I.INIVERSITI SAINS MALAYSIA
Peperiksaan Kursus Semasa Cuti Panjang Academic Session 200812009
Jun 2009
JIM 104 - Introduction To Statistics [Pengantar StatistikJ
Duration: 3 hours
lMasa:
3jaml
Please ensure that this examination paper contains THIRTY TWO printed pages before you begin the examination.
Answer ALL questions. You may answer either in Bahasa Malaysia or in English.
Read the instructions carefully before answering.
Each question is worth 100 marks.
fSila pastikan bahawa kertas peperiluaan ini mengandungi TIGA PULUH DUA muka surat yang bercetak sebelum anda memulakan peperilcsaan ini.
Jawab SEMUA soalan. Anda dibenarkan menjawab sama ada dalam Bahasa Malaysia atau Bahasa Inggeris.
Baca arahan dengan teliti sebelum anda menjawab soalan.
Setiap soalan diperuntukkan 100 markah.J
L
at
[JrM 104]
(a) The weights of 40 male students at a local university are recorded to the nearest pound and are given below:
138 t64 150 r32 t44 r2s 149 1s7
146 158 140 147 1,36 148 152 t44
168 t26 138 t76 163 119 r54 16s
146 173 142 147 135 153 140 135
I6T t45 135 t42 t50 t56 t45 128
(i) Construct a frequency distribution and a histogram for the above data using the interval 118 - 126 as first class.
(ii) Construct an ogive based on the frequency distribution obtained in (i).
(iiD Find the percentage of students that weigh more than 170 pounds based on the constructed ogive.
(iv) Find the mean, median and mode for the grouped data.
(60 marks)
(b) A student received a grade of 84 on a final examination in mathematics for which the mean grade was 76 and the standard deviation was 10. On the final examination in physics, for which the mean grade was 82 and the standard deviation was 16, she received a grade of 90. In which subject was her relative standing higher?
(20 marks)
(c) The average cost of a certain type of fertilizer is RM4.00 per package. The standard deviation is RM0.10. Using chebyshev's theorem, find the minimum percentage of data values that will fall in the range of RM3.82 to RM4.18.
(20 marks) a
1.
2
2.
-3- [JrM 104]
(a) Box I contains 3 red and2 blue marbles while Box II contains 2 rcd. and 8
blue marbles. A fair coin is tossed. If the coin tums up heads, a marble is chosen from Box I; if it turns up tails, a marble is chosen from Box IL
(i) Draw a tree diagram for the above experiment.
(ii) List the sample space.
(iii) Find the probability that ared marble is chosen.
(40 marks)
(b) Four different mathematics books, six different physics books, and two different chemistry books are to be arranged on a shelf. How many different arangements are possible if
(i) the books in each particular subject must all stand together,
(iD only the mathematics books must stand together?
(40 marks)
(c) If at least one child in a family with 2 children is a boy, what is the probability that both are bovs?
(20 marks)
(a) The probabilities that a text book page will have 0, 1,2, or 3 typographical elrors arc 0.79, 0.12, 0.07, and 0.02, respectively. If eight pages are randomly selected, find the probability that four will contain no errors, two will contain 1 error, one will contain 2 errors, and one will contain 3 errors.
(30 marks)
(b) Ninety percent of all people between the ages of 30 and 50 drive a car. Given a sample of 20 in that age group, find the probabilities of
(i) Exactly 20 drive acar.
(ii) At least 15 drive a car.
(iiD At most 15 drive a car.
(35 marks)
...4t_
3.
3
-4-
(c) The average credit card debt for college seniors
normally distributed with a standard deviation
probability that a senior owes
lJrM 1041
is RM3262. If debt is of RM1100, find the
at least RMl000,
more than RM4000,
between RM3000 and RM4000.
(35 marks)
4. (a) The mean weight of 15 year old males is 142 pounds and the standard deviation is 12.3 pounds. If a sample of thirty-six 15 year old males is selected, find the probability that the mean of the sample will be greater than 144.5 pounds. Assume the variable is normally distributed. Based on your answer, would you consider the group overweight?
(30 marks)
(b) Women comprise 83.3 % of all elementary school teachers. In a random sample of 300 elementary school teachers, what is the probability that more than 50 are men?
(30 marks)
(c) A study of 40 English professors showed that they spent, on average, 12.6 minutes correcting a student's assignment.
(i) Find a point estimate of the mean.
(ii) Find the 90 % confidence interval of the mean time for all assignment when o :2.5 minutes.
(iii) If a professor stated that he spent, on average, 30 minutes correcting an assignment, is he/she your typical english professor?
(40 marks)
(i) (ii) (iii)
4
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5.
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(a) Find 95 Yo conftdence interval for the variance and standard deviation respectively for the time it takes a customer to place a telephone order with a
large catalogue company if a sample of 23 telephone orders has a standard deviation of 3.8 minutes. Assume the variable is normallv distributed.
(30 marks)
(b) A manager states that in his factory, the average number of days per year missed by the employees due to illness is less than the national average of 10.
The following data show the number of days missed by 40 employees last year. Is there sufficient evidence to believe the manager's statement at a. :
0.0s?
Use the P-value method.
(40 marks)
(c) Two groups of students are given a problem-solving test, and the results are compared. Find a 90 Yo confidence interval of the true difference in means.
Mathematics majors Computer Science majors
rr = 83.6 sr = 4.3 nr =36
Vz =79'2
sz = 3.8
nz=36
(30 marks)
5
0 6 12
aJ a
J ) 4
1a 9 6 0 7 6 a f 4
7 4 7 I 0 8 I2 a J
2 5 t0 15 J a 2 5
a J 11 8 2 2 4 I 9
...6t-
t. I
-6- [JrM 104]
(a) Berat badan 40 pelajar lelaki di sebuah universiti tempatan direkodkan pada paun terdekat dan diberikon seperti berilcttt:
138 r64 150 132 144 125 149 157
146 Is8 140 r47 r36 148 152 144
r68 126 138 176 163 119 154 165
146 173 142 147 135 153 140 t35
I6I 145 135 142 150 156 t45 128
(i) Binakan taburan frekuensi dan histogram untuk data di atas dengan menggunaknn selang kelas pertama I 18 - 126.
(ii) Binakan ogifnya berdasarkan taburan frekuensi yang diperolehi daripada bahagian (i).
(iii) Dapatkan peratus pelajar yang mempunyai berat badannya melebihi 170 paun berdasarkan ogifyang telah dibina.
(iv) Dapatkan min, median dan mod bagi data terkumpul.
(60 markah)
Seorang pelajar memperolehi gred 84 untuk peperilcsaan akhir dalam matematikyang mana min grednya ialah 76 dan sisihan piawainya 10. Untuk peperilrsaan akhir dalam Jizik yang mane min grednya ialah 82 dan sisihan piawainya 16, pelajar tersebut memperolehi gred 90. Dalam subjek manakah
keduduknn relatifnya adalah lebih tinggi?
(20 markah)
Min kos sejenis baja tertentu ialah kM4.00 untuk satu paket. Sisihan piawai kosnya ialah RM0.10. Dengan menggunakan teorem Chebyshev, dapatkan peratus minimum datayang akan berada dalam julat RM3.82 ke RM4.I8.
(20 markah) (b)
(c)
6
.,.7 t-
-7 - [JrM 104]
2. (a) Kotak I mengandungi 3 biji guli merah dan 2 biji guli biru manakala Kotak II
mengandungi 2 biji guli merah dan 8 biji guli biru. Sekeping syiltng adil dilambungknn. Jika kepala yang muncu| sebiji guli akan diambil daripada Kotak I, iika bungayang muncul, sebiji guli akan diambil daripada Kotak II.
(, Lukiskan gambarajah pohon untuk elcsperimen ini.
(i, Senaraikan ruang sampehya.
(ii, Dapatkan kebarangkalian guli merah dipilih.
(40 marks)
(b) Empat naskah bulan matematik yang berlainan, enom naskah buku /izik yang berlainan dan dua naskah buku kimia yang berlainan disusun di atas satu rak.
Berapakah susunqn berlainanyang mungkin jika
(i) buku untuk setiap subjek mesti disusun bersama,
(it) hanya buht mqtematik sahaja mesti disusun bersama?
(40 markah)
(c) Jikn sebuah keluarga mempunyai 2 orang anak dan sekurang-kurangnya seorang ialah anak lelaki, berapakah kebarangkalian bahawa kedua-dua anak tersebut ialah anak lelaki?
(20 marknh)
3. ( a) Kebarangknlian bahawa suatu muka surat sebuah buku telcs mempunyai 0, I, 2, atau 3 kesilapan taip, masing-masing adalah 0.79, 0.12, 0.07, dan 0.02.
Jika I muka surat dipilih secaro rawah dapatkan kebarangkalian bahawa 4 daripadanya tidak mengandungi sebarang kesilapan, dua akan mengandungi
I kesilapan, satu akan mengandungi 2 kesilapan, dan satu akan mengandungi 3 kesilapan.
(30 markah) ( b) Sembilon puluh peratus daripada semua orang yang berumur di antara 30
dan 50 adalah pemandu kereta. Bagi sampel seramai 20 dalam kelompok umur ter
sebut dapatknn keb ar angkalian
(i) tepat 20 merupakan pemandu kereta,
(it) selatrang-kurangnya I5 merupakan pemandu kereta,
(iii) paling ramai l5 merupakan pemandu kereta.
(35 markah)
...8/-
rt I
-8- urM 1o4l
(c) Min hutang knd lcredit untuk pelajar kolej tahun akhir ialah RM3262. Jika hutang tersebut bertaburan secara normal dengan sisihan piawai sebesar RM I
1 00, dap atkan keb ar angkal
ian
bahavv a p el aj ar t er
sebut
(, berhutang paling sedikit RMI000,
(ii) berhutang lebih daripada RM4000,
(iii) berhutang di antara RM3000 dan RM4000.
(35 markah)
4. (a) Min sampel berat badan lelaki berusia 15 tahun adalah 142 paun dan sisihan
piawai adalah 12.3 paun. Jika sampel tiga puluh enam orang lelaki berusia
I5 tahun dipilih, dapatlan kebarangkalian bahawa min dari sampel tersebut
lebih beSar daripada 144.5 paun. Anggap pembolehubah bertaburan secora normal. Berasaskan jawapan anda, adakah kumpulan tersebut terlebih berat badan?
(30 markoh)
Peratus guru wanita di sekolah rendah ialah 83.3%. Dalam sampel rawak 300 orang guru sekolah rendah, berapa kebarangkalian bahawa lebih daripada 50 adalah guru lelaki?
(30 markah)
Suatu kajian mendapati 40 profesor Bahasa Inggeris memerluknn pada purata, 12.6 minit untuk menyemak tugasan para pelajar.
(i) Dapatkan anggaran titik untuk min.
(ii) Dapatkan 90 ok selang keyakinan untuk semakan tugasan bila
o: 2.5 minit.
(iii) Jika professor tersebut menyatalmn bahawa dia menghabisknn masa secarq purata, 30 minit untuk menyemak tugasan, adakah ia ahli Iatmpul an pr ofe
sor yang dip erihalknn ini?
(40 marlmh) (b)
(c)
B
...9t-
lJrM 1041 Dapatkan selang keyakinan 95 % untuk varians dan sisihan piawai masa yang diperlukan oleh pelanggan untuk membuat tempahan melalui telefon dengan syarikat berkatalog besar jika sampel yang dipitih terdiri daripada tempahan telefon yang mempunyai sisihan picnvai 3.8 minit. Anggap
p emb ol ehub ah
bertabur an
s e car a normal.
(30 markah) seorang pengurus mengatakan bahawa dalam kilangnya, min jumlah hari dalam setahun yang tidak dihadiri pekerja kerana sakit adalah kurang daripada min nasional I0 hari. Data berikut menunjukkon jumlah hari yang tidak dihadiri oleh 40 pekerja pada tahun lepas. Adakah terdapat cukup butai untuk mempercayoi pernyataan pengurus pada a : 0.05?
Gunakan kaedah nilai-P.
(40 marknh) Drya kelompok pelaiar diberikan ujian menyelesailmn masalah dan hasilnya dibandingkan. Ujikan samo ada terdapat perbezaan di antara min kedua-dua
major. Gunakan a:0.10.
Major Matematik Major Sains Komputer Xr = 83.6
sr = 4'3 nr =36
7z =79.2
sz = 3'8 Lz =36
(30 markah)
-9 - (a)
(b)
(c)
I
0 6 I2 3 3 5 4 I
3 9 6 0 6 3 4
7 4 n I 0 B I2 3
2 5 I0 5 I5 3 2 5
3 tI 8 2 2 4 I 9
...r0/-
i0
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Important Formulas
Chapter 3 Data Description
Mean for individual data: F = Yx L"
n
- | f 'x^
Mean for grouped data: X - L'/r
n Standard deviation for a sample:
lIx'-lft '- xY lnl
^ _.l/-/-' L\4,'^ ) l'"
-)I n-l
Standard deviation for grouped data:
Range rule of thumb: s
^, ry
4 Median for grouped data:
(nl2\-cf .
MD =\.-' -/ -r (w)+ L.
f\
where
n : sum offrequencies
c/ : cumulative frequency of class immediately preceding the median class
w : width of median class
f : frequency of median class
L.:lower boundary of median class Chapter 4 Probability of Counting Rules Addition rule 1 (mutually exclusive events):
P(A or a)= r(d)+r(a)
Addition rule2 (events not mutually exclusive):
P(A or n)= r(e)+ r(n)- p(A and B) Multiplication rule 1 (independent events):
r(.n and n)= r(d).P(B)
lJrM 1041
Zr x:,-l(>r ,-f hf
1l-
...1U-
- 11- urM 1041
Multiplication rule 2 (dependent events):
P(A and a)= r(t).r(al,a)
Conditional probability : r (Al O) =
W
Complementary events: P(E)=I- P(E)
Fundamental counting rule: Total number of outcomes of a sequence when each event has a different number of possibilities: h
.k,
.k,
. . .kn
Permutation rule: Number of permutations of n objects taking r at a time is
1t -- n!
n-'r
@-r)t
Combination rule: Number of combinations of r objects selected fromn objects is
/1 nl'
n -r
(n-r'yt'Y1
Chapter 5 Discrete Probabilify Distributions
Mean for aprobability distribution: p=ZlX P(X)]
Variance and standard deviation for a probability distribution:
o' =Zlx' .r(x)l- r'
o' = {I[ xz
.P (x)f- p'
Expectation: E(x) = I[x.P(x)]
Binomial probability : P (X\ ' = (n- , X)lXl 1-)r, ,=.. p* . qn-*
Mean for binomial distribution: p = n. p
Variance and standard deviation for the binomial distribution:
o: =n.p.Q o=inl.q
L2
...r2/-
- 12 - urM 1041
Multinomial probability:
P(x)=:*
-.p{'
.p{, .p{, ...p{r
xtlx2lx3l...xo! ''
Poissonprobabitity : r(x;r) ='-::- \'rxl whereX : 0, 1,2, ...
Hypergeometric probability: P (X) = *#:
Chapter 6 The Normal Distribution
Standardscore: r=X -F o, * -X o^c
Mean of sample means: Ft* = lt
Standard error of the mean: oy =|
{n
Central limittheorem formula, - ,=F olJi r-L
Chapter 7 Confidence Intervals and Sample Size z confidence interval for means:
X -ro,,t +l < p<E +zo,,t+l
\.,1n 1 \^ln 1
/ confidence interval for means:
v (s ) -- (")
X -torzl
- | < p<X +to,rl i
t\'Jn 1 \'ln )
Sample size for means: n =( '"''='o\' where E is the maximum error of estimate
\E )
Confidence interval for a proportion:
13
...r3t-
-i3-
Sample size for a proportion: , = pa(4)' -\ E )
where fr =: X and Q =I- i)
n
Confidence interval for variance:
(n -l\ s' , (r- 1) r'
, \v - t
Ii$t bf"n
Confidence interval for standard deviation:
urM 1041
Chapter 8 Hypothesis Testing
z testl s =:-ff X-u for any value n. If n < 30, population must be normally distributed.
ol 4n
,=+ for o unknown and n>30
,s/ vn
f test :
7=ff X-u for n <30 (d.f. : n - l)
s/ r/n
ztestforproportions:, =+
I pqln
Chi-square test for a single variance: Z"
(d.f.: n-l)
_(n-r)s' o'
14
...14t-
-t4- lJrM 1041
Chapter 9 Testing the Difference Between Two Means, Two Variances and Two Proportions
Ztest for comparing two means (independent samples);
(x,- x,\-(n- p,)
lo? o?
.t--r-.:
!', n2
Formula for the confidence interval for difference of two means (large samples):
Note:
(x'
sl
-\ tri' -\ tr t:
- N,) -'* lt * r- . t4 - ttz . (& - x,) *'r,17.7
and s2, can be used when nr>-30 and. nr>30.
a
F test for comparing two variances: F = +
S;
a ) , ..
where s,' is the larger variance and d.f.N. =\-t, d.f.D =nz-l
/ test for comparing two means (independent samples, variances not equal):
(X,-x,)-(p,- ry,)
t- t7--;
ls,- s"'
l:-+-:
\ry n,
(d.f. : the smaller or1ft1- | or n2- l)
i5
...t5t-
Formula for the confidence interval for difference of two means (small independent samples, variance unequal)
:-15-
(a.r. = r-t)
[JrM 104]
r, ,
l,s;
^s"-\ry t-T- n2
(d.f. : smaller of n1- 1 and nz - l)
t test for comparing two means (independent samples, variances equal):
(d.f.: h+ nz-2)
Formula for the confidence interval for difference of two means (small independent samples, variances equal)
:(N, - X,)+t",, (r\-t)s? +(n,-t)si
14+nr-2
and d.f. : ltL * nz-2.
/ test for comparing two means for dependent samples:
,=D ,Pp where D=ZD
soltln n ^d
X,- X,)-(r,r- p,)
(n, -r)sf +(n,-r)sl F J
rD'-[(r n)'l"f
lri
...161-
-t6- [JrM 104]
Formula for confidence interval for the mean of the difference for dependent samples:
,s_ .s
D-t-,r+
vt--''- tln < p^ <D1y^,,$ *'- tln
(d.f. = n-l)
/ test for comparing two proportions:
-(r r)
pql-+-
I\ur nz ) r _ X,+X"
where F=T b,
nL+ n2
1/
_T -Al
X,
n2
q =l-F t,
Formula for the confidence interval for the difference of two proportions:
E-;---;-;-
lr ^ \ lhQ, Pr4. /t r \
\h- Pz)- zartf?+' \,ry ry '':=- t h- Pz <lA- iir)+ z"t,
{
;.1L(
...17/-
-t7 - FrM 1041
0.9-i 0.8 0.9
0,7 0.6 0.4 0.5
0.2 0.3 0,05 0.1
The Binomial Dhtribution
0.902
().$l0 0,640
0.4900.09_5 0.180 0.320
0.4200.001
0,0t0 0.040
0,0900.857 0.729
0.5I2
0.-1430.
t35 0.343
0._384.
0.4410.007 0.017 0.096
0. r 890.00t 0.008
0.0370.8r.5 0.656 0.4t0
0.1400.t7r 0.293' 0.4t0
0.4130.0
l4 0.049
0. |5!t
0.1650.0u 0.0?6
0.0760.002
0.00t10.77+ 0.590 ' 0._12{t
0. t680.204 0.338
0.4 |0
0.3600.01 0.073 0.?05
0.3090.001 O.(xlt|
0.0_5|
0. 133()In6
3:3ii
0.73.5 0..5.11
0.2(12().33? 0..3.s4
0.3930.03| 0.098
o.u+o0.002
0.0t5
0.0820.00
|
0.0 t-50.002
0.698 0.478
0.2 t00.1-57 0.-i73
0.367 0.04|
0. |24
0.27.50.004 0.0?3
0. I I 50.00.1 '
0.0390.0040.0?6 0,008
0.(x)?0.154 0.076
0.0260.346 0.?65
0. t-540.346 0.412
0.41(l0.130 0.240
0.4 | 0 0.0t0
0.002o.lt77 0.02tt
0.0060.230
0. |32
0.05 |0.346 0.309
0.?0_50.259 0.360 .0.4t0 0.078
0.168
0.32t10.004
0.00 |0.037 0.010
0.003(1.t38 0.060
0.0t-s0.276
0. Ir{-5
0.0820.31
I 0.324
0.24(r0.187 0.303
().393t).(147 0.118
0.?(r2 0.0020.0t7
0.0040.017 0.025
0JX)40.19.{ 0.097
0.0290.290 0.?27
0. Il5
o.iot 0.3|8
0.3750.131 0.14?
0.3670.028 0.083
0.2 r0 0.0010.008
0.0010.0+t
. 0.010
0.0010.124 0.047
0.0090.332 0.l3(r
0.0.160.?79 0.154
0.1-170.109 0.196
0.29-+0.090
0.r98
0.-13(r0.017 0.058
0. 168{l.S|0
(t0n1 fr,!)(.) I0.{)17
0.0071).1+3-.-0.
t35{J.7:9
0.8.s?(1.(xt+
{
).049
0.0t{
().19t
0.1710.656 '
0.lt t5
{).(x)H
(}.0010.()7,i
0.03 t0..1tri
0.1{,+0..sq(r
0.77-l 0.()0 |(1.0t5
0.0()l0.091i
0.01I0..154 O.l.il
0.53
|
().7-'150.00.j
0.0?.1
0.{)t,J0.
124
0.{}.+ l0.372
0.1.i70.478
(r.6qto.0o-s
0.0_33
0.0050.149
0.05 |0.38-1
0.1790.4i0
0,6630.1 t8 0.303 0.324 0.1 85 0.060 0.0 t0 0.00 |
0.0tt2 0.?4'7 0.3 r8 n 141 0.09?
0.025 0.004
0.360
0.1500.480
0.5000.t60
0.1500.3r6
0.t350.43?
0.3750.188
0.3750.064
0.1350.130
0.0620.346
0.1500.346
0..3750,154
0.2500.026
0.0620.078
0.03 |0.359
0.1.s60.346
0.-31?0.230
0.3 t20.077
0. t560.0t0
0.0.110.047
0.016 0.ltr7
0.0940.31
|
0.2340.276
0.3120.t38
0.3340.037
0.0940.004
0.0 t60.028
0.0080. t3
I
0.0550.26t
0.t640.290
0.2730.r94
0.2730.077
0. 16l0.017
0,0550.002
0.0080.017
0.0010.090
0.03 |0:309
0.1090.?79
0.2 r90.232
0.373 0.t:.r
0.? r90.04r
0,t090.009
0.0310.001
0;0040.t60 0.090
0.04(l0..180 0.420
0,3100.360 0.490
0.6-100.064 0.027
0.(x)8 0.288'
0. I
tig
0.0960.4-12 0.tl4l
0.3t{.10.216 0,343
0.s 120.0t0
().()0:{). |
80
0.095t)
I
:
4 -) (r 7 S
0,(163 0.430
0.168
0.05802'19 0.383 0.3-j6
0. 1980.0.5
|
0.t49 0.294
0.2960,005 0.03.3
0.t47
0.2540.005 0.04(1
0.1360.009
0.0470,00
|
0,0100.00 |
0 .l 2
.1 n
6
1B
...18/-
-18- lJrM 1041
(continueid)
0.05 0.1 0.2 0.3 0.4
0.630
0.3870.299
0.3870.063 0.t72
0.008.
0.0450.00
r
0.0070.001
0.134
0.0400.302
0.1560.302
0.267a.176
0.2670.066 0.t72 0.017 a.w4 0.003
0.0210.004
0,002
0.018
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29
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.0001Reprinted with pcmi$ion frcm Wi H. Beyq. Itardbook oJTthlq lor Ptubsbrlirl ard Std,r'.vtct 2nd cd. Copyright CRC Ptrss. Bo@ Rlton. Fh.. 1986.
30
...30/-
