Regression Analysis
~::::======:::::::::::::::=:::::::::::::=::::::::;~~~====~===
tNiRODUCTION
5,1 We know that the correlation studies the relaf h' b
. hies X and y In this
h II consid th I
tons tp etween two vana
·
pter, we s a k
er e re ated problem of prediction or estimation of the value of one
ch~iable from a no~n v~l~e of other variable to which it is related. When there are_ two
viiaria
. bles X and• y and tf y is. influenced by X, i·e1
., ·r y depends on X, then we get a simple hnear
1
v"ression or s1mp _e regres.s1on equation of 'Y on X'. Here y is known as dependent variable or
re;ression or _exp lamed va~table and X is known as independent variable or predictor or explan~tor.
~~~case of a sunp le regression model if y depends on X, then the regression line of y on X is given
b(
Y = a + bX
... (A)
st
}-{ere a an~ b arc two con ~nts and their values are obtained by solving normal equatio.ns.
th
J-Jere a is e mtercept an~ b is the slope of the line. They are also known as regr~sSion
9111eters. Furthermore, b ts also known as the regression coefficient of Y on X and is also
::~oted by bP. A_nother regression line in which X depends y is : X = a + bY. Here b is the
ession coeffic1ent of 'X on y , and is denoted by b
~r
~
.
1011
Regress
!.'hows a relationship between the average values of two variables. Thus
,essiull is very helpful in estimating and predicting the average value of one variable for a
'~~ell
be made with the help of a
gt1 mlue. of the
I . other
I
I variable. Tire estimate or predictio11 mau
J
regression /me w uc,, s wws tire average value of one variable x for a given value of the other
,,ariable y. The best ave~age value of one variable associated with the given value of the other
t'ariable may also be estm,ated or predicted by means of an equation and the equation is known
as Ree.oression equatio11.
s.2 TYPES OF REGRESSIONS
Simple Regression.
The regression analysis confined to the study of only two variables at a
time 1s called the simple regression.
Multiple Regression. The regression analysis for studying more than two variables at a time is
known as multiple regression.
Linear Regression. If the regression curve is a straight line, then there is a linear regression
between the two variables under study In other words, in linear regression the relationship
between the two \'ariables X and Y is linear. In order to estimate the best average values of the
two variables, two regression equations are required and they are used separately. 011e equatio11 is
used for estimating the value of X variable for a given value of Y variable and the second
equation is used for estimating the 1•alue of Y variable for a given value of X variable. In both
cases, the assumption is that one is an independent variable a11d the other is a dependent variable
and 1•ice 1·ersa.
5.3 LINES OF REGRESSIONS
A line of regression is the line which gives the best estimate of one variable X for any
given \'a)ue of the other variable Y.
5.1
fcSJS {i,
bat aot
y) and if
p
aadb = l!..xbP
..
q
sip, Le., either botb
essioa coefflden
are neeative.
IILll[ii=-=~11a
l!!I"
,; J., jP 1111 lutH the su,e
.....
•zl./tfal/lwlfto• tJ,e rep
,.._
- - . . 111¥ ,,.,,.,&ldiu to e,td,
=.Ji.ixoM = .Jo.368 =0.606.
a. Here :t
-2+3+2+
.. 4+9+4+
re,t:~
M
1.7 IIITtt00 o, L,!AST SQUARES
,tell! Jr,/a /I /J used for obtalr,ir,g tJie
~
~
TIN MeeW el Lalt
iJ a ,nat/rlflllJ
of a cww wl,ic/s fiJJ bat tlJ a gtW'I set of oblerY"''°;;bt 1qa•rts of dlrferencet bet
'
It IS bated OD the . . .mptJoa tb•t tbt ,11111 0' the d•t• if ,ninfmum. The
0
atlrMted v•ltNt •ad the •daal oblerved value; to tbt ~ea data and consist, le
\
method h wed to ol,taJa the belt ntda& 1traf&bt :.ent variable for dependent va,~'tbe bat flttJac ,tralpt line to the values of lodt9'
bt obtained by means or I
Mathematic.ally, the bttt ntdn& ,tral&bt 1111;1:nWbtll the relatlon1hlp l~ht~'1 - .
ie-.."'"-
~
Ille.-"
•bow•
"hldl
the rdadon t,etwttll the two vari\ainin, the 1,ett valutt or the con,,1 • ~
llnnr, the method of least .quaret Is uttd for ob f the constants in the rnunatrng llta i_
appropriate equation. for ol,tauung the beSI val°:n~ in the esiunating equation, arc
1
,et of equauons, depending on the number of cons
ed for obtaiffiffK the values of cocq...,,~
I11
be 10lved Tise syim" of equat/JJffS required
be JO
" 'h1,
-~ItoUnuatlOIII•
t/fe ntu,,llling equ111/JJ11 l• knowff as,N or.. - ""f
t'on
for fitting a rtv
"'
1
The method of least ..quares 11 used in obtaining !he equa
~
~~ti,,.'-
"''k,,i ~-
5.7.1
Direct Method
E.qua11on of a wa1ght line 1s : J' • 11 + bx
ts a and b arc:
Nor,,,al Equations for obtairung the values of the con,uin
{/) l: y • Na + b!. x and
aix + b!.x1.
x • Independent variable,
where
y • Dependent variable,
I• Sign ,,f summati""•
a • Con,tant,
(11) 1:xy •
b - Conitsnt,Of
.
Of Ob
N • Number pairs
·r he equation of the beil fit fs the ,tralght line : Y • a + bx,
where the v:.luc:1 l>f a and h are obtained by ,,,lving the norm.al equatlom.
•ervaurJn,
•
The method Ii illustrated by the following er.ample.
t..rample 5. Pit a Jtra1ghl fine of Yon X from the following data
--X
L-_r.:..:_
O
I
2
3
4
5
-;,--.._.
3
1
_:2 _ _ _:_1 _ __::.3:___ _ _
2_ _ _4_ _ _ __ _ _
-.::."?---
--........
Solutfon. The regrtisi"n line of Yon X:
Y•a+bX.
The two normal equations to evaluate a and b are •
I Y• aN+hIX
I:XY aT,X + bI:)(l.
y
X
2
I
3
2
4
3
5
()
I
2
3
4
5
6
.EX• 21,
I: Y
=20,
xi
0
CJ J
v.... -11
... ,
(n'.
1
rhe equation of regression line Oxonyls:
f
x-x = b (y
"'
or
X
-
- Y)
CJx
-
-x = r-(yCJ
y;;-\,.
,Ex:arnple 6. Calculate the re-e,s
,,. · ton 'coefficients fi
=r
. I: X 50, I: Y = 30, I: XY = I 000, I: X !
the fol~ng informatum:
5 1ut10n. Regression coefficient or y
.
.
3000,
on X 15 given by : t Y • = /80• N. JO•
0
Table : Computation of Fitting a Straight Line
r
h~"' ---.!.::_
ro
t JC)'-11i;
5irnilarlY,
XY
()
I
I
4
9
16
25
36
I: X2 =91
6
b,x= ,CJ'=I){Y-NXY
CJx r,xz-N(X)z
6
Here
16
15
30
IXY
X
=
tx
_50
. .
N -io=5,
Subst1tutmg these Values m (I), we get ;
74.
- l:Y 30
Y = N=10 =3.
b = 1000-10(5){3) 1000-150
yx
3000-10(5)2 = 3000-250 -0.309.
... (I)
iS
~ ~ T e s t QuanMative
s;s
~
. tiJ)g (iii) in (,J, we get: 3x + 2 (3l
3x - l2x = 26
-6:r)•~
So"5ptil
. o coeflideot or X on Y JS, given by •
R,cresao
-y- 1000-10(5)(3) 1000 - 150 -0497
IXY-YX _.:...:.----:--=--..
bx,= IY1-N(Y)z - 1800-l0(3)2
1800-90
.
. ll d
_,1 record the following data are ava,lable :
Eumple 7. In a parM y estroyeu
sr
1111'5·
~
.
( )
Sol Ution. 0
1,1"
•
x and y
lllcd ~
•
7
= 22
.•. OJ
and
64 x-45y = 24.
Multipl}iog the equation (1) by 45, we get :
225x-45y = 990.
Putting
x = 6 in ( l ), we get :
Hence
x = 6 and y = 8.
30- y
.•• (2)
... (lJ
= 966 ⇒ x =6.
Subrractmg (2) from (3), we get: 161.i
tJJe p-0iot of mtenection of{')....
= 22 => y = 8.
p
2
.
A gam
.
.
regression equation of x on y
⇒
.
lS :
8 64
y = --+-x.
15 45
Jte'l'"red correlation coefficient is given by:
~(
' · t~b~z, =
=
, .. b ii
th
le 9. for I 00 students of a class the • ~ aep e, IO' wOI also buepdvel
is:aJJ]p
. regresm,n equatio .,
ks ,n commerce ( Y) is 3Y- 5X + 180 = o. Th
. n o, maria in statistics (X) on
# ,nark5 in statistics is (4/9Jth of the variance of em;::. maria ,n commerce iJ 50 and varuuu:e
of,,,a~ and the coefficient of correlation between rna L ~ commerce. Find the mean maria in
siansr1cs
marll.J m the two subjects.
solution, Given 3y - 5x + 180 = O or 3y + 180 = 5r
L,et •x' represents ~rlcs m Statistics and 'y' represents marks in Conmerce.
W}!en y = 50, x will be given by 5x = 3 (50) + 180
22 1
Sx - y = 22 or x = 5 + Y.
5
Calculation of Coefficient of Correlation: b
b
x,
x,
== -
s·
·. bxy = 0.6 ;
.!.
.
15
::::)
r
But we know that :
b
TX
=
is'
=
r CJ Y
(J
=>
64
45
8 CJ y
=iiX5
::::)
(Jy
40
=)= 13.JJ
Hence. the standard de,iation of y : l3.3J.
E~ample 8. Given rhe regression lines as 3x + 2y = 26 and 6x + y = 31.
x andFind
y. their point of intersection and interpret 11. · Also find the correlation coefficient between
Solution. 3x
+
2y = 26
... (z)
6.r+y=31
... (it)
From (ii), y = 31 _ 6x
... (iir)
330 :66,
5
ax
a,
r~ = r
o,
~ (given). Thus 0.6
~9
=
r [~)
~9
⇒ 0.6 = r ( ½) ⇒ 2r = 1.8 or r = + 0.9.
= 5•
64
byx =45 '
8
Also
(JX
= ,
-½·
Regression coefficient of x on y from the given equation is:
5X = Jy + 180 ⇒ X = 0.6y + 36
(+ve sign with r is taken as both the regression coefficients bx, and by,: are positive.)
(c) ~ow it is given that vanance of x = o 2X = 25
X=
Hence, the mean marks in Statistics are 66.
8
Hence, the coefficient of correlation r =
±vl-¾Jl-¼)
5x = 150 + 180 or
:;::,
64
b)'X = -45 •
r = ±Jbx, xbyx = ±)~ x~ = 15 .
But
. ,, .... .,u.,
6
(b) The regression equation of y on x is :
64
24
64x- 45y = 24 or y = - x - 45
45
•
(U) II ..,_ ltJ (4, "'
lie on the regression lines and are obta·
soh·ing the given regressmn equaoons.
5i-y
k
tbe
fbUS•
The mean nlues
-62-9.t.
regression line of y on x be • h
'
+Z1•1'Le., l•U - -x
3
2
at of x ODY be : 6x +y = 31 ; e .. 31 l
aodt..
. ., .. = -6- - ,
b
3
1
6
b :::: _ - and bx,= - -
1tl
Variance of x = 25.
· .r n y·
5x - y = 22.
Regression equa11on o, x o ·
· .r on ...
64x - 45y = 24.
Regression equa11on o, Y .,..
.
Find (a) mean i·alues of x and y; (b) coefficient of correlation between x and y.
(c) Standard deviation ofy.
. ..
~. tiJ)gx:::::4m(11,J,weget: 1 • 31
~Le~~•4;.
.1,dJ(l.l
•
-24•7. 7
Example 10. The equations of the regression lines between two variables are exprened as
lX - 3Y = 0 and 4 Y - 5X - 8 = 0. Find X and Y. the regression coefficients and the correlalion
coefficient between X and Y.
Solution. The two regression equations are :
2X - 3Y = 0.
-CJ>
and
4 Y - 5X - 8 = 0.
- (M)
The pomt ( X, Y) hes at the intersection of both the lines of regression equabGIII, ~ lwo regression equations.
(1) x 4 ⇒ 8X-' 12Y = 0
(i1) x 3 ⇒ - l 5X + I 2 Y - 24 = 0
common proflclenc
sis
Test: Quantitative
assume that the line •et Us
~ ...
1Oy "' Bx + 66 ==>
l,\' 3>' . ().
Also
-
~r - o
:2-4 )
... , '
,AlS'' f
(---7 -...
40x
= 18y
+ 214
,knowthatr =
l)llf \\IC
n'
(yos
If tl:t• \\iriulll'<' <!/) ,:,- / 5, Jind tht• stand,ml dt'1·iat1011 ofx.
Solution. Stllnng the given equations, we get x = 3 and .I' = I.
• ,.
~ ~,,,,., ,
·
1
--
=-
ftf fC sign is taken because both b
2
'i
•
7
7
.-•
,Jl',,x '
-5
= - -,
7
IO
2
3y+ 3 : therefore, bx,,= - 6.
[ ·: bf). 1s negative, so r 1s negative)
=- {w =- /5.
{ii fii
Also ox+ 2y = 2y = 20 => .r = -
'\o" r=- - ~. xb09
J
-I (Hi)-J
I/¥'
. CJ ~ 'is b = r-'-=>-=lcr,
~ = ✓7 = 2,646
Agam
xcr.r
- => crx = - - x vl5
' v D, w
CJ."
3
21
CJr
3 5
Hence the standard de,i ation of x is 2.646.
Example 12. Given that the i:iriance of x = 9 and the regression equations are
8t- /Oy + 66 = O. 40.i:- 18y = 214 Find (a} mean \'Glues ofx and y; (b) coefficient ofcorrelation
berween x and _i, (c) standard del'iatio11 of y.
Solution. (a) The mean values of x and y, i.e., (x, y) he on both the lines
8x-lOy = - 66.
•.. (I)
40x- I8y = 214.
Multiplying (I) by 5 and subtracting it from (2), we get ;
32y =544 or y =17.
Substituting y = 17 in (I), we get: x = 13.
Hence, the mean values of x and y are
5 20
Jr/
-9
••
-6
(i
100=+-.
10
and 6~ are ,-..uye.)
-.1..a...
x = 13, y = 11.
•.. (2)
(Given)
~wx-=>a,. =4.
_
owing data:
X = 20, y = I5, cr = .,
·
·
tain the regression
equations
and find th·' " • a>• - 3• r = 0•7·
.
Ob
.
e most like/11 val 0.r Y. I,
solution, The regression coefficients are :
:J
ue , • w en X = 24.
£%11
a,.
.r .. 1.
,~
)',t
l
cr>' 4 6
b"' = r-=>-a,.
AlsO
.
cr.r 5
3
e the standard deviation ofy Is 4
J-fcllC
,
mple J3. Two variables gave thefioll .
3_ 2.1
- = o.s2s and b = CJ,
4
4
IC1 r -=0.7x-=0933
:,:
3 ' '
0x
The regression equation of X on y: x _ X = b (Y -Y)
::::>
X - 20 = 0.933 (Y _ 15)
,,,
933
or
_
~ = o. y - 0.933 x 15 + 20 or X = 0.933Y + 6.
JJence the regression equation X on y is : x = 0_933 y + ,
6
b,.r.
v = .::. _.:... \, Therefore, b1.•
48 ==>t-'
cr.t = 9 ::::> cr.t = 3•
1 .S S ~, r .. o71 (r is posltiw since b1.1 und b . are po .
' •
·
•'ii
Sttj ,
..• r : • hf l ' /I ,, • _:,..
J ':'
:, "' -t.5 'e)
'-'
I 11 F' I tit . ,11l' lll rnl11t•1· of two random mriablt's x and y and the Correl
~.:ump,, . mi <
•
• •
.
'
.• , b ,.
a11
<<'<1fkicnt h<'fWf<'JI them wht·n the• nm /inl'.1· of n-gt't.'sS1011 ,m: gn t" J.
o~
.fa + iy _ 2: = O,md 6x + 2y - 20 ~ 0.
•·'I
IV
21
x~-.H-!
J;;;;--;;;; =!R;9
-><-::
.$
5
Hence .\' • J 11nd
==>
ny Is
18
40
7
• •20 11 I }"';" • - 2,28(1,
Tims .\: • - -'•"
I
10 +
, .. - , t
t,c regression equation or x O
l'•- ~ --- 2.ltl(I.
Q
'f
::=
r -
(1
= 07
. X4
The regression equation of Yon X: y _
~
y = byx
(X -
X).
Y - 15 = 0.525 (X - 20)
or
Y = 0.525X + 15 - 20
When X = 24, the most likely value of y is :
x
0.525 or Y =0.525 X + 4.5,
f = 0.525 X 24 + 4.5 = 17,1,
Example 14. From the data given below estimate the most likely height of a brother whose
sisters height is 70 cm
Brother's · mean height is 67 cm with a s.d. of 3.5 cm
Sister's. mean he1glu is 65 cm with a s.d. of 2.5 cm
The coefficient of correlation between the heights of brothers and sisters is + 0.8.
Solution. Let x, y be the variables corresponding to the heights of sisters and brothers.
:. Then
x = 65,
ji = 67, cr, = 2.5, crr = 3.5, r
=
0.8.
rcr,. (0.8) (3.5) I.l2.
b,.x= - - 25
CJX
,
The equation of line of regression of y on x ls:
Now
y-y
= byx(x - x)
⇒ y- 67 = 1.12(x-65) ⇒ y=l.12r-U
,. ll)i
,,\,,\ \' '''lllP\,.,r-,,
l \
t 1.1 l
'" (l)
''
l•>\tt)IN
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's
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t I IJ /(I (1/> \1 ' \•,.1: ,.,,o.~I'
, ''" /'/ll'i l 1) 1111c/
l"\llllll'k I>, r ,
c,/>t,11111,/
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o;,,.,,,
, , •11,1 /t' 111111')
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h l
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th,· ,,,,,,.
,,w1d
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" ' • • , " hr 11· c,
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I 1 , •1
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~01111 ''"· 1II' l\'l'll'SSl\111
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S1,lvi11~ (ll
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II II I 1.01 -, , .
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1'1<:111111111
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,
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(:'/\'I'll
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Sc'l'/1'1
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'II 11h,
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,
I \( )11
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u
I' 11·/, 1•11 I
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289
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, jl)
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~
64
!'i
l'/0
~
25
170
56
25
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..
(I
ltiO
I~()
"
I
•I
170
(;
l~
,,,,
17~
172
5
0
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1/1
m
0
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11
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l(t•.
I'
Ill
()
m
,.
r
.,
1/ll
I I~
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,/\:
1(, /
I
7, (,
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...
;~ 17(}
11, ~
•~. " ·
1/11 1'11/11 1•
I ',l//1/11//1
I'
\
\ \(/(l,
''.
(1
• . ' •I /\',\:'''"' • 1 Pf
"·'
1/w /pl/p11·111,1: ""'" 11~•1r
"'1'1' 11 { l'l,
ll
};
,,,, ..
~2(,
t I'•
16110
225
25
25
0
9
0
~
ttly•
60
4'$
0
JO
0
al
ollcloncv Tosi: Quon . n_ vo -
P
c~.J-_;J..-_
9,12
-------
j( . 10
•
r.dwli• (l:1fr!_J~1/y)/N
hll
·
- -r.,1l
p;~d IN
.
u1:~1•
h,.1
~ 168,8; Y
IX
X
lkre N 10,
( 12)x( 60)__/lo_
~ 09
- ----- - ~ - - O
806 ( 60) 11
,
409 72
i2)2
526 (-
-(t1M(r.,1yl1-_!t'_ -
t,/x1 (f.1L\·)2/N
°
·
v
}:)'
N
~11,1~
1.13
1690
s • ~(X.:..x
)z
-10--169,
XY
409 72
337
806 360 - 446 "'0,756,
337
s2614.4
no
'P\!!~Ue
St
337
siT6
.,,,c,e "
r
0
,659,
~~
-
,, n h (Y-Y)
'J'
Rcgrrsslon ~:quntlon f X on Y ' ,\ - ,\
)
X 168 8 0.756 (Y 169
or
)
75 6)' r 168 127.764 ⇒ X= 0.756 }' +4
or ,\' 0 756Y + 168 (0. 756 ) (169 0·
_
°'236
r.
y ji ,.. b _(X X)
Regression •:quation of yon , . J'.1
y I69 0.659 (X 168.8)
y o. 65 9x + 169 - (0.659) ( 168.8) or Y 0.659X + .
57 76
:henX "' 164, y 0.659 x 164 + 57.761 108.076 + 57.761 or Y= 165.837.
I
I
510 VARIABILITY IN REGRESSION ANALYSIS
. Let X and y denote the estimated (or computed) val~es. of X and y obtained from lhc
c . c Then we know that the sum of the deviations of Y's and X's from th elr
regression equations.
regression lines are zero.
Therefore
l: (Y- Ye)'= Oand l: (X - Xe)= O
Variability: Variability is the measure of spread or scatter of actual values around th,
computed values Ye. It 1s given by
(Y-Yc)2
Variability - - - 11
. . . _ (X-Xc)
2
or Variab1hty - - - - II
STANDARD ERROR OF ESTIMATE
Tire standard error of estimate is similar to sta11dard deviation. The sta11dard error of
estimate is a measure of the variatio11 or scatteredness about the li11e of regression, whereai
sta11dard deviatio11 measures the variatio11 or scattered11ess about the arithmetic mean.
Standard error of Estimate indicates how precise the prediction of Y is based on Xor
conversely. There are two types of standard error of estimates.
(a} Standard Error of Estimate Yon X. It is denoted by Srx· It ts denved from the formula
5.11
1:(Y-Yc}2
2
1:Y -al:Y-bIXY
or S = . , - - - - - - YX
n-2
rx
(11-2)
where a and b are to be obtained from the normal equations and
a = intercept, b = slope of line.
S
Also
=
Srx=
cry ✓t-r 2 ,
where r is the coefficient of correlation X and Y; Srx predicts the values of Y based on X.
(b) Standard Error of Estimate of X on Y. It 1s denoted by S .. It predicts the values ofX
based on Y
n
~
11-2
sXY .. 0 XVr--:11-. rl2
O
= (_
s • Jtx 2 -otx-btxY
or
b w b< rhcevaluatcd Iron, then
lip d
or111a1 equatJ
"
)
•
" 2
ons ao d 11 • Intercept, b • slope of line.
,,AIS tiIe coefficient of correlation betw '
r 1s
een X and Y.
~
.
e)(PLAINED A.ND U~EXPLAINEo VARIATIONS
6'1Z c.,pta jpcd Variation. It ts the variation wh·tc h 1s. exp! ·
db
v". giVCP by 1: (Y.C - f)2 •
aine Ythe variable X.
JI JS
• •
•
lained Variations. It 1s the variati·o h.
V11e%P
n w tch is u
I·
.
•
_ due to some other factors (variables) tli .
nexp atned by the variable X. This
0n 1s
·
a
ecttng
the
t
ta!
·
•
.
v,riau is .
b !: ( y _ y )2.
o vanatton m Y-values.
c
_
11 given Y
2
Total Variation: 1:(Y -Y) = r [(Y -Y. ) (Y. - 2
J,JoW
C + c-Y)]
== !: (Y-Yc)2 +t (Yc-Y)2 +2L(Y-Y. )(Y.,-Y)
== 1:(Y-Yc)2+1:(Yc-Yl
c c
== Un-Explained Variation + Explai· d "
. .
ne .-ar1atton.
.
"' .
Thus, Explamed .-anance =
1:(Yc-Y) 2
n
COEFFICIENT OF DETERMINATION
; Unexplained Variance= 1: (Y -Ye)
n
2
513
· TIie coefficie~,t of determi1tatio~ is th e sq11are of the coefficient of correlation. It is equal to
1
• t'ton 111
• y •
l · db~
r or r,J· The maximum
. value of r2 ts unity and in that case all th e varta
1s exp ame
the variation in X It is also defined as :
. .
Explained Variance
Coefficient of Determmatton
= --:::-----Total Variance ·
.
D
.
.
Unexplained
Variance
Coefficient of Non- eternunation = ::------Total
Variance
It IS denoted by k2• Also k1 = 1 - r
Coefficient of Allienation. It is the square root of coefficient of non-determination. It is
denoted by k.
5.14 UTILITY OF REGRESSION ANALYSIS
1. TIie cause and effect relations are indicated from the study of regression analysis.
2. It establishes the rate of change in one variable in terms of the changes in another
variable.
3. It is usefttl i11 economic a11alysis as regression equation can determine an increase in
the cost of living index for a particular increase in general price level.
4. It helps in prediction and thus it can estimate the values of unknown quantities.
5. It helps i11 determining the coefficient of correlation as: r = ✓b,x x bxy •
6. It enable us to study the nature of relationship between the variables.
7. It ca11 be usefttl to all natural, social and physical sciences, where the data are in
fu11ctional relationship.
Anal : :.:.si~s_ _ _ _ __
·c,·ency Test: Quantitative Aptitude
Common Pro f1
5.14
,e~
botb (A) and (B).
(C) None of these.
MULTIPLE CHOICE QUESTIONS
·
Write the correct answer out ofthe given
o,,es·.
(A) measuring the extent of association
between two variables.
(B) establishing
a
mathematical
relationship between two vanables.
(C) predicting the value of . the
dependent variable for a . given
value of the independent vanable.
(D) Both (A) and (C).
2. If there are two variables x and Y, then
the number of regression equations
could be:
(A) 2
(B) I
(C) Any number (D) 3.
3. The method applied for deriving the
regression equations is known as :
(A) C"ncurrent deviation
(B) Least squares
(C) Product moment
(D) Normal equation.
4. What are the limits of the two regression
coefficients ?
(A) Must be positive.
(B) No limit.
(C) One positive and the other negative.
(D) Both positive or both negative.
5. Two regression lines coincide when :
~)r=O
~)r=2
(C) r =+I
6. The two lines of
identical when :
(A) r = 1
C r=O
(D) None of these.
regression become
(B) r = - I
(D) (A) or (B).
- . <lit' difference between the observed
~=
~=
the estimated value in
JS known as :
(BJ error
:!:'f:!$SJD.!: ~-sis
s.
In the line Y = 19 - (5/2) X, b i
~(A) 5/2
J:1
s equ
~
(B) 15/2
{C) - 5/2 .
(D) None of the
9. In the equat10n X = 35/8 - (2/5) y se.
,b
equa I to :
l}IS
(A) - 2/5
(B) 2/5
(C) 7/12
(D) 5/2.
to. The line X = 31/6 + Y/6 is the reg
.
ress1
equation of :
on
(A) Y on X
(B) X on y
(C) both
(D) None ofth
. 1·
ese
I 1. The regress10n mes are perpendicul ·
each other if r is equal to :
ar 10
(A) 0
(C) - 1
.. . .
fbe rntn1nusat1on of horizonta~
(13) distances in the scatter diagram,
EXERCISE - 5.1
J. Regression analysis is concerned with
10J1
(B) + 1
(D) I.
12. The li~e Yf= 13 -(3/2) Xis the regression
equation o :
(A) Y on X
(B) X on y
(C) both
(D) None of these.
13. The errors in case of regression
equations are :
(A) positive
(B) negative
(C) zero
(D) All these.
14. The line Y = a + bX represents the
regression equation of :
(A) Y on X
(B) X on Y
(C) both
(D) None of these.
15. Two regression lines always intersect at
the means.
(A) True
(B) False
(C) both (A) and (B)
(D) None of these.
I 6. The line X = a + bY represents the
regression equation of :
(A) Y on X
(B) X on Y
(C) both (A) and (B)
(D) None of these.
17. The regression line of y on x is derived by
(A) the rrururrusation of vertical
distances in the diagram.
.,
. 1'be slope of the
y is :
resreaaion line of x on
(A) b
(C) 1;
(B) b,.,
(O) . ear equations Y = a + bX and
27,
If
b
·'>·
(D)
111
lib,....
Jll 1 ' bY 'b' is the:
Yt and b are
.
J8• .,, a+
A)
·
.
.
·'>·
negative,
then r is :
(
)(,.. . tercept of the line.
Positive
(B)
.
(A) in
(C) zero
negative
pe of the line.
1
28 F h
(D) None of these.
0
(J3) ~:th (A) and (B)
· xr I e regression equation of Y on X
2 +3y.
(C)
'
A
.,. SO = 0. The value of b is :
None of these.
(
(D)
.
) 213
{B) - 2/3
y.,
fbe Jine of regr~ss1on passes through
(C) - 3/2
19· t1te points, bea~mg ......... number of
(D) None of these.
29 s·
·
mce
Blood
Pressure
.
ts
of a person
on
both
sides
:
i
potJl
dep.:o<ls
on
age,
we
need
consider
:
(Al equal
(B) unequal
(A) th e regression equation of Blood
C) zero
(D) None of these
Pressure on age.
~be slopes of the regression line of y on
th
(B) e regression equation of age on
zo.
Xis:
Blood Pressure.
(A) b,y
(B) by.,
(C) Both (A) and (B)
(C) b.«
(D) byy.
(D) Either (A) or (B).
36· A small value of r indicates only a .......
The equations Y = a + bX and
ZJ. X== a + bY are based on the method of
hnear type of relationship between the
variables.
(A) greatest squares
(A) good
(B) poor
(B) least squares
(C) maximum
(D) highest.
(C) both (A) and (B)
31. Feature of Least Square regression lines
(D) none of these.
is that the sum of the deviations at the
22 . brr is called regression coefficient of :
Y's or the X's from their regression
(A) x on y
(B) y on x
lines are zero.
(C) both (A) and (B)
(A) True
(B) False
(D) None of these.
(C) both (A) and (B)
23, In linear equations Y = a + bX and
(D) None of these.
X =a+ bY, 'a' is :
32. The square of coefficient of correlation
'r' is called the coefficient of :
(A) intercept of the line.
(A) determination (B) regression
(8) slope.
(C) both (A) and (B)
(C) both (A) and (B)
(D)
None of these.
(D) None of these.
33.
The
coefficient of determination is
24. The regression coefficients are zero if r
defined
by the formula :
is equal to :
explained variance
2
(A) 2
(B) - 1
(A) r
total variance
(C) 1
(D) 0.
unexplained variance
2
25, bxy is called regression coefficient of :
(B) r =I
total variance
(A} x on y
(B) y on x
(C) both (A) and (8)
(C) both (A) and (B)
(D) None of these.
(D) None of these.
I
1
2'
(~
'CC) bodi(A)and
(D) Ndntof
42. If llopa of two
equal. tbea rfl
(A) I
(C,
ANSWERS
5. c.
... o
,a. A.
20.8.
28.8.
3'. A
13. o.
21. 8
29. A.
37, A.
0
6. D.
14. A.
22, B.
30, B.
38. B.
7. D
15. A
23. A
31, A
39. C
ient1 b..,
llbxr • hbuv
b
IXERCISE - 5.2
. . ,, - , lllmHtS.
ofabyilpven
• (lc/h) b
11
b • (hlk) b
'YI
11 cbanacd
cdtob+k
Wrill ti,, co~ 1111nwr :
(C) y-y=r(a,lay)(x-i')
(0) None of these.
3, The Rpaaon coefficacnt of_
pvea by:
(A) , (01 I a,)
(B) , (a.
(C) (a,' 0 1 )
(D) NOif
4, Tbe two rcgJ'ClllOD
;.aq:..mdlculartocteh
(A) r• I
(C)
,-o
..
Test: Quantitative A titude Stu..,
Common Prof1c1enc
~
YL--------~~(A) more thin I
(B) less than l
D) None of these.
(
(C) less than mo
• for I b,vanate d1Stnbu11on (x,. Y,),
13
,. • J. 2• .... n •. hn 1s defined as :
rcr,
Cov(x,y)
(A)
(C)
(8) -
,
(crJ
er'
rx,y, +[(Ex,)(1:y,)lnJ
L x,2 -(Ex, )2 I 11
rrx, -.r')(y, -.flJ 2
(D)
L(X,
-)'
-x.
24. lfthe two regression coefficients are 0.8
and 0.2. then coefficient of correla11on r
is :
(8) - 0.4
(D) None of these.
(A) 0.4
(C) 1. 6
JO. If p (x, y) = 0.4 and b,,. = 0.2, th
equal to :
en b"I:&
(A)_ o8
(B) 0.2
·
(C) 0.8
. (D) ± 0.8.
Out
of
the
two
Imes
31 •
d of regress,· on D~
by x + 2y = 4 an 2x + 3y _ 5 ,, c,,v~
regression line of x on y is :
O, ~
(A) 2x + 3y- 5 = 0
(B) x + 2y = 4
(C) x + 2y = 0
(D) The given Imes can't be regr
Imes.
ess~
32. If the correlation coefficient between
two variables 1s 1, then the two 1. ~
Ines 0f
regression are .
(A) parallel
(B) perpendicufa
1
(C) coincident
(D) None ofth
esc
33. Coefficient of correlalion 1s :
ion
,Anal sis
15 ' y =80,cr,=2,thea1 ,.1,,I6'
JI·
15 , t~en the :sti~t~d value of y
p
onding to x - 25 is .
corresP
(B) 140
rf: o.
,,,,. ::::
(C)
.
f,v"'
. en the following data : b'>. = 2.33,
0.39, th.en the value of correlation
,~ ffjcient r 1s :
coe
(A) 0.39
(B) 0.79
(C) 0.95
(D) 0.85.
. n the following data :
41 • Give
r(x- J)(y- y) = 3900, (x-x) 2 ==
,IO,
r
(A) G.M. offththe coeffiffic~ent of regression.
6360, t (y - y)2 = 2668, then the
(B) AM. o e coe 1c1ent of regressi0tt
regressIOn coefficient by., is :
(C) H.M. of the coefficient of regress~~
(A) 0.5
(8) - 0.5
(A) 0.613
(B) 0.363
(D) Product of G.M. and A.M. of lhc
(C) 0.25
(D) None of these.
(C) J.363
(D) 2.363
regress10n coefficients.
26. If b-'). = - (413), by, · - (1/3), the value
34. For a b1vanate data, the two lines of · 42 , Given the following data of a bivariate
ofr is:
regress10n are 4x - Sy + 33 == Oand
distribution · b,y = 1.36, byx = 0.613,
(8) (2/3)
(A) (213)
2x - 9y + 127 = 0. For this data r""
then the coefficient of determination is
(D) None of these.
(C) (1/9)
(A) 2/9
(B) 4/5
given by·
27. Regression equatwn of Y on X is
(A) 0.634
(B) 0.834
(C) 5/4
(D) ✓10/6
Bx - IOy + 66 "" 0 and cr, ~ 3. Hence
(C)
0.
734
(D)
0.534
35.
Slope
of
the
regress10n
equation
of
y
00
Cov(X, Y) is equal to ;
XIS.
43. Given the following data for a bivariate
(A) 11.25
(B) 7.2
d1stnbut1on . bxy = 0.756, b,y = 0.659,
(A) b,>
(B) by,
(C) 2.4
(D) None of these.
then the coefficient of non(C) llb,,.,
(D) lib,>.
28. The coefficient of correlation (r) and the
detemnnat10n 1s given by :
36. If the two Imes of regression for a
two regression coefficients byx• bx; are
bivariate data coincide, then :
(A) 0.502
(B) 0.402
related as :
(A) the two variates are independent.
(C) 0.602
(D) 0.702
(B) there is a perfect correlation bet•
44. For a bivanate data, the coefficient of
(B) r = b<>' x br,
ween the two variates.
correlation r = 0.91, then the coefficient
(C) there is not a perfect positive correla•
of alhenation is given by :
(C) r = b<>' + br,
tion between the two variables.
(A) 0.5 12
(B) 0.412
(D)
None
of
these.
(C) 0.312
(D) 0.712
(D) r = (Sign by,) Jbxyhyx .
4s. If the coefficient of correlation between
37. If x = IO, ji = 50, crx = 3, cry= IS,
29. The two lines of regression meet at :
X and Y is 0.65, then the coefficient of
P = 0.9, then the estimated value of x
(A) (X, Y)
determination is :
corresponding to y = 100 is :
(8) (a,, er,)
(A) 0.48
(A) 19
(B) 20
(C) (a,2, a/)
(B) O.Si
(D) None of these.
(C) 18
(C) 0.42
(D) 21.
(D) 0.32
25• If b,. ==- (312), b1 ,
will be:
...
(116), the value of r
~i.~
150
(D) 145
130
'
(C)
the following data : n == 7
· en
,
39. oiv ""- 15, r dy =-0.13, 'f.dxdy== 18,
i ~ "" o.89. The value of b·'>' is :
i 18 67
(B) 19.67
(M .
67
(D)
29.6.
20
(A)
(A)Ut
(C} 0.74
47
•
(B) OM
(l))OM
~ 1be foUowaig raallt of 11ae twO
lineaor...._tioma..,._dala
(X, Y); CJ-'= 2.5, r = -0.8. The mndanl
error of estimate of X on Y is :
(A) 2.5
(q
1.5
(B) O.S
(D) 1.2S
48• Given the followiug bivariale data (..r,y) ;
b_., == 415, h,. = 9120, then the coefficient
of detennination is given by :
(A) 0.26
(B) 0.36
(C) 0.46
(D) 0.56
49, For the following data of a bivariate
distribution (x, y) : (a/a) = 2/3,
bxy == 315 the coefficient of determination is:
(A) 0.9
(B) 0.79
(C) 0.83
(D) 0.81
SO. For the following data; r = 0.6, a,. = 4
the standard error of estimate of y on x
is:
(A) 0.84
(B) 0.74
(C) 0.64
(D) 0.54
St. Given the following data of a bivariate
distribution : hp = 9120, hJ« = 415. Then
the coefficient of allienation is :
(A) 0.3
(B) 0.2
(C) 0.4
(D) 0.48
S2. For a perfect correlation between the
variable x andy, the line of regression is
ax + by + c = 0 where a, h, c > 0 ; then
p (x,y) =
(A) 0
(C) I
(B) -1
(D) None.oflbese.
.r:.
53, The two Jines or r e ~
x + 2y = 1 and 2x + Y • 1. •~
regression equation of Y OD.~ a.
(A) 2x + y = 1
(B)
(C) X + 2y = 0
(f)) ~
~tt?i•"l
i
Test· Quantitative A p t h ~
.
.
... - ~
.
en
the
following
data
t )(
01
,1. 2: ; .. 300, 2: XV = 7900, t )(2,..111 ",
650oI
and N • 10I
2: y2 -- 10000
I
then r is :
O8
(B) - 0.8
(A) .
(D) 0.6.
(C) 0.6
If
b
.
and b . are both positive th
62,
)'l
.IJ
I
C!I:
I
I 2
I
I
(A)-+-<-:- (B)- +-.~2
common Proficiency
M
(b + b )/2, then
54- lt!p>-OIJld1tt • "'(8) ffl"' a p
~) "' 2! p
(C} '" s p
None of these.
· are 2.t - 1y +
55. The two lines ofrcgrcss10~ What is the
6 • 0 and 1x - 2y_ + 1 .. . xand I'?
Co-lation
11
coefficient between
·
(D)
"
(A) - 2/7
(B) 2/7
4/49
(D)Noneofthese.
(C) r
f regression of Y on X bY
Th
56. e me o
h following
least square method for t e
data
5
4
3
2
X
1
II
12
10
y 9
9
is :
(A) y
(B)X
(C) X
8.1 ·t 0.7X
0,718.IY
y
(D) none of these
57. The statistical method. which helps us to
estimate or predict the unknown vulue.
of one variable from the known vuluc of
the rcluted variubles. is called ;
(A) C'orrclntion
(B) Scatter diugnnn
(C) Regression
(D) Dispersion.
58. lf the two lines of regression arc
J.\' - I' - 5 0 und 2x y - 4 0, then
'i:' 11 ,;d .I' n:spccllvcly un· :
(A) I nnd - 2
(B) · I Ull(I 2
(C) 2 and I
(D) - 2 and I.
59, For two v11ri11hlcs .,· und y with sumt'
mean, the two regression cquut1011s urc
/,
y•
llX ➔
/,,
I +a
(A)T;;;
l - 11
(C) ~
x • ay
1
13, tht'n
~
is :
I ➔ a
(B) T;-;;
1- a
(D) 1- (I .
60. The regression coefficients of y on x 1s
2/3 end of x on y is 4/3. If the acute
angle between the regression lines is 0,
then tan 9 is :
2
1
(A) (B) 9
9
1
(C)
(D) none of these.
18
brx
l7•Y
1
111·1
b.1; .. '
I
I
r
- (D) None of,i...__
(C) -b + -t , <2
~~
\'. I
I)
In titting of u regression of' Y on)(
63,
.b .
. . to
bivariate distn u11011, c~ns1s1ing or 1
observations, . the cxplumcd and u:.
cxplatncd vunat10ns WCJ'C computed
24 and 3(J, respeclivc Iy. ·1·1le cocfiicicni,,
of detcnrnnut1on 1s :
(A) 0.4
(B) 0.J
(C} o.2
(D) 0.1.
M. l'hl' likely production corresponding lo
u rainf'ull of 40 ems. from the followi11g
clatu :
Ruinfitll (111 ems) Oulpul (in quintmlt)
Avcrugc :
30
so
S. D
5
10
,. • 0.8 (in cm) ts :
(A) 56
(fl) 66
(C) 76
(D) 46.6,
65. The regression cquolions of two
vnriablcs X and Y urc us follows : 3X +
2Y 26 O; 6X I Y 31 0. The
coefficient of corrclution between Xand
Y is:
(A) 0,4
(B) 0.5
(C)
0.6
(D) 0.65.
66, For some bivariate duta, the following
results were obtained for the two
variables x and y : x 53 .2, y "' 27,9,
by.r .,, - I •5, b,y
- 0.2. The, most
probable value of y when x "' 60 1s :
(A) 17.7
(B) 16.6
(C) 15.7
(D) 26.7.
ion
observations Oil' P1f
10
for
(.Y), the followina
1
' ' sllP~~ed (in appropriate unitt
181
ob ~ 13 0, I:y = 220, tr
t~2 "' 5506, I: _xy = 3~67: Thhupp
t) the price 1s 16 uruta 11 :
whel125 04
(B) 26.04
A) .
( 21.04
(D) 15,04,
(C) orrelation study of two vattablea
68·
In: t the following values are obtained :
811
r ..• 65
I
ji
•
1
~ 67, a, .. 2.S, of • 3 s
•
o I
· 0_8. The two regression coefficients
r I ) are
(b ' 1"
(A,) (0.57. 1.12)
(8) (1.12,0,57)
C) (2. 12, 0.67)
(D) None of these.
( . 11 i~ the following information :
(jtVC
69·
X
y
Arithmetic mean
6
8
Standard dcviauon
~
40/3,
C ffic 1cnt of corrclatton between X
an.ode y is 8/ I5. The. most likely value of
y when X • I00 1s :
(A) 140.67
(D) 141.67
(C) 241 .(,8
(D) 94.68,
76 The follow1n g cu lculations have been
· nude fot prices of 12 stocks (X) on
I
'
Iiange on a certain
Bornbay Stock be
day along with the volume of sales in
thousands of f; harc (Y). 2:X • S80,
l: ,.\'2 41658, 2: Y 370, 2: Yl 17206,
2: XY 11494 From thc~e calculations,
!ind the regression equation of prices of
stock s, on the volume of' the sales of
share~ .
(A) Y I. I 02X l 82.33.
(B) X
1.102Y I 82.331.
((') X 4.4 l 0.2Y
(D) None of these.
71. The following figures relate to the years
of service and income in hundreds of
rupees of the employees of an
organisation.
x
8, Y 5, b,x • 0.7S.
The initial start for a person applyina for
a job after having served in another
factory for a period of 12 years in a
similar capacity is :
I
w
bet
~
be
most lilt,
heiaht"
(A)IOJ5
(C) 73 7
74, Oivenx
llne1 of re
re1pectivety, If M
of correladc,n
(A) 0.4
(C) 0.6
75, Which otthe fol
true
(A) The rep
applyma tbl
lq\llrll,
(8) TbeCOffela
(C) b11 1114 b
potJtlvo m
(D) Tbo,.1ipt0
b.,or
76, You~
'"'
cortt
fa 0,
(~
. . . obtained
9f' dafl ,!D tw0
Y· f •20, y • 15,
. ot X • 4, Standard
~y • ); , • the coefficient of
• o,7. The likely value of Y,
•Uu:
(.l) 17.9
(B) 17.8
(C) 17.1
(D) 16.1.
71. In a partially destroyed la~oratory
record of an analysis of correlat10~ data,
only the following results are legible :
Variance ofX = 9
Regression equations :
8X-10Y+66=0
... (1)
40X - 18Y = 214
... (2)
On the basis of the above information,
the value of cry is :
(8)3
(A) 2
(D) 5.
!C) 4
79. The following data, based on 4~0
students, are given for marks m
Statistics ~nd Economics at a certain
examination.
Mean marks in Statistics = 40
Mean mars in Economic = 8
S.D. of marks (Statistics) = 12
Variance of marks (Economics)= 256.
Sum of the products of deviations of
marks from their respective mean =
42075. The average marks in
Economics of candidates who obtained
50 marks in Statistics is :
(A) 54.5
(8) 45
(C) 52
(D) 49.
80. From the following data : cr.r = 3. b.r, =
0.85 and by.r = 0.89, the value of cry is :
(A) 3.57
(8) 3.07
(C) 3.97
(D) 2.07.
81. In order to fmd correlation coefficient
between two variables X and Y from 12
pairs of observations, the following
calcu!Jtions were made :
I X • 30 I X2 · 670, I Y = 5,
I Y2 = 285, I XY = 344.
On subsequent varification. I
discovered that the pair (X = 11, y ~
was copied wrongly, the corre~
)
being (X = 10, Y .= I~). The
correlation coefficient 1s :
'
(A) 0.61
(B) 0.71
(C) 0.81
(D) 0,67.
cor:~
82. The following ~ta ~hows the nutnber
motor registrations m a certain tc . Of
for a term of 5 years and the s~~
motor tyres by a finn in that terntoa c or
. d.
ryf0r
the same peno
Take dx = (X - X) = (X - 700),
dy = (Y - Y) = Y - 1300, then
r. dx = 8, r. dy = 0, >.: dx 2 = 27,Soo
r. dy2 = 8500; >.: dxdy = 41500 ',,,_
· •oe
estimated sale of tyres when the mo
.
. 1s
. 850 1s
. :
tor
registration
(A) 1424
(B) 1324
(C) 1524
(D) 1624.
83. The following calculations have been
made for pnces of 12 stocks (X) 0
.
n
Stock Exchange on a certam day alon
with the volume of sales in thousands 0~
shares (Y). From these calculations the
regression coefficient of prices of
stocks, on the volume of sales of shares
is :
r. X = 580, >.: Y = 370, >.: XY = 11494
r. x 2 = 41658, r. v 2 = 11206.
·
(A) - 1.10
{B) - 2.10
(C) - 3.10
(D) - 0.10.
84. For 100 students of a class, the
regression equation of marks statistics
(X) on the marks m Economics (Y) is
3Y - 5X + 180 = 0. The mean marks in
Economics is 50 and variance of marks
in Statistics is 4/9 of the variance of the
marks in Econonucs. The mean marks
in Statistics is :
(A) 56
(B) 55
(C) 66
(D) 65.
85. A departmental store gives in service
training to its salesman which is
followed by a test. The management is
considering whether it should terminate
,ervicos of the sa
lfle well in the test. The
dO tes to the test s~
;~e by the salesman :
"' ,:::. 20. y = 40, .% ...
;c y _ y ; t d x2 = 120, t f d.t
r
dY::::-
11 ""9,
the correlation between the til.1.·
1 .
~t.and the sa es 1s :
score
.
5
(B) 0.85
(A ) - 0.9
(C) _ o. 75
(0) 0.9S.
some bivariate data, the folloWing
86· for
Its were obtained; Mean of Variable
resu
X"" 53.2 and of Y = 39.5; Regression
a-:cient of Yon X = - 1.5 and ofX
coe111
Y
on ""_ 0.38. What should be the most
likely value of X when Y = 50 ?
(A) 39.21
(B) 49.21
(C) 48.21
(D) 59.21.
The lines of regression of a bivariate
87, distribution
. are as 10
"' IIows :
5X- 145 = - IOY; 14Y - 208 = - 8X.
fben
5
The mean values (X, Y) is :
(A) (5, 12)
(C) ( 12, 3)
S8, While calculating the coefficient of
correlation between two variables X and
y the following results were obtained :
The number of observations N = 25,
E X = 125, I: Y = 100, :E X2 = 650,
E Y2 = 460, >.: XY = 508. It was
however, later discovered at the time of
checking that two pairs of observations
(X, Y) were copied (6, 14) and (8, 6)
while the correct values were (8, 12) and
(6, 8) respectively. The correct value of
by.r is :
(A) 0.6
(B) 0.7
(C) 0.8
(D) 0.65.
89. Given : Unexplained variation= 19.22,
explained variation = 19.70, then the
coefficient of correlation, is :
(A)± 0.71
(C) ± 0.61
t
(B) (12, 5)
(D) (3, 12).
(B) :!: 0.75
(D) :c 0.65,
If
~)
(C)
95. There
and y fi
Ix=SS,
ty2.= 11
(A) 1
(Gl2
"·
eommon proficiency reel: Quantitative Aptit~'-
•
(A) 4r
ti. la I panfally desiroyed l1bor11ory
record of ■n 1nalysi1 of • correl11i~n
data. the only information remained is
••-•lo
l
--..-..- n equal on• :
•
- 9 - I07 "' 0.
4.r Sy + 33 0, 2x Y
Then lhe means
_ values of x and Y are :
x • 13, y • I 7
(A)
(8) -
12 • • 13
' ~
(C) ;r .. 15, )' a 13
(D) None of these.
:
99. For the following data, x = 36, )' -= 85,
a.• 11. a, = 8, r = 0.66, the regression
line y on x is :
(A) y = 0.48x - 67.72
(B) y = 0.48x + 67. 72
(C) y = 0.91y- 41.14
(D) None of these.
100. If the lines of regression in a bivariate
distribution are given by x + 2y = 5 and
2x + 3y = 8. then the coefficient of
correlation is :
(B) 0.866
(A) - 0.866
(D) 0.666.
(C) - 0.666
101. 4x + 2y = 1 and 2x + 3y = 4 are two
lines of regression of x on y and y on x
respectively. Then the coefficient of
correlation is :
-11 ✓3
(A) 1/ ✓
3
(B)
(C) 11 ✓
2
(D) -11 ✓
2.
102. For the following data :
L x = 110, Ly = 70, L x2 = 2500,
L XJI = I00 and n = 20, estimated value
of x when y = 4 is :
(A) 5.42
(C) 5.3 2
(B) 5.62
(D) none of these.
103• Let X and Y be two variables with
correlation coefficient r. If the values of
X and y series are changed such that the
Cov (X, Y) remains unchanged while
~e variances of X and y become 4
hmes ~eir original values, then the
correlation coefficient between X and y
becomes:
(B) (r/4)
~
l6''
(D)
(C)
.
(r/16).
104. If the coe~c1cnt of cor_relation b
X and Y 1s 0.28, covariance b cti,,e...
• 6 d h
ct.., -ti
and y is 7. an t e variance o ten~
then the S.D. of Y series is . f )( ~ g
(B) 9.J ·
'
(A) 9.05
(C) 9.08
(D) 10
.Os.
105. If the two lines of regress 10
x + 4y '" 3 and 3x + y == 15 n arc
th
value of x for y = 3 1s :
' en !ht
(A) - 4
(B) 4
(C) - 3
(D) 3.
J06. If 0 1s the angle between th
regression lines with
e ~o
correla1·
coefficient r, then
ion
(A) sin 0 ~ 1 - r2 (B) sin 0 :s; ,.i
(C) sin 0 ~ r2 + I (D) sin 0 > I - I
- -~
107. Let x = 15, Y = 80, cr. == 12
·
r
' (J "' 12
r = 0.75. Then estimated valu ,
y correspondmg to x = 55 15 :
e of
(A) 110
(B) 120
(C) JOO
(D) none of thes
108. Two random variables hav e.
th
regression Imes 3x + 2y == 2e6 c
_
and
6x +I [ -b 31 . The coefficient of
corre at10n etween x and y is given b .
(A) - 0.5
(B) 0.5
y.
(C) 0.25
(D) None of these
109. The lines of regression of y on X and
X on Y make angles 30° and 60,
respectively
with the positive direcn ~
.
of x-axis, then the correlation coefficient
between X and Y is :
(A) 1
(B) - 1
(C) (11 ✓3)
(D) ✓
3.
110. The correlation coefficient when the line
of regression are 2x - 9y + 6 = Oand
X - 2y + 1 = 0 is ;
(A) 2/3
(B) _ 2/3
(C) ± 2/3
(D) None of these.
111. 3x + 4y - 7 == 0 and 4x + y - 5 = 0 are
the equations of two regression 1ine1.
The correlation coefficient between x
and y is :
f.)
o.43
(B) - 0.43
(D) -0.34.
(c, o.34
( between the lines of regrCllion
1J. Jf r rid y on x, is± l, then the: Ob
I oriY
. 'd
Jjoes comc1 e
(Al lines are perpend'1cular
(B )
lines are not parallel
(C )
J'lone of these.
(D )
and cr are standard deviations f
Jf (Jr.
\
0 X
.
113· ao d y· senes and r the correlabon
y
at
(A)_!_
1-a
2
coefficient, then cr x- .l (i.e., Variance of
t _ y) equals :
·
(C)
cr/ + cr/ + 2r cr. cry
a,2 + cr/ - 2r cr.r aY
11 (a/- cr/ + 2r s, cr)
(D)
none of these.
(A)
(B)
If u = x + y, v = x - y and x and Y are
11 4. independent then cr.-' 1s
. equal to :
(Al
a/
(C) cr/ - cr/
(B)
cr/ + cr/
(D)
cr,2.
JtS. If x and y are related by y = mx + c, m
and c being constants, then coefficient
of correlation r between them is :
(B) 0
· (A) 1
(D) none of these.
(C) 2
!16, If the values ofx; and Y; are transfonned
to u, and v, by the relations u; = ax, + b,
v; = cy; + d, then which of the following
is not true ?
(A)
cr. = acr,
(B)
cry= cay
(C) Cov (u, v) = 2c Cov (x, y)
(D) p (u, v) * p (x, y).
117. Angle between two lines of regression is
given by:
(C)
-1..
1-•
t,
(D) - .
119.
.
1 ex
The regresnon equations af'y mt-.r l1id x
on Y are respectwely Y <Oil x ad
4x - Y = 3, then the correlation
coefficient between x and y is :
(A) 0.5
(B) - 0.5
(C) 0.7
(D) - 0.7.
l20. The standard error of estimate of X on Y
is given by:
(A) cr.r (1- r2)112 (B) a_. (1 _ r2)
(C) cr, (1 _ r2)112 (D) 01 (1 _ r2).
121 • The Cov (x, y) between x andy for the
following data is :
x: 3 4
5
6
7
y: 8
7
6
5
8 is:
(A) 2
(C) 3
(B) - 2
(D) 4.
122. The coefficient of correlation between x
and y for the following data
x: 1 3
4
S
1
y: 3
7
9
11 15 is :
(A) 2
(C) l
(B) 3
(D) 0.
123. The regression coefficient b.\)· for the
following data :
{xy: (5, 2), (7, 4), (8, 3), (4,
(6, 4)}
,>..
is:
(A) 0.5
(C) 0.7
124. The regtession
following
{(.xy) : (1, 6
is:
(
~
Common Profic1enc
S.21
(B) 0
(A) 014
(B) 0.64
(D) 0.74.
.
(C) 0.36
. of determination
.
the
coefficient
157. G1ven
• .
= 0 9025 the value of r is .
.
'
0 95
(A) 0.85
(B) .
(D) 0.65.
(C) 0.75
. data : b ' == 0.39,
. !'- o9025
158 Given the following
• coefficient of dete~
· atIOn IS •
'
then the value of b_ry is :
(B) 2.314
(A) 3.314
14
(D) 4.314.
(C) OJ
· tion is
159. If the coefficient ofnon-de~e ~
0.502, then the value of r is .
(A) 0.69
(B) 0.71
(C) 0.73
(D) 0·8 1.
. .
. of non-detennination
160 If the coefficient
. . 1s
. 0.502, then the coefficient of alhenat1on
is
(A) 0.71
(B) 0.61
(C) 0_51
(D) 0.81:
. .
161. Given the coefficient of alhenatto~ is
0.4, then the coefficient of correlatton
is :
(A) 0.91 7
(B) 0.817
(C) 0.617
(D) 0.717.
162. Given r == 0.4, cry = 0.6, then ~he
standard error of estimate of Y on X IS :
(A) 0.45
(B) 0.65
(C) 0.55
(D) 0.35.
l63. Given cr, == 4, r = 1, then the standard
error of estimate of X on Y is :
(A) 1
(D) 3.
(C) 2
Given
byx
=
3,
CJ>=
1!,
and r:::, 1 th
164
· tandard error of estunate of)( ct\ 11.
s
011 y .,
(A) 4
(B) 3
is :
(C) 2
(D) 0.
2. B.
10. A
17. A.
18. C.
25. C.
33. A.
26. B.
34. D.
42. B.
50. C.
58. A.
66. A.
74. B.
41. A.
49. D.
57. C.
65. B.
73. B.
3. B.
11. B.
19. A.
27. B.
35. B.
43. A
51. C.
59. C.
67, A.
75. B.
4. C.
12. C.
20. B.
28. D
36. B.
44. B.
52, B.
60. C.
68. B.
76. C.
91·
s 13.
,o , 13.
113· 13.
1il-
13.
,i9· p.
13"
3 ;..
14 .
(D) none of th
. wh1c
. hi
. a ':ays r1es on theese•
166. The pomt
lines of regression 1s :
liiio
167. If the tw~ l!nes_ of r~gr~ssion of
bivariate d1stnbulion coincide, the a
correlation coefficient p satisfies n the
(A) p = 0
(B) P > 0
(C) p < 0
(D) P "' 1 or _ I
168. The regression line of Y on X i
y = 3x + 10 and that of X on y is
s
X
1
3
10
= - y - - · If (Jx = 4, then
3
(A) 3
CJ ::
>'
(B) 48
(C) 12
(D) none of these.
169. In a bivanate data :Ex = 30. :E y "'400
:E x2 = J96, :E xy = 850 and n == 10. Th~
regression coefficient of Y on X is :
(A) - 3.3
(B) - 3.2
(C) - 3.5
(D) - 3.4
1
,s j. ;..
1S9• 13.
1 v.
16 •
82, C.
90. D.
98. A.
106, A.
114, A.
122. c.
130, C.
136, C.
144, D.
152, A.
160, A.
168, C.
5. A.
13. C.
21. C.
29. A.
37. A.
45. C.
53. B.
61. A.
69. B
77. C.
6. D.
14. C.
22. C
30. c.
38. B.
46. B.
54, A.
62. B.
70. B.
78. c.
7. B.
15. C.
23. B.
31. A.
39. B.
47. C
55. B.
63. A.
71. A.
79. A.
8. A.
16, B.
24. A.
32. C.
40. C.
48. B.
56. A.
64. B.
72. A,
80. B.
83, A.
91. C
99. B.
107. A.
11S. A.
123. D.
131. A.
137. D.
14S. C.
153. C.
161. A.
169. C.
84. C.
92, C.
100. A.
108. A.
116. D.
124. D.
132. A.
138, B.
146. D.
1S4. C
162. C.
85, D.
16. a
93, A.
101. B.
fl, B.
95. -A.
103. B
111. B
119. A.
127. A.
133, D.
141. D.
149. B.
157. B.
165. A.
94.e,
102. A.
110. A.
118,A.
126, B.
132. D.
140. B.
148. A.
156. B.
164, D.
109, C.
117. B.
125, A.
132, D.
139, A.
147. B.
155. C.
163. B.
& &a
~A.
*·
A.
tu.~120. A.
128. B.
134. B.
142, B.
150. C.
158. B.
166. A.
SOLUTIONS/H1NTs
ee Text.
t,.)' S
l( ): see Text.
z(JJ : See Text
3(1J)· see Text
4(C):
e Text
5(;\): Se
. If x and Y are two independent
6(0)· variables, then the correlation
oefficient r between
them is zero.
C
•
•
Putting r = 0 m equations of the two
Jines of regression, we get Y = Y and
X = X which are of the type
x = const. and y = const.
byx + bxy
>r
7(B): We have to prove that ~
Or
ANSWERS
1. A.
9. A
A·
t}· A·
(C) 4.71
(B) (b_ty, by,)
(D) (0, 0).
ssiO
13·
sl•
165. If r = 0.5, b.'9' = 0.6 a~d cry::: 2.6, the
ndard error of estimate of)( tq~
0 l\y .
sta
(A) 2.71
(B) 3.7}
Is.
(A) (x, y)
(C) (crx, cr)
~
~
)
(ray l a x)+(ra x lay)
--'--------->r
2
cry2 + crX2 > 2crX crY
or cr/ + cr/ - 2crxcry > 0
⇒ (ay- aJ2 > 0 which Is true.
S(B): The lines of regressio~ of Y on X
and X on Y are respectively
a1x + b,y + c1 = 0 and
azr + b-Jl + C2 = 0.
Therefore, byx = slope of the line of
regression of Yon X =- (a/b,) ...(l)
(1/bxv) = slope of the line of regression
of X on Y = -(a/bJ
x-a
byx
= (rcr/cr) = r · (kcrjhcr.,)
= (klh) rcr/ o.) = (lc/h) b,,,..
lO(A): The regression coefficients are
independent of origin but not of
scale.
ll(B): Here m1 = 2/5, m2 = 8/5
ffl1 - ffl1
:. tan 8 = 1 + m1m1
(8 / 5)-(2 / 5)
30
= 1+(8 / 5)(2 / 5) =
41
12(C): See Text.
13(B): Since G. M. > H.M.
:. G.M. of (byx and bzy) > H.M.
of (byx and b..,)
⇒
r>
!.
2byx · bxy ⇒ byx + bxy >
b
bxy + bxy
by_.. xy
r
1
1
2
⇒ -+->-·
b"' b"' r
14(C): bzy = r (a/a,) and by,= r (a/az>•
. bzy x bYJ = r (a/a)
..
., x , (a/aJ
•r.
:. bxy = - (b/ aJ
Since bvx · bxv ~ 1, therefore
y-b
v = - - where a, b,
h '
k
.h and k are constants.
9(A): Let u = -
tS(C): See Text.
Ion llnts .,· ,m J' and .I'
TIit two N1rffl
ar rhrlr nit•ans
on x always lnrtrst'C I
(.r,)'),
~
• ,.
or
line
or
=- ,.~ 41(
s 41( :S I ( .'. 0 :i , J :S I)
41(
1•,y
....:d)
.;;;:;:/I-:; ~ b,, x 0.2
.
(0.4)i
=> h.t''
0.H.
23(8 ): Sec Text.
0 8 x 0,2 • 1 6
=> r • 0.4.
26(8): r •
I
J2(C): Since ,J
I
⇒ Cov( x, y) =
a/
i
5
9 X 4 36
Cov (x, y) =- - - ..,
5
5 1.2,
2
28(0): p = byxbry. But br:r, h,Y' p Cov (X, Y)
are all of same sign.
:.
29(A):
p = Sgn hrx Jbyxbx, •
(X, Y) , by definition.
JO(C): We know that b,,Jxy = p2
J
'
0
b11 h1.,, when,.
80
,
X h,,
..
33(A): Smee ,:i
(,, =r-
I
II
-:-._
I,
'>
18 _'( 15)( O.IJ)
7
89
b1·1,
. , r G.M. of lht• cocfflclll
rcgrc,slon.
11h or
34(1>): Ilcrc, I\, 2/9 ond
11/,,y slope or 4.1 - Sy., 33 • 0
I
4
5
or - •
h,v • /Jn,
4
18 0,279
11
,o,n:
p•
4
JJO
.!:(x
(Note that p , 0 as both /, and 11
•·t
11.arc
posillve)
·
35(B): Regression equation of y on x 18
y y hv., (x x)
Its slope by.,· ·
36(8): The two lines of regression coincide
If there Is a perfect (direct or
Indirect) correla tlon between the
two variable~.
27
9
•
~m • 19,97.
21
0,9 ( ~)
1
0.18.
150
50
Hence, regression equation of X on Y
jg: X 10 0.18 (y 50).
Fory= 100,x 10 + ,18 (100-50)
=- 19.
)<
x)(y-y)
l.(x
o.•w • 0,897
O.9S.
-2
x)
3900
·6360
-
0.613.
42
rl •
43 cM:
s •r•
0,834,
o 1s6 x o.6s9
('oclflclcnt of ll<JU-dctermh111tlon
(k 2) J r2 I 0.498 0.502.
44(B): Coefllclent of ulllcnatlon (k)
,.
(0.,,1. o.a1.
SO{C): 11nd•rd Error eatimate of y on x 11
given : s,... a, (I - r)llt
51 C:
4 )( ( I 0.6)2 4 )( (0.4)Z - 0.64.
• ( ): ,i h,y >< br, (9/20) >< (4./S) • (9/25).
Coernc1ent or Alllenatlon • (i)
• (I (I 9/2S>''z
• (16/25)' 12 4/S 0.4.
5
l(R): The !incur rclation1hip between the
lw<, voriuhlcs 1s
"x ~ by
I ,. 0 with alopc - !!. < 0.
.
b
So, t~c two variate~ arc in perfect
negative correlation und hence
p (X, Y) • -1.
53 (H): If firnt line x 1 2y 7 is taken as
rcgrcsswn equation of Y on X, then
Wt wrilt it a~ :
I
Ji-{0.91)
2
• .ft- 0.83 • Ji 7 ., 0.412.
4S(C): Coefflclent of determination
r2
(0.65) 2 0.42.
46(8): Coefficient of non-determination
1 - r2 I (0.25) 2
0.063 0.94,
47(C): Standard error of estimate of X on Y
b
)'I
2
Al~u, the Becond equation (which will
ht rtgrc~sion equation of X on Y) can
ht written a~
I
7
x• --y+2
2 -_, hty
I
--.
2
1
1
So,hrAr• (~ )(~
)-¾st.
Hence regre11lon equation or Y on
X lsx+ 2y • 7.
54(A): Since p > 0, therefore by, and b.ry ari:
also positive.
byx +hxy fi:'"T
Hence - -- ~ ,.;byxhxy
2
( ·: A.M.
~
hyx +hxy
G
--i-t?vfr
~
~ c!! p
s. .
is given by : y a,.. (l - r)i/l
2.5 (1 0.64) 112 • 2.5 >< 0.6"' l.!,
7
--x+::::)
2
2
0498
.. ✓(1-r)2
u,.
1
3
09
5 2 "
Coefficient or determlnadon
y
bry X hy,,
r ",y x h,.t
~--,><3
5
,.!)(!
(,i}: ('ocfflclcnl of d_t·lcrml1111tlon
1.36 x (t613
6 .
37(A): Here, bxy = p ( :·~]
h,,,
>'.
Jo.k97
., r
5=
10
36
,.2 • "•Y
• r • •,, - •,. • (415) (9120).
49(1)): b • , !.._
.,
0
,
r>"1 .
( () IJ)l
7
17,72 1
89 00241
48(8), Cotmc..tor~
~
140.
l:,(v2 _ ~dy)
h
2 5
15)
>:.dw(v- (l:tlt) (l:dy)
(Y - Y),
-x-
15),
C,(.\
2!11,
y · HO I 6(25
.... j
I, I,
0.75)( 16
~
I
i,or x
2,l' ......
2
H•
Ii
II' I
2
6,
rciir,·.
regression
equation
of
y
1
on X
,s
I'
S
.....
Rt'gtl·~smn lim·s arc
y y hl'I (:\ X)
⇒
( •: r has same sign as of b,r or b>•x)
27(8): Given regression line is :
Bx - IOy + 66 = 0
8
66
4
33
⇒ y= -x+- ⇒ y=-x+10
JO
5
5
'" s
und /,
.,
9
= J(-4l3)x(-(l/3) "- (2/3).
i
5
II C,c,
Rt•gn·s,1011 l'ct1111t1 011 of\' 2
l.,· I· Jy ~ 0.
. 1111J'li:
0.25.
,p;;;x h,,x
b =
J\'
,\
2S(C): r • Jbx, x hp - J(3/2)X (l/6)
(1/4)
f
Thl'Y un• ,•ohll'ld,•111.
22(8): If one of the regression rocfflclcnts
ls greater 1111111 unity. then the other
Is less than unit)',
?- • b,, x h,,
0,8
111t•
--2 ·f }. und .1
-
,t8(p):
.·.
und X -
:::, 0SK!i(l/4),
ll(('): r •
linl:s
g1\'l'11
Clcndy h11
(),bl,
0.2
+ 2y 4 llllll l1
.Jo.s X (),4(l
h,,
,.
JI(,\): The
(OA)l
""' -""'
i.1•, .''
s I,,pc
20(8): We h11,·e
rtgrcssion ofY 011 X I(: :md (_lll:w)
sll1pc: ,if line i,r rrgn·ssi1111 lit :,; on
y ( !/4). Thnd'im·,
,,,,
;,
~
----
p2
I
17(A): Sff Texr.
18(C): Sff Tell,
19(A): r .. .fl,., x b,,.
24(A):
r~i
~111,,
uo
bvx + hxv
2
=> "' ~ P•
;?
G.M.)
---------------:~~cv'1ei!~~~~~~jj~~=--,~1~0~:;::::.-,~~-:-:;:;::::~::~:~~~~i~=-~
~
· ·:~~~~~-~:!:!=f.S¼:
(sincewe bsign ofb)'X must be negative,l
=
=t
ion are
I
1be
ofresress
...
:e regression equation of yon
Q ..
.<)'
twO Hid
li- 1y + 6 • 0
5'(11>:
.,
(1)
... (2)
2t1+ 1 • 2,
.
and ,x - -;
the regression
If we take <I) x•'then (2) is that of
· ofY on ,
cquabon W an write these as :
X on Y. cc
2
I
2
6
..
-y-,. -x+- andx
7
> 7
7
7
respectively.
2
2
d h•J "' -7
b=-an
)'T
7
4
2 2
b
-x-=--<1
⇒ bP .o,. 7 7
49
4
Now, P2bb-- " •Y - 49
⇒ p
Total
9
18
tu
30
48
2
= -7
yi
81
81
4
9
100
144
16
12
121
25
55
II
5
IS
SI
160
SS
527
l:Y "'Na + h'f.X, l:XY = al:X + b"i.X2
51 =Sa+ !Sb and 160..., !Sa+ 55h
Solving, we get: a 8.1. b - 0.?
Regression equation : is Y "" a + hX
or Y 8.1 + 0.7X
S7(C): Regression
S8(A): Solving the equations 3x - y - 5 = 0
and 2x-y-4 = 0, we get: x= 1,y =-2.
S9(C): Since y = ax + b and x ay + P,
111
.ai
:. y = ax+ b, x
But x = y.
00
1-a
60(C): by, •
2
b
ii
x b,,.. =
~
= + 0.8.
G.M > H.M.
G.M. of bJ., and b." > l-l i ,
.
.•
and
.
·••·· Of b
.•,
b_"<Y
2 byxbx_v
r
b b
⇒ r >
b ⇒ - > -:--!.:...-1J>
b1•x + xy
2 bYx +b'
b
byx + xy
2
1
1 ~
⇒ --~>- ⇒ - + - 2
bvxbxy
r
b,-x
b,,x >-..
63(A): Coefficient of deternunation .,
Explained Vanation
r'- "" Total variation
64(B): Regression of Yon X :
=
+ bxy
,_aY = 0.8 x _3.5 = 1.12;
a,
2.5
a.r
0 8 2.5
b = r - = • x- = 0.57.
xr
aY
3.5
aY 8 40
u
b - r- = -x69(1J): nere >•x - ax
15 3 x 5
68(B) byx
= 1.422.
Regression hne Y on X is
Y - 8 = 1. 422 (X- 6)
O' A
When X = 40 cm, then
y - 1.6 (40) + 2 66 cm.
65(8): Solve the given hnes to get X . 4,
y 7.
Assume 3X 2Y 26 0 as the
regression line of Y on X.
p,
3
3
⇒ y
- -X
2 + 13 ⇒ byx = --2 .
Similarly, from the second line, we
70(B): b,y
-6
.... - 0.5.
ljl..,.,,._~~.;;.:;...;;,
Sitrilarty,
b .. -10-0
• 33-(-1)1 /8 • -0.34.
,-~
== ✓(-0.278) X (-0.304) =-0.29.
7
3(1): Regression equation of y on X :
o,
Y -Y
== r-(X-X)
o..
⇒ y - 80 = 0.6
Y = la + y ⇒ bx,, == k.
Now r2 = bx, x bJIJC = 4k.
1
If k =
, then the result r2 = 4k
16
l
1
⇒ r2 = or r = ± = ± 0.5.
4
5
0.75 (X - 8)
⇒ Y = 0.75 ~ -1.
When X = 12, then
Y - 0.75 x 12 - 1 = SorRs.800.
2
As both b..~, and byx arc positive so r
must be positive, r = 0.5.
75(B): ·: r = ..}b.., x b1" , the sign of r
will be as that of b..,
76(C): Regression equation X on Y :
-
ax (Y - -Y)
X - X = Y-
o,,
⇒
X - 36 = 0.66
X
11
8
(Y - 85)
⇒
X = 0.9075Y - 77.1375 ••• (1)
Put Y = 75 in (1) to get X = 26.93.
77(C): (Y -
Y by., (X - X)
Y
(X-165)
X == 170 cm, then Y = 83.75 kg.
74<B): x = 4y + 5 ⇒ b = 4 and
.:ey
17236-12(30.833)2
:. Regre~sion line of X on Y:
X 48.333 ( 1.102) = (Y - 30.833)
⇒ X = - 1.102 Y + 82.331.
7l(A) Regression line of Y on X :
10
8
X
⇒ Y == 0,75X - 43.75 When
11494 - 12 (48.333 X 30.833)
= - 1.102.
Y
Ji,;xbx,•
""Yl
-N(Y)l
~
=
⇒
lb,.b,, - 11
byx
3467 = 130a + 2288b ⇒ a= 8.8 and
b"' t.015.
'fhe line is y = 8.8 + l.015x.
put x = 16 to get : y = 25.04.
⇒ y = 1.422 X - 0.532
When X = 100, then Y = 141.67,
:EXY-NXY
I
... 8 •
:::: 107.7 - 1.5x
or y
Wbell x == 60 , then
J' "" 107.7 - 1. 5 x_ 60 : 17,7,
rhe line Y on x_ is Y - a + bx, where a
d b are given by the nonnai
61(.-\): ::uation
220 = 10a + 130b, and
24
⇒ Y 50 - l.6(X 30)
⇒ Y 16 X + 2.
r=
<0
xis:
60 "' 0,4,
Y _ Y ,. _
a y (X _ X)
:.
r
_ y "" b>" (x - x)
~ y- 27.9 = (- 1.5)(x- 53.2)
ll
have b.ty ' - -.
6
3 b.ry "' 3
6
"' o.4
1
4
,;_.-\)·
a. y +p.
i(l - a) = b, x(I - a.) =
1-a •
, = Jbxy
< 0,
.
62(8): Since
(Note that b.. > 0), so P > O
y
XY
x2
S6(A): X
9
9
I
2
3
4
Similarly,
1ox 7900 - 250 x 300
b
.___
xy = 10 x 10000- (300)2
:.
So our choice is valid.
'
(8/9)-1
6/ 3
18 ·
NIXY -(tX) (tV)
1
6l(A): b,,x = N(tX ) - ( t ~
10 X 7900 - 250 x 30o
=
~
10 x 6500 - (250)2
"'l,
⇒
Y)
Oy
= r-
o ..
-
(X - X)
· 0.7x3
Y-15= -4-(X-20)
= 0.525 (X - 20)
or Y • 0.525X + 4.5,
When X •24,
so
or
Hence
R
~
t,91
t,..
100
Cerreded J:
66.
Jt:
6SO
193
9x(l20 9)
IHI Jf l
e,•Jrl =J346
"
9 •
t..,• Eq _
•o/
. ,1 •
~r•
Jt,r
193
9x(346 9) •
x bq
193
12
R .._
330
or
lt(D): a.
180 (
⇒ r2
I
J120xk6 = 203
Cerrected %
S08 6xl
520.
96(A): The regression line : Y - Y • b,,
(x _ x) or y- 8.8 "" 1.24 (x - 5.5)
⇒ y• J.Z4x+ 1.98.
97(A): 4x + ~1• - I = 0 ⇒ y (-- 416)
⇒ blt). = - (2/3).
4.x - Sy + 33
2.x - 9y - 107 = 0,
we get ; x "' 13, y "' 11.
98(A): Solving
ray
99(8): b_..,,
t03(B): Herc r ::::
X
= 0 and
(2<Jx)(2cry) - ~ 1
X Oy '",
4t,
104(A) r =
⇒ 0.28
7,6
<1x <1y
~
cry =9.05.
(J _
=
,2
)2 + ,2
(~]2 II
r2 = hxy
/
+ ,2
x = (I 10/20) = 5.5,
y = (70/20) = 3.5
II
[crt +~]
sin 2 8 s;; (1 - r2)2
⇒ sin 8 s;; 1 - r.
⇒
/
107(A): hxy =r(aylax)
= 0.75 X (12/12) = (3/4).
Regression line y on x is :
⇒
y=
~
hyx
X
⇒
bxy
I
r
= - (1/4).
[(y-y)2 ±2(x-i)(y-ji))
a,.2 • a,+ay±2ro
2
2
a
~02_2
X (
J/4)
(3/16)
= - 0.43 nearly.
I in
1-,2
Ou::o2+ 2
x o,, +2r CJ• CJ and
2
,
• -o, +oy -2rc, c,
N
•
ow: and Y are independent,
so r• 0
y
... C1 u = 02J(
+ c,2y
Also
r = (IIN)l:(x-x)(y-ji)
[✓(IIN)F.(x-x)2
° --X--Y-,
r
0 2 +02
tan 8 = 0 => 8
=o
✓(II N) (f.y- ji)2]
:::) r= (IIN)mI:(x - x')2
(II N) Jm 2 (l: (x _ x) 2
±m ± I
:. u-u ;;;. a(x-x)
... (I)
Similarly, v - v = c (y _ y)
...(I)
:. a; =(II n)
2
= a
a;.
r.
(u -'ii)2
=(1/n)I: a 2 (x-x) 2
(using (1))
Similarly, cr! .. (1/n)l:(v- v,2
a x CJ
X
•y
o2- 2
= - ( .jj /4) = - (1.732/4)
=±
2
,. -o, + oy -2, ax a,.
2 m Queatlon No. 113, we have
114(A): Fro
116(0): u = ax + h ~ 'ii= ax + b.
x bxy
}'X
ll2(A): Putting r
tan 0
~
=
Since r, h,y and b,y have same sign
:.
0/N) t (X -
m
(l/4)y+(5/4)
=b
X)2
•
X
(3/4).
=± (✓314)
:. r
• (x - ?) :I: (y - Ji)
(II N).t(w - w)2
(3/4)x+c114)
=-
(3/4)
⇒ y- 80 = (3/4) (x
15).
Putting x = 55 in it, we get : y = JOO.
108(A): The line 3x + 2y = 26 can be written
)' = (-3/2) X + (26/3).
,., (l)
The !me 6x + y = 31 can be written as
x = - (116) y + (3116).
...(2)
Now byx = - 3/2 and b.IJ' = - 1/6.
:. r2 = (- 3/2) (-1/6) = - (1/4)
= - 0.25 ⇒ r = - 0.5
(·:bzy<O,so r <0). /
-
- W• (x:l:y)-(f :1:Ji)
:. Y - ji = m (x - x).
Equati~n of the line of regression of
x on y 1s : 4x + y - 5 O
~
141
l 15(A)·. y. mx+c::) Y=mx+c
t1HB): Equati~n of the ltne of regression of
y on x 1s : 3x + 4y - 7 = o
Now r2
I
o. 64
hYIC - (219) (219)
r = (2/3) as byx > 0.
y-y=bxy (x-x)
20 X 100 - I 10 X 70
= 20 X 2000-(70) 2 =- 0.1 6•
X
~ r = ± .J4;9 = ± (213)
Ox Oy
Regression line X on Y is :
X - 5.5 = - 0.16 (Y- 3.5)
⇒ X= 6.06 - 0.16Y.
When Y = 4, then X = 6.06 _
= 5.42.
xy
X Oy
~
(I_ ,2)2
_
(219) x + (6/9)
)' _. (2/9) X + (2/3)
Equation of the line of regre ssion
. of
2y I
Regression coefficient of
.
b = 2.
x ony is
x cry
sm 2 0
(I _ ,2 )2
nl:/-(I:/)
Also,
0
{1/ ..fj ).
x on y ls : x
cr 2
(l-,2)2
Now
. Regression coefficient of
Is hyx "' (2/9).
y on x
I
[at
+ ]2
--t
,2
(I-,2)2
= - (2/3).
I02(A): b = n.Exy-(..tx)(I: y)
i e.
cosec 2 0 = I + co12 8
= 1+
+ (4/3)
(sign of r 1s to be taken negative as
both b-cy and byx are negative).
or y
CJ X
:.
... iii•x :t Y
b>•x
(·: r > 0 as b
•Y > 0)
)' equation Of the line or regres
110(.4 · .Y on x Is : 9y 2x + 6
Slon or
.I
0
X
::= ((II ✓
3) X (1/ ✓3))'12
)
1-,2)
tan0 =± (- , - ~
+ a:
⇒
=-(1/ ✓3).
), I.et" .
:. r:::::. Jbxy
105(B): Putting y - 3 in the line of
x on y, i.e., 3x+y 15 w regressj
, e get, o~
106(A): We know
·-t"'4
Now r =Jbxyx hyx
xy
1~
Cov (X, Y)
~
⇒
~'.n10: sJopc of regression line .,., on .t , ~
---.__
Ht
,"''
hyx • tan Joo {I/ .,/j '
ore ••
'
)
sJope of regression line .t on Y
I IJ(B . .._, are ; , : : ~• coblddua or
"" (J lb,y) = tan 600 "' ,fj
X+u
"'
.,,u • x-yancf
• x :l:y
::J b,y (1/ ✓
3).
4
, __ Cov (X, Y) _ Cov ()(
⇒ r = (-✓
312) = - 0.866
(:. byX'bXY <0sor<0)
101(8): 4X + 2y = 1 ⇒ X = - (1/2) )' + (1/4)
= -J(-2/3)x(12)
. L et r
1-
Jlney on x:
,
y- 85 = 0.487 (x - 36)
⇒ y = 0.48x + 67.72.
IO0(A): Let the lines of regression of Y on X
and X on Y be x + 2y = 5 and
2x + 3y =8 respectively. Then
br.r= - (112) and bxr = - (3/2).
r 2 = brx· bxr =(3/4) ⇒ r2 = (3/4)
⇒ b,'X
'
correlation coefficient b I be
new values of X and y ,,::t"'cch
+ (116)
0.66x8
X
(1 y (1
t
' •11e11 '' I~
=~=---ii-= 0.48.
bxy = - (1/2).
2x + 3y = 4 :⇒ )' = - (2/3)
Cov (X Y)
y
,
we get :
• cz ai
Cov (u, v) = (1/n)
r (u - ii) (v - v)
.
Common Proflc1enc
5.31
Test: Quantitative Aptitude (S
la!it.
Ex= 20, Ly= 45,
E xy == 220, L x 2 = 100.
Ey2 = 485, n = 5
J22(C): Here
=
-
•c :t (x - x)(y ,re co,· (x, y)
(1/11)
fl
Cov (u, ,,)
ac Cov (:c, y)
=
= Cov (x, y)
x~(~
~.v)i
I
5 x 220 - 20 x 45
a <1x X c <1y
= p (x,y)
yon x
·
117(8): Slope of line of regression
= b = m1 (say)
'>
Slope of line of regression x on y
I
=-=m2 (say)
bXl'
Let 0 be the angle between them, then
b -(11 bw)
-
T/11 - "'2
= ✓100
123(D): Here L x = 30, Ly= 15,
Exy = 94, Ex2= 190
"I: xy- (I: x) (:t Y)
~
2
'C'
II ,1., X -(:t x)2
bxy
x (1 I b,y)
5x94-30x15
5 X 190 -(30)2
bxy + b;-,,·
e"' tan-•
b)'X bXJ' - ] )
~~-
(
bXJ'
470-450
950 - 900
.
+ bJX
I= f
(given). Smee the two
lines of regress10n pass through
(X, Y), therefore.
118(A): Here
⇒
b
-
X=-=Y
1-a
119(A): Here b1 , = 1
and bx,,=
1
4 (·:
⇒
b
X=--=Y.
-
1-a
( ·: y
= x)
:c=(1!4)y+(3l4)
Now r= ✓b..n· xbyx
=
R
10
= 50 = 0.2.
I
2
120(A): See Text.
12l(B): Here :I: xy = 140, Ex= 25,
l:y = 30, n = 5
=_!_ [300 -(52 X 64) / 11)
II
3300-3328
= - 0.23.
121
126(8): The mean
x, y
1s the pont of
intersection of these lines. Solve the
equations to get : x = 5, ji = (1/3).
⇒
⇒
y = (8/)0)
X
+ (66/10)
= (4/5). Sunilarly,
= (18/40) )' + (214/48)
bxy = (9/20)
b,,x
X
25
=-
C:E x) (t J')}
127(A): 10y =Bx+ 66
5 X 140 - 25 X 30
50
= - 25
(l/11)
⇒
2•
r
= ✓byx
X
b_.,.
"" 0 .64 x o.83
-b
X
138(B):
= o.sl3.
3
b-3b=]__
n==y.,
2 ⇒ bxb
yx- 'xy
2
So, the
5
l
=- y +3
3
is the regression equation of X on y
and the equation y = 2t - 4 1s that of
= 3X -
5 ~
=3
⇒ p (X , Y)
⇒ {p {X, Y))·
= mJ • (·:
:. b
y.t
:::3
byx and
(A): Sy= 4x + 30
5
4
b}.t = -.
5
~
20x = 9y + 107
⇒
X "'
r2
9
107
9
20 y + 20 => b'I)' = 20 .
=b
-')'
== 0.36.
x b
yx
==
~x~=~
✓0.36 = - 0.6.
100
= 0.9
140(B): r2 == b"J' x by,= 1.5 x 0.6
Coefficient of allienation
p
= ✓I -0.9 == 0.32.
14l(D): Since the two lines of regression
therefore,
always intersect at (
x. _v ).
2:x + 9y- I79=0.
=-
5
20
4x=Sy-137=0
13S(D): In the given data b,., = - 0.!J is
negative and b,y = 0.4 is positive
which 1s not possible as bn and b,,
are always of same sign. Hence the
given data is i~consistent.
136(C): {p(X, Y)}2=b<>x by_,=(- 0.4)(-0.9)
2
⇒ y = ix+ 6
bn. are positive, therefore, p (X, Y) >
132(A): Cov (X, Y), b,,. b,y ,and p (X, Y) all
are either positive, or O or negative
s1mutaneously.
133(0): See Text.
134(D): See Text.
~.
,'I)'
0)
⇒ p (X, Y)
=.!.2
= 0.75.
139
X
== = 2 and b~.
Y on X. So, bn
··• 3
, 2
2
⇒ by_, x b.'CJ'
y=.!_x +25
2
Coefficient of determination
= r2 = bp x b,,, = (1/2) x (3/2) = (3/4)
If we take y = 3x - 5 as regression
1Jl(M equation of Y on X and y = 2x _ 4 as
that of X on Y, then
J
X-
l(
2
== 1 - r2 = 1 - (0.14)2 = 0.02
. Standard error of estimate of x on y
'
112
j.iO(C)• ~ <1x (1 - r2) = 11 X [I - (0.6)2]1'2
~ I I x 0.8 = 8.8.
not possible,
.6 = 0.9 < 1.
SO => 2y = X + 50
•
_yi
SitniJarJy, x = ~ Y - 5 => b =
coefficient of non-determination
.
O
b b =15
~
b
J"'
f{ere r - XJ'
== 0 24 x 0.58 = 0,14.
1l9(JJ):
which in
equation
t
:)I
I
125(A): Cov (x,y)
= (1/11) p: xy -
:s1.JJ)=
1
50 "'0.4.
5 X 107 - 15 X 35
5 X 55 - (15) 2
bxy
·-=±-
= ± 0.5.
r = 0.5 as both b_9 • and byx are positive
C ov ( x,y )
6 :::=4 •
20
124(D): Here Ex= 15, E y = 35,
.Exy= 107,Ex2 =55
V = aX + b ⇒ X =aX + b (': X = Y)
-
-( 4S)l
✓40Q = 10 X iQ "" J.
X
b."CJ' bp. -1
~
200
lfere, P OC. Y) is.negative u b b
pt "7
p (X, Y) all ha
ve the same sign.
137(1)): Note
alterua~t b,. b..,, :S 1. Out of given
tives only (D) is such Chat
. ·
>'
-;::;, f{icient
o fd e t ernunation
coe
"" ,:i == bxy x b}'X
..1
5 (485)
X
1100 - 900
--
.I)'
1 + b1"
tan 9 - 1 + m1 m2
= ✓5 (100) - (20)
2
== (6/IOf.
:::> (4 /5)=(6/I0)x(cr>'/3)
•. r - ~ )
-----...___
[✓ n I: x- - (.t x) 2
p(u,v)-Jvar(u) xJVarM
.fi_4 I 5) x (9 I 20)
rJoW yx
" I: xy - (.t x) (l:
-
.
:=
b ==(r<J,l<Jx)
Solve these two equations to get : x
and jias 13 and 17.
142(B): Since p2 $ I, therefore, byx b-'>. :St.
143(A): The two lines of regression are
••• (1)
X + 4y = 3
••• (2)
and 3x + y = 15
If we take (I) as the regression
equation ofY on X, then (2) is that of
X on Y. These two equations can be
written as :
y
=-
1
3
I
15
4 x + 4 and x = - 3Y + 3
common Proficiency Tes :
MD
JSO(C):
1
__ .!..
~ b)T = - -4 and bx,· - 3
x,·
12
is valid.
To find x, wheny =3, wer are to use
the regression t>quation of X on Y.
1
3
144(D): Given data is inconsistent ~s b,.x < 0
and b > o. For any bivariate data
both
and b should be of same
6p
X1
sign or both should be 0.
5(C): If 4x _ 5y + 33 = 0 is the regression
14
equation of yon X, then b,, = Slope
= -4
5
1
and the = slope of the other
bxy
-line =
2
9
⇒ bxy =
9
36
10 > I.
So, our selection is not correct. The
regression equation of Y on X is
1x- 9y • 127 = 0 and hence
byx = slope = 2/9.
I 46(D): Though p2 = b,x x b_ry, therefore,
⇒
Thus,
if
if
IpI= Jbyx bxy
b,, >0
b,, <0
is not correct.
147{B): Regression line of Y on X is
y-
y = byx (x - x)
Clearly, slope of this line = byx·
148(A): Solving the lines, we get x = 130,
y = 90 as their point of intersection,
which is their means (x, y) .
x = 130, y = 90.
rcry
149(B): b = Y-'
cr-'
⇒
-3
(- ✓3)
-=
4
4
=> cr-' = fj
or o/
,
'
rines
,_::I
5x 46.'ji,.So
Solving then, we get : x "" 1 , ' I
I
and
, y"'
I52(A): 4y- 5x = 15 ⇒ Y == (514) X4 'I '
.. b. = 5/4 = 1.25. Also ...,
(Is,~
>·•
r"" b ')
⇒ (0.75) 2 = bry x 1.25 ~ b -t> ~ b
t53(C): The two lines of regressio -9""
n are
.
5x + 3y = 55
and 7x + y = 45
··· (1)
If we take hne ( l) as the
··· (?j
equation of Y on X, then (i;e&ressi0ii
151
X on Y. We can wnte these as.hat 0r
5
55
.
y= --x+-
2
- - x2
crx
= 16/3.
3
1
45
= - 7Y +7
-5
=3
=-)jx~=-Ji.
156(B): r2 = b_-ry X byx = 0.64.
Coefficient of Determination = ,i
=0.64.
157{B): Coefficient determination :
r2 = 0.9025 ⇒ r = 0.95.
158{B): r2 = byx X b-'Y = 0.9025
⇒ 0.39 x b-ry = 0.9025
bxy = 0.9025/0.3 9 = 2.314.
159(B): k2 = 1 - r2 ⇒ 0.502 = I - r1⇒ r2 = 0.498 ⇒ r = 0.11.
160(A): k 2 = 0.502 ⇒ k
161(A): k = 0.4 ⇒
⇒
= 0.11.
k2 = 0.16
1-r2 = 0.16.
I"" 1-0.16=0.84
,-;:.-;;;
,::;
161(C>:, •
✓ 0,84
r""
= 0.9170
(l-,2)1'2
CJ
,-ch ♦
61'-c>: Sr% ""o.~ (1 - o.16)"2 =0.6 x .Joii.
84
1
""o.6 x o.917 = o.ss.
· ·
2
.,, Ox (1-r2)1'
63(JJ): S,cv ""4 x (I - I)= 0.
1
cry
ra
,::; , - ⇒ (JJC =--2'..
6'(1'):
bxy
1
<J x
-
byx
== erX (1 - r)112 == o•
<Jx
b "" r 165(,.\):
11
⇒ 0. 6 = (0.5)
x
<J Y
-::::1 <J x ==
:)I-
⇒
cr
-..!.
OY
a,= 3a, = 3 x 4 = 12.
850
(I - r)IIZ
= 3.12 X (1- 0.25)112
CJ X
6(/\): Since the two lines of regression meet
in (x, y) . therefore, the point (x, ji)
always lies on the two regression
Jines.
(0): The two lines of regression coincide
167
if pz = I, le., if p = - 1 or I.
(A) (130, 90)
(C) (8~, 120)
(B) (90, 130)
(D) (120, 80).
2. The equations of two regression lines
are as follows :
3X + 2Y - 26 = 0; 6X + Y - 31 = 0.
The regression coefficients; (b,y, byx) are:
(A) (- 1/6, - 3/2) (B) (1/6, 3/2)
(C) (3/2, - 1/6)
(D) (-- 3/2, J/6).
3. You are given the following data for the
variables x and y. x = 36, y = 85,
cr = J I cr = 8 and r = 0.66. The value of
X
> y
x when y = 7 5, is :
=-3.3.
t 70(D): Given bxy = 0.25 = .!_
<l
4' Also byx bxy⇒
1
-4 byx <- I
⇒ byx~4.
Also, by, and b,y are of same sign,
therefore b > 0 i.e., O < b S 4.
QUESTION BANK - 6
MULTIPLE CHOICE QUESTION
Select the correct alternative out of the
given ones :
t. For the regression lines : 2y-x - 50 = 0,
3y - 2x - IO = 0, the mean values
(x, y) is :
_ 30x400
10
196- (J0)2
10
= 3. J2 X 0.87 = 2.7}.
16
ax
169(A): Here,b = Ixy-(txty)/n
P
Ix2-(tx)2 /n
(0.6 x 2.6) I 0.5 = 3.12
:. S,cv ==
0/3.
Nowfrom(l)b,..=3 ⇒ pa, =3
3
S ,CY
=> b*•3
Also the
- (I)
rear-ion
Yis x' = 1/3
l equation of X OD
:. b = 1/3·
F z,
,
-{l)
0
~ (1) ~ (2), we get:
,. bxy - I ⇒ p2 = 1 ⇒ p = 1
12
-Jx-=4-
respectively.
and bxy ::: .:2_
7
p = • .Jbyx bxy (·: bY.t < 0)
Hence, byx
⇒
3
154(C): When the two• lines are at right anges
1
then there 1s no lmear correl . '
.
ation
between
.
_ the two Imes of regres s1on,
z.e., p- 0.
lSS(C): If p (X, Y) = 1, then the two lines 0I
regression coincide.
!pl =.Jbyxbxr
-{ Jb,, b')
p - -Jb,. b,y
2x + 3y = -
and x
2•
This would mean byx x b_"> =
✓J - p
or
O~s
. x=--x3+5=4.
••
CJJ'
t51(A): (x, y) lies on the given
I < l. So, our choice
b xb =r,
S,,.. =
2
'"
'"
(A) 25.93
(B) 26.93
(C) 27.93
(D) 24.93.
4. The following data give the correlation
coefficient, means and standard
deviation of rainfall and yield of paddy
in a certain tract :
Annual rainfall
Yield per
in cm.
hectare in kgs.
18.3
Mean
973.5
2.0
38.4
S.D.
Coefficient of correlation = 0.58.
The most likely yield of paddy when the
annual rainfall is 22 cm, other factors
being assumed to remain the same, is :
(A) 1024.7
(B) 1014.7
(C) 1114.7
(D) 914.7.