^S" EJ
l\
<
1r, n2
ltr
-
Fz <
(it -ir)+
tal2, v
S" EJ
,, _(nr-l)sl +(nz--l)sl and v =nt*fl2_2,
nr+nr-2
1r,
where
nz
DSM 2932
FINAL EXAMINATION
SEMESTER / SESSION: SEM
I l20l0l20Il
COURSE NAME : ENGINEERING
STATISTICS
PROGRAMME
2DFTI3DFTI
3DDTI3DET/3DEE
COURSE CODE
DSM2932
(ir-L)-torz," .,8(r,'*rr1 . pr- pz < (i,-rr)+
Yn"
where degree of freedom,
(r,
v:2(n
tat2,v
-l)
t',-
-ir)- torz,,,ltlnl *€n2 . lt t - ttz < (x-, - ir)*
where degree of freedom,
W
Gi 1",
v:
nt-l
+
|("l
to12,,
t',
-Ir)-
ta/2,
-ir)+
v
ta/2, v
)
,si . s;
r,-i
!
l"rf
nz
-l
j6l .";)
(7t
.";)
"/{.,'4
\nt n2
and,
and v
= Gll"'*tllnrl
Gi l,,Y *G; l",l
nr-l
7=
(t, - F)-- (ttt- tt), 7 = (x,
loi , "i
1,1
n,
nz
t'rE+
)
, = (v, - 7i
-
Q=,
tl
.
./+
\l n' n2
- p,)
where
u
tl
= ,9? l"r\* l",l ,,
Gi l,,l *G: l,,Y
nr-l
nr-l
=2(n-l),
l=ftt+nr-2,
nz-l