Part 1.1 Famous Statisticians Stati sticians : Pafutny Chebyshev. Chebyshev. Pafnuty Chebyshev (1821-1894) (1821-1894) is a Russian mathemacian who is well known fo Chebyshev!s "heoem# "heoem# which e$ten%s the &o&ees of nomal %istibuons to othe# non-nomal %istibuons with %istibuons with
the fomula
( ( )) 1−
1
2
k
# as lon' as the %istibuon!s %istibuon!s scoes scoes absolute value is less than o e*ual to
k and the stan%a% the stan%a% %eviaon is moe than 1+ "he ine*uality was oi'inally known as the ,ienaym-Chebyshev ,ienaym-Chebyshev ine*uality a.e lin'uist /ene-0ules ,ienaym# the autho of the oi'inal theoem+
Career Pafutny Chebyshev was bon on 1 ay 1821 in 3katovo# alu'a Re'ion# Russia an% 5ie% on 8 5ecembe 1894 in 6t Petesbu'# Russia+ 3ve the couse of his caee he &o%uce% many notable &a&es# inclu%in' &a&es on stascs# calculus# mechanics an% al'eba+ /n 1847# he was a&&ointe% to the nivesity of 6t Petesbu' a.e submin' a thesis tle% On integraon by means of logarithms + /n 18:;# he was &omote% to e$tao%inay &ofesso at 6t Petesbu'+ Pafutny Chebyshev is &eha&s the most famous Russian mathemacian an% is consi%ee% the fathe of mo%en Russian mathemacs+ mathemacs+
Contributions to Mathematics Pafutny Chebyshev is &obably most famous fo the theoem that!s name% a.e him+
"he Chebyshev ine*uality (not to be confuse% with his "heoem) which states that if > is a an%om vaiable with stan%a% %eviaon ?# then the &obability that the outcome of > is no less
than
[email protected]? away fom its mean is no moe than
Chebyshev &olynomials+
Chebyshev ,ias
1 2
a
+
de!iaon measures how concentrated the data are around the mean and the more concentrated" the smaller the standard de!iaon.
c.
#he two students with the grade $ will made a huge impact on the performance of the sub%ects in SMK &. #he standard de!iaon from '. become '. and the percentage from *+.,, drop to *+.-.
Part 2 a+
P ( SMKP )=
¿
15
2
1Z
6 \
;
Z
9
6 R
1
:
6 6
1
2
C: ¿ 80730 4 C2 × 2ZCZ ¿ 10626 Z CZ × 24C2 ¿ 276
i+ ii+ iii+
27
i+ ii+
:
¿ 120
2
×
P1
¿ 12
2
Z
2
1
1
A+
A
A
A
A+
Z
2
2
1
1
A
A+
A
A+
A
Z
PZ
1
×
iii+
¿ 3 ×2 ×2 ×1× 1 ¿ 12
Part 3 n= 27, p = i+ ii+
18 27
9 27
1
2
3
3 2
= , q=
× 100 =66
3
Standard Deviation =√ npq
¿
27 ×
1 3
×
2 3
¿ 2.449
b+
n= 9, p = 0.3 , q =0.7 P ( X = 3 )=¿ i+ ii+
A
6 P
c+
a+
A+
17 5
b+
P: P1
School
CZ (;+Z)Z (;+7) ¿ 0.2668 P ( X ≤ 2 )= P ( X =0 ) + P ( X =1 )+ P ( X = 2)
P ( X =0 )=¿
9
9
C; (;+Z); (;+7)9
¿ 0.0404
P ( X = 1)=¿ P ( X =2)=¿
9
C1 (;+Z)1 (;+7)8 9 C2 (;+Z)2 (;+7)7
¿ 0.1556 ¿ 0.2668
P ( X ≤ 2 )= P ( X =0 ) + P ( X =1 )+ P ( X = 2)
¿ 0.0404 + 0.1556 +0.2668 ¿ 0.4628
Part 4 a+
b+
X N ( 46,225 ) mean = 46 , standard devition =√ 225 z=
¿ 15
X − μ σ
z =
52−46 15
=0.4
f (/)
c+ i+
(
P ( X ≥ 52 )= P z ≥
)
52 −46 15
¿ P ( z ≥ 0.4 ) ¿ 0.3446
/
f (/)
ii+
(
P ( X < 30 )= P z <
30 −46 15
)
¿ P ( z ← 1.067) ¿ 0.1430 /
%+
P (30 ≤ X ≤ 52 )= P
(
30− 46 15
≤ z≤
52 −46 15
)
f (/)
¿ P (−1.067 ≤ z ≤ 0.4 ) ¿ 1− P ( z > 0.4 ) − P ( z > 1.067 ) ¿ 1−0.1430 −0.3446 ¿ 0.5124 /
f (/)
e+ ]et minimum scoe is m "otal stu%ents [ 49 Pobability of to& ten stu%ents [
10 469
P ( X > m )=0.0213 m− 46 =0.0213 P z >
(
15
m −46 15
00213
=0.0213
/
)
= 2.028
m=76.42
f+
(
P ( X ≥ 40 )= P z ≥
40 − 46 15
)
¿ P ( z ≥ −0.4 ) ¿ 1− P ( z ≥ 0.4 ) ¿ 1−0.3446 ¿ 0.6554 Numer of students = Proai!it" ×#ota! numer students f (/)
¿ 0.6554 × 469 ¿ 307.38
/
$ 307 n
'+ "he minimum scoe [
00"
P ( X > n ) =0.92
(
P z >
−n −46 15
n− 46 15
)=
0.92
=1.406
n= 24.91
Part 5 a+
b+
Planum # % =
Yol% # % =
6ilve # % =
,one # % =
48
65 60
× 100 = 108.33
× 100=106.67
45 40
× 100=100
40 36 35
× 100 = 102.86
092
Planum Yol% 6ilve ,one
% =
¿
/n%e$ 1;8+ZZ 1;+7 1;; 1;2+8
^ei'hta'e 2 4 Z 1
%& 21+ 42+8 Z;; 1;2+8
∑ %& ∑ &
1046.2 10
¿ 104.62 c+ e%al
Planum
,one
% =
100
100
6ilve
% =
108.33 × 110
106.67 × 100
Yol%
%+
/n%e$
100 × 95 100
% =
2
=106.67
4
=102.86
∑ %& ∑ & 2 ( 119.16 )+ 4 ( 106.67 )+ 3 ( 95 ) + 102.86 10
¿ 105.29
e+
100
=119.16
=95
102.86 × 100
' 16 ' 14
' 16 455
× 100
× 100=105.29
' 16 =479.07
^ei'hta'e
Z
1
