22536177 mathematics and statistics unit iv v

7/21/2019 22536177 Mathematics and Statistics Unit IV V

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BHARTHIDASAN UNIVERSITY

UNIT-IV

THEORY OF SAMPLING AND TESTING OF HYPOTHESIS

4.0 OBJECTIVES4.1 NEED FOR SAMPLING4.2 ELEMENTS OF SAMPLING PLAN4.3 TYPES OF SAMPLING

4.3.1 Rand ! " P" #a#$%$&' Sa!(%$n)S$!(%* Rand ! Sa!(%$n)S&"a&$+$*d Rand ! Sa!(%$n)S',&*!a&$- Rand ! Sa!(%$n)C% ,&*" Sa!(%$n)

4.3.2 N n/Rand ! " N n/P" #a#$%$&' Sa!(%$n)C n *n$*n-* Sa!(%$n)

J d)!*n&a% Sa!(%$n)&a Sa!(%$n)

4.4 SAMPLING AND NON/SAMPLING ERRORS4.4.1 R*a, n, + " ,a!(%$n) *"" ",4.42 R*a, n, + " n n/,a!(%$n) *"" ",

4. TESTING OF HYPOTHESIS4. .1 Sa!(%$n) D$,&"$# &$ n4. .2 S&anda"d E"" "4. .3 N %% A%&*"na&$ * H'( & *,$,4. .4 E"" ", $n &*,&$n) + '( & *,$,4 .5.5 C"$&$-a% R*)$ n4. .5 T6 &a$%*d and On* &a$%*d &*,&4. .7 La")* and S!a%% ,a!(%* &*,&

4.5 PROCEDURE FOR TESTING OF HYPOTHESIS4.7 TESTS OF SIGNIFICANCE

4.7.1 T*,& + " ,$n)%* !*an4.7.2 T*,& + " d$++*"*n-* + &6 !*an,4.7.3 T*,& + " &6 ,&anda"d d* $a&$ n,4.7.4 T*,& + " S$n)%* P" ( "&$ n4.7. T*,& + " d$++*"*n-* + &6 (" ( "&$ n,

4.8 Ana%',$, + Va"$an-*4.8.1 A,, !(&$ n,4.8.2 On* 6a' ANOVA4.8.3 A((%$-a&$ n,

4.0 Objectives

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Sa!(%$n) $, #*$n) ,*d $n " * *"9'da' %$+* 6$& & :n 6$n) a# & $&.F " *9a!(%*,;a - : &*,&, a ,!a%% < an&$&' + "$-* & ,** 6 *& *" $& a, #**n 6*%% - :*dand a )"a$n!*"- an& d *, n & *9a!$n* *a- )"a$n + 6 a& * $n&*nd, & ( "- a,*; # &

$n,(*-&, n%'a ,!a%% < an&$&' + )"a$n,. M ,& + " d*-$,$ n, a"* #a,*d n & **9a!$na&$ n + a +*6$&*!, n%'.

In a ,&a&$,&$-a% $n *,&$)a&$ n; & * $n&*"*,& , a%%' %$*, $n & * a,,*,,!*n&+ )*n*"a% !a)n$& d* and & * ,& d' + a"$a&$ n 6$& "*,(*-& & n* " ! "*

- a"a-&*"$,&$-, "*%a&$n) & $nd$ $d a%, #*% n)$n) & a )" (. T $, )" ( +$nd$ $d a%, " n$&, nd*" ,& d' $, -a%%*d ( ( %a&$ n " n$ *",*. T , $n,&a&$,&$-,; ( ( %a&$ n $, an a))"*)a&* + #=*-&, " n$&, nd*" ,& d'. T *( ( %a&$ n !a' #* +$n$&* " $n+$n$&*.

Sampling and Sample

Sa!(%$n) $, a !*& d + ,*%*-&$n) n$&, + " ana%',$, , - a,,* %d,; - n, !*",; - !(an$*, *&-. +" ! & * "*,(*-&$ * ( ( %a&$ n

nd*" ,&a&$,&$-a% $n *,&$)a&$ n. T * & * "' + ,a!(%$n) $, #a,*d n & *principle of statistical regularit . A-- "d$n) & & $, ("$n-$(%*;a ! d*"a&*%' %a")* n !#*" + $&*!, - ,*n a& "and ! +" ! a %a")* )" (a"* a%! ,& , "* n an a *"a)* & ( ,,*,, & * - a"a-&*"$,&$-, + & * %a")*")" (. A ,!a%%*,& n n/d$ $,$#%* (a"& + & * ( ( %a&$ n $, -a%%*d aunit . A n$& , %d #* 6*%% d*+$n*d

and , %d n & #* a!#$) ,. F " *9a!(%*; $+ 6* d*+$n* n$& a, a ,* %d& *n $& , %d #* d*+$n*d & a& a (*", n , %d n & #*% n) & &6 ,* %d,n " , %d $& %*a * & (*", n, #*% n)$n) & & * ( ( %a&$ n.

A +$n$&* , #,*& + a ( ( %a&$ n $, -a%%*d a ,a!(%* and & * n !#*" +n$&, $n a ,a!(%* $, -a%%*d $&, ,a!(%* ,$>*. B' ana%'>$n) & * da&a - %%*-&*d

+" ! & * ,a!(%* n* -an d"a6 $n+*"*n-* a# & & * ( ( %a&$ n nd*" ,& d'.

!arameter and Statistic

T * ,&a&$,&$-a% - n,&an&, + a ( ( %a&$ n %$:* !*an ?!@; a"$an-* ?, 2@;and (" ( "&$ n ?P@ a"* &*"!*d a, parameters. S&a&$,&$-a%measures li"e!*an ? x@; a"$an-* ?, 2@; (" ( "&$ n ?(@ - !( &*d +" ! & * ,a!(%*d

#,*" a&$ n, a"* :n 6n a, statistics . Sa!(%$n) $, *!(% '*d & & " 6 %$) &n & * ( ( %a&$ n (a"a!*&*". A statistic $, an *,&$!a&* #a,*d n ,a!(%*

da&a & d"a6 $n+*"*n-* a# & & * ( ( %a&$ n (a"a!*&*".



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4.# N$$% &O' S()!*IN+S (( ,* & a& & * "a6 !a&*"$a%, d*(a"&!*n& $n a - !(an' "*-*$ *,

$&*!, $n % &, and $,, *, & *! & & * (" d -&$ n d*(a"&!*n& a, and 6 *n"*< $"*d. B*+ "* a--*(&$n) & *,* $&*!,; & * $n,(*-&$ n d*(a"&!*n& $n,(*-&,

" &*,&, & *! & !a:* , "* & a& & *' !**& & * "*< $"*d ,(*-$+$-a&$ n,. T ,

?$@ $& - %d $n,(*-& a%% $&*!, $n & * % & "

(ii) $& - %d &a:* a ,a!(%* and $n,(*-& & * ,a!(%* + " d*+*-&$ *,S&a&$,&$-, + " Mana)*", and & *n *,&$!a&* & * & &a% n !#*" +d*+*-&$ *, + " & * ( ( %a&$ n a, a 6 %*.

T * +$",& a((" a- $, -a%%*d complete enumeration ,census . It as t/omajor d$,ad an&a)*, na!*%'; & * &$!* - n, !*d and & * - ,& $n % *d $n $&.

T * ,*- nd a((" a- & a& ,*, ,a!(%$n) a, &6 !a= " ad an&a)*,.

?$@ I& $, ,$)n$+$-an&%' %*,, *9(*n,$ *.

?$$@ I& &a:*, %*a,& ( ,,$#%* &$!* 6$& #*,& ( ,,$#%* "*, %&,.

T *"* a"* ,$& a&$ n, & a& $n % * d*,&" -&$ n (" -*d "* 6 *"* ,a!(%$n) $,& * n%' an,6*". A 6*%%/d*,$)n*d ,&a&$,&$-a% ,a!(%$n) !*& d % )' 6 %d)$ * a-- "a&* "*, %&, and a& & * ,a!* &$!* 6$%% "*, %& $n - ,& "*d -&$ n and%*a,& &$!*. T , ,a!(%$n) $, & * #*,& a a$%a#%* & % & d*-$,$ n !a:*",.

4. $*$)$NTS O& S()!*IN+ !*(N

T * !a$n ,&*(, $n % *d $n & * (%ann$n) and *9*- &$ n + ,a!(%* , " *' a"*

I) Objectives T * +$",& &a,: $, & %a' d 6n $n - n-"*&* &*"!, & * #a,$-#=*-&$ *, + & * , " *'. Fa$% "* & d*+$n* & * #=*-&$ *?,@ 6$%%

-%*a"%' nd*"!$n* & * ( "( ,* + -a""'$n) & & * , " *' $&,*%+. F "*9a!(%*; $n a na&$ na%$>*d #an: 6an&, & ,& d' ,a $n), #an:a-- n& %d*", (*"-*(&$ n + & * ,*" $-* < a%$&' "*nd*"*d *" a(*"$ d + n* '*a"; & * #=*-&$ * + & * ,a!(%$n) $,; *"*; & ana%'>*& * (*"-*(&$ n + & * a-- n& %d*", $n & * #an:.



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ii) !opulation to be covered Ba,*d n & * #=*-&$ *, + & * , " *';& * ( ( %a&$ n , %d #* 6*%% d*+$n*d. T * - a"a-&*"$,&$-,- n-*"n$n) & * ( ( %a&$ n nd*" ,& d' , %d a%, #* -%*a"%'d*+$n*d. F " *9a!(%*; & ana%'>* & * (*"-*(&$ n + & * ,a $n), #an:a-- n& %d*", a# & & * ,*" $-* "*nd*"*d #' & * #an:; a%% & *

a-- n& %d*", $n & * #an: - n,&$& &* & * ( ( %a&$ n & #*$n *,&$)a&*d.

iii) Sampling frame In "d*" & - *" & * ( ( %a&$ n d*-$d*d ( n;& *"* , %d #* , !* %$,&; !a( " & *" a--*(&a#%* !a&*"$a% ?-a%%*d& *frame @ 6 $- ,*" *, a, a ) $d* & & * ( ( %a&$ n & #* - *"*d.

T * %$,& " !a( ! ,& #* *9a!$n*d & #* , "* & a& $& $, "*a, na#%'+"** +" ! d*+*-&,. T * ,a!(%$n) +"a!* 6$%% *%( , $n & * ,*%*-&$ n

+ ,a!(%*. A%% & * a-- n& n !#*", + & * ,a $n), #an: a-- n&%d*", $n & * #an: a"* & * ,a!(%$n) +"a!* $n & * ana%',$, +

(*"-*(&$ n + & * - ,& !*", "*)a"d$n) & * ,*" $-* "*nd*"*d #' & *

#an:.iv) Sampling Unit F " & * ( "( ,* + ,a!(%* ,*%*-&$ n; & * ( ( %a&$ n

, %d #* -a(a#%* + #*$n) d$ $d*d ( $n& ,a!(%$n) n$&,. T *d$ $,$ n + & * ( ( %a&$ n $n& ,a!(%$n) n$&, , %d #*

na!#$) ,. E *"' *%*!*n& + & * ( ( %a&$ n , %d #*% n) & = ,&n* ,a!(%$n) n$&. Ea- a-- n& %d*" + & * ,a $n), #an: a-- n&

$n & * #an:; + "! a n$& + & * ,a!(%* a, a%% & * ,a $n), #an:a-- n& %d*", $n & * #an: - n,&$& &* & * ( ( %a&$ n.

v) Sample Selection T * ,$>* + & * ,a!(%* and & * !ann*" +

,*%*-&$n) & * ,a!(%* , %d #* d*+$n*d #a,*d n & * #=*-&$ *, +& * ,&a&$,&$-a% $n *,&$)a&$ n. T * *,&$!a&$ n + ( ( %a&$ n (a"a!*&*"a% n) with & *$" !a")$n + n-*"&a$n&' a"* , !* + & * $!( "&an&a,(--&, & #* + %% 6*d $n ,a!(%* ,*%*-&$ n.

vi) 1ollection of data T * !*& d + - %%*-&$n) & * $n+ "!a&$ n a, * d*-$d*d; :**($n) $n $*6 & * - ,&, $n % *d and & * a-- "a-'a$!*d a&. P ',$-a% #,*" a&$ n; $n&-" *6$n) "*,( nd*n&, and- %%*-&$n) da&a & " ) !a$% a"* , !* +& * !*& d, & a& -an #*+ %% 6*d $n - %%*-&$ n + da&a.

vii) (nal sis of data T * - %%*-&*d da&a , %d #* (" (*"%' -%a,,$+$*dand , #=*-&*d & an a((" ("$a&* ana%',$,. T * - n-% ,$ n, a"*d"a6n #a,*d n & * "*, %&, + & * ana%',$,.

4.2 T pes of Sampling



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T * &*- n$< * + ,*%*-&$n) a ,a!(%* +" ! a ( ( %a&$ n , a%%' d*(*nd, n& * nature of & * da&a and t e t pe of en3uir . T e procedure of,a!(%$n) ma #* #" ad%' -%a,,$+$*d nd*" & * + %% 6$n) *ad,

1) !robabilit sampling or random sampling and

Non-probabilit sampling or non-random sampling.

4.2.# !robabilit sampling

S&a&$,&$-, + " Mana)*", Probability sampling $, a !*& d + ,a!(%$n) & a&*n, "*, & a& * *"' n$& $n & * ( ( %a&$ n a, a :n 6n non- ero c ance +#*$n) $n-% d*d $n & * ,a!(%*.

T * d$++*"*n& !*& d, + "and ! ,a!(%$n) a"*

,a Simple 'andom Sampling ,S'S

Simple random sampling $, & * + nda&$ n + (" #a#$%$&' ,a!(%$n). I& $, a,(*-$a% -a,* + (" #a#$%$&' ,a!(%$n) $n 6 $- * *"' n$& $n & * ( ( %a&$ n a,an *< a% - an-* + #*$n) $n-% d*d $n a ,a!(%*. S$!(%* "and ! ,a!(%$n) a%,!a:*, & * ,*%*-&$ n + * *"' ( ,,$#%* - !#$na&$ n + & * d*,$"*d n !#*" +

n$&, *< a%%' %$:*%'. Sa!(%$n) !a' #* d n* 6$& " 6$& & "*(%a-*!*n&. I&!a' #* n &*d & a& 6 *n & * ,a!(%$n) $, 6$& "*(%a-*!*n&; & * n$&, d"a6na"* "*(%a-*d #*+ "* & * n*9& ,*%*-&$ n $, !ad*. T * ( ( %a&$ n ,$>* "*!a$n,- n,&an& 6 *n & * ,a!(%$n) $, 6$& "*(%a-*!*n&. I+ n* 6an&, & ,*%*-& n

n$&, +" ! a ( ( %a&$ n + ,$>* N 6$& & "*(%a-*!*n&; & *n * *"' ( ,,$#%*,*%*-&$ n + n n$&, ! ,& a * & * ,a!* (" #a#$%$&'. T , & *"* a"* NC n( ,,$#%* 6a', & ($-: ( n n$&, +" ! & * ( ( %a&$ n + ,$>* N. S$!(%*"and ! ,a!(%$n) ) a"an&**, & a& a ,a!(%* + n n$&, a, & * ,a!*(" #a#$%$&'1NCn + #*$n) ,*%*-&*d.

$5ample

A #an: 6an&, & ,& d' & * Sa $n), Ban: a-- n& %d*", (*"-*(&$ n +& * ,*" $-* < a%$&' "*nd*"*d *" a (*"$ d + n* '*a". T * #an: a, &("*(a"* a - !(%*&* %$,& + ,a $n), #an: a-- n& %d*",; -a%%*d a, ,a!(%$n)+"a!*; ,a' 00. N 6 & * (" -*,, $n % *, ,*%*-&$n) a ,a!(%* + O & + 00

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and $n&*" $*6$n) & *!. T $, - %d #* a- $* *d $n !an' 6a',. T6 - !! n6a', a"*

,# Lottery method : S*%*-& 0 ,%$(, +" ! a # 9 - n&a$n$n) 6*%%, ++%*d 00 ,%$(, + a-- n& n !#*", 6$& & "*(%a-*!*n&. T $, !*& d -an

#* a((%$*d 6 *n & * ( ( %a&$ n $, ,!a%% *n ) & and%*., Random numbers method: *n & * ( ( %a&$ n ,$>* $, *"'

%a")*; & * ! ,& ("a-&$-a% and $n*9(*n,$ * !*& d + ,*%*-&$n) a ,$!(%*"and ! ,a!(%* $, #' ,$n) & * "and ! n !#*" &a#%*,.

,b Stratified 'andom Sampling

Stratified sampling $, a &6 /,&*( (" -*,, $n 6 $- & * ( ( %a&$ n $,(a"&$&$ n*d $n& , #/( ( %a&$ n,; " ,&"a&a. T * ,&"a&a , %d #* ! & a%%'*9-% ,$ * and - %%*-&$ *%' *9 a ,&$ * $n & a& * *"' ( ( %a&$ n *%*!*n&, %d #* a,,$)n*d & n* and n%' n* ,&"a& ! and n ( ( %a&$ n *%*!*n&,, %d #* !$&&*d. N*9&; *%*!*n&, a"* ,*%*-&*d +" ! *a- ,&"a& ! #' a"and ! (" -*d "*; , a%%' SRS. T*- n$-a%%'; n%' SRS , %d #* *!(% '*d $n,*%*-&$n) & * *%*!*n&, +" ! *a- ,&"a& !. In ("a-&$-*; , !*&$!*, ,',&*!a&$-,a!(%$n) and & *" (" #a#$%$&' ,a!(%$n) (" -*d "*, a"* *!(% '*d. S&"a&$+$*d,a!(%$n) d$++*", +" ! < &a ,a!(%$n) $n & a& & * ,a!(%* *%*!*n&, a"*,*%*-&*d (" #a#$%$,&$-a%%' "a& *" & an #a,*d n - n *n$*n-* " = d)!*n&. A!a= " #=*-&$ * + ,&"a&$+$*d ,a!(%$n) $, & $n-"*a,* ("*-$,$ n 6$& &$n-"*a,$n) - ,&.

T * a"$a#%*, ,*d & (a"&$&$ n & * ( ( %a&$ n $n& ,&"a&a a"* "*+*""*d& a, T * "' + Sa!(%$n) and ,&"a&$+$-a&$ n a"$a#%*,. T * -"$&*"$a + " & *,*%*-&$ n + & *,* a"$a#%*, - n,$,& + T*,&$n) + H'( & *,$, ! )*n*$&';

*&*" )*n*$&'; "*%a&*dn*,,; and - ,&. T * *%*!*n&, 6$& $n a ,&"a& ! , %d#* a, ! )*n* , a, ( ,,$#%*; # & & * *%*!*n&, $n d$++*"*n& ,&"a&a , %d#* a, *&*" )*n* , a, ( ,,$#%*. T * ,&"a&$+$-a&$ n a"$a#%*, , %d a%, #*-% ,*%' "*%a&*d && * - a"a-&*"$,&$- + $n&*"*,&. T * ! "* -% ,*%' & *,* -"$&*"$a a"* !*&; & *)"*a&*" & **++*-&$ *n*,, $n - n&" %%$n) *9&"an* , ,a!(%$n) a"$a&$ n. F$na%%'; & *

a"$a#%*, , %dd*-"*a,* & * - ,& + & * ,&"a&$+$-a&$ n (" -*,, #' #*$n) *a,' & !*a, "* anda((%'.

?-@S stematic 'andom Sampling

In systematic random sampling, & * ,a!(%* $, - ,*n #' ,*%*-&$n)a "and !,&a"&$n) ( $n& and & *n ($-:$n) * *"' $ & *%*!*n& $n , --*,,$ n +" ! & *,a!(%$n) +"a!*.

T * ,a!(%$n) $n&*" a%; i, $, d*&*"!$n*d #' d$ $d$n) & * ( ( %a&$ n ,$>* N #'



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& * ,a!(%*,$>* n and " nd$n) & & * n*a"*,& $n&*)*". F " *9a!(%*; & *"* a"* 100;000*%*!*n&, $n& * ( ( %a&$ n and a ,a!(%* + 1;000 $, d*,$"*d. In & $, -a,*; & * ,a!(%$n)$n&*" a%.i, $,

100. A "and ! n !#*" #*&6**n I and 100 $, ,*%*-&*d. I+; + " *9a!(%*; & $,n !#*" $,23; & * ,a!(%* - n,$,&, + *%*!*n&, 23; 123;223;323;423; 23; and , n.

Systematic sampling $, ,$!$%a" & SRS $n & a& *a- ( ( %a&$ n*%*!*n& a, a:n 6n and *< a% (" #a#$%$&' + ,*%*-&$ n. H 6* *"; $& $, d$++*"*n& +" ! SRS $n& a& n%'& * (*"!$,,$#%* ,a!(%* ,$>* n & a& -an #* d"a6n a, a :n 6n and *< a%(" #a#$%$&' +,*%*-&$ n. T * !*a, "$n) ,a!(%* + ,$>* n a, a >*" (" #a#$%$&' + #*$n)

,*%*-&*d.

F " systematic sampling, 6* a,, !* & a& & * ( ( %a&$ n *%*!*n&, a"*"d*"*d $n

, !* "*,(*-&. In , !* -a,*,; & * "d*"$n) $, n"*%a&*d & & * - a"a-&*"$,&$-+ $n&*"*,&.

S',&*!a&$- ,a!(%$n) $, a - n *n$*n& 6a' + ,*%*-&$n) a ,a!(%*. I& "*< $"*,%*,, &$!* and- ,& 6 *n - !(a"*d & ,$!(%* "and ! ,a!(%$n).

,d 1luster 'andom Sampling In cluster sampling, & * &a")*& population $, +$",& d$ $d*d $n&

! & a%%' *9-% ,$ *and - %%*-&$ *%' *9 a ,&$ * , #( ( %a&$ n; " -% ,&*",. T *n a "and !,a!(%* + -% ,&*",,*%*-&*d; #a,*d n a (" #a#$%$&' ,a!(%$n) &*- n$< * , - a, SRS. F " *a-,*%*-&*d-% ,&*"; *$& *" a%% & * *%*!*n&, a"* $n-% d*d $n & * ,a!(%* " a ,a!(%* +*%*!*n&, $,d"a6n (" #a#$%$,&$-a%%'. I+ a%% & * *%*!*n&, $n *a- ,*%*-&*d -% ,&*" a"*

$n-% d*d $n & *,a!(%* & * (" -*d "* $, -a%%*d one-stage cluster sampling. I+ a ,a!(%* +*%*!*n&, $,d"a6n (" #a#$%$,&$-a%%' +" ! *a- ,*%*-&*d -% ,&*"; & * (" -*d "* $, two-stage clustersampling. F "& *"! "*; a -% ,&*" ,a!(%* -an a * ! %&$(%* ?! "* & an &6 @,&a)*,; a,$n ! %&$,&a)* -% ,&*" ,a!(%$n).



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T * d$,&$n-&$ n #*&6**n -% ,&*" ,a!(%$n) and ,&"a&$+$*d ,a!(%$n) $, & a&$n -% ,&*",a!(%$n); n%' ,a!(%* + , #( ( %a&$ n, ?-% ,&*",@ $, - ,*n; 6 *"*a, $n,&"a&$+$*d,a!(%$n); a%% & * , #( ( %a&$ n,?,&"a&a@ a"* ,*%*-&*d + " + "& *" ,a!(%$n).

4.2. Non-!robabilit Sampling

T * + nda!*n&a% d$++*"*n-* #*&6**n (" #a#$%$&' ,a!(%$n) and n n/(" #a#$%$&',a!(%$n) $, & a& $n n n/(" #a#$%$&' ,a!(%$n) (" -*d "*; & * ,*%*-&$ n + & *,a!(%* n$&,d *, n & *n, "* a :n 6n - an-* & & * n$&, #*$n) ,*%*-&*d. In & *" 6 "d,& * n$&, a"*,*%*-&*d 6$& & ,$n) & * principle of probabilit . E *n & ) & * n n/(" #a#$%$&'

,a!(%$n) a, ad an&a)*, , - a, "*d -*d - ,&; ,(**d and - n *n$*n-* $n$!(%*!*n&a&$ n; $& %a-:, a-- "a-' $n $*6 + & * ,*%*-&$ n #$a,. N n/(" #a#$%$&' ,a!(%$n) $, , $&a#%* + " ($% & ,& d$*, and *9(% "a& "' "*,*a"- .

T * !*& d, + n n/"and ! ,a!(%$n) a"*

,a 1onvenience sampling6

Convenience sampling a&&*!(&, & #&a$n a ,a!(%* + - n *n$*n&*%*!*n&,. T * ,*%*-&$ n + ,a!(%$n) n$&, $, %*+& ("$!a"$%' & & * $n&*" $*6*".O+&*n; "*,( nd*n&, a"* ,*%*-&*d #*-a ,* & *' a((*n & #* $n & * "$) &(%a-* a& & * "$) & &$!*. E9a!(%*, + - n *n$*n-* ,a!(%$n) $n-% d*

?1@ U,* + ,& d*n&,; - "- )" (,; and !*!#*", + , -$a% ")an$>a&$ n,;

?2@ !a%%/$n&*"-*(& $n&*" $*6, 6$& & < a%$+'$n) & * "*,( nd*n&,;

?3@ D*(a"&!*n& ,& "*, ,$n) - a")* a-- n& %$,&,.

Convenience sampling $, & * %*a,& *9(*n,$ * and %*a,& &$!* consuming +a%% ,a!(%$n) &*- n$< *,. T * ,a!(%$n) n$&, a"* a--*,,$#%*; *a,' & !*a, "*;

and - (*"a&$ *.

,b 7udgmental sampling6

Judgmental sampling $, a + "! + - n *n$*n-* ,a!(%$n) $n 6 $- & *( ( %a&$ n *%*!*n&, a"* ,*%*-&*d #a,*d n & * = d)!*n& + & * "*,*a"- *".

T * "*,*a"- *"; *9*"-$,$n) = d)!*n& " *9(*"&$,*; - ,*, & * *%*!*n&, &



9/61


#* $n-% d*d $n & * ,a!(%*. B*-a ,* * " , * #*%$* *, & a& & *' a"*"*("*,*n&a&$ * + & * ( ( %a&$ n + $n&*"*,& " a"* & *"6$,* a((" ("$a&*.C !! n *9a!(%*, + = d)!*n&a% ,a!(%$n) $n-% d*

?1@ T*,& !a":*&, ,*%*-&*d & d*&*"!$n* & * ( &*n&$a% + a n*6 (" d -&;

?2@ P "- a,* *n)$n**", ,*%*-&*d $n $nd ,&"$a% !a":*&$n) "*,*a"- #*-a ,*they a"* - n,$d*"*d & #* "*("*,*n&a&$ * + & * - !(an';

?3@ E9(*"& 6$&n*,,*, ,*d $n - "&.

,c 8uota sampling6

T $, $, a "*,&"$-&*d &'(* + = d)!*n& ,a!(%$n). T $, - n,$,&, $n,(*-$+'$n) < &a,

+ & * ,a!(%*, & #* d"a6n +" ! d$++*"*n& )" (, and & *n d"a6$n) & *"*< $"*d ,a!(%*, +" ! & *,* )" (, #' = d)!*n&a% ,a!(%$n). &a,a!(%$n) $, 6$d*%' ,*d $n ($n$ n and !a":*& "*,*a"- , " *',.

4.4 S()!*IN+ (N% NON S()!*IN+ $''O'S6

A ,a!(%* $, a (a"& + & * 6 %* ( ( %a&$ n. A ,a!(%* d"a6n +" ! & *( ( %a&$ n d*(*nd, n - an-* and a, , - a%% & * - a"a-&*"$,&$- + & *

( ( %a&$ n !a' n & #* ("*,*n& $n & * ,a!(%* d"a6n +" ! & * ,a!*( ( %a&$ n. Any statistical measure say, mean of the sample, may not beequal to the corresponding statistical measure of the population from whichthe sample has been drawn. hus there can be discrepancies in thestatistical measure of population. $.*.. parameters and !he statisticalmeasures of sample drawn from the same population. $.*.; statistic. T *,*d$,-"*(an-$*, a"* :n 6n a, E"" ", $n ,a!(%$n). E"" ", $n ,a!(%$n) a"* + &6&'(*,

?$@Sampling Errors

?$$;@on-sampling Errors

4.4.# Sa!(%$n) E"" ",

Sampling Errors $, $n *"*n& $n & * !*& d + ,a!(%$n). Sa!(%$n)d*(*nd, n

MATHEMATICS AND STATISTICS Pa)*


10/61


- an-* and d * & & * *9$,&*n-* + - an-* $n ,a!(%$n); & * ,a!(%$n) *"" ",-- ".

E"" ", $n ,a!(%$n) a"$,* ("$!a"$%' d * & & * + %% 6$n) "*a, n,

!. "aulty selection of the sample. T $, !a' #* d * & ,*%*-&$ n +

d*+*-&$ * ,a!(%$n)&*- n$< *, 6 $- !a' $n&" d -* & * *%*!*n& + #$a,; *.).; ( "( ,$ * " = d)!*n&a%,a!(%$n); $n 6 $- $n *,&$)a& " d*%$#*"a&*%' ,*%*-&, a n n/"*("*,*n&a&$ *,a!(%*.

#. Substitution. S !*&$!*, an $n *,&$)a& " 6 $%* - %%*-&$n) & * $n+ "!a&$ n+" ! a(a"&$- %a" ,a!(%$n) n$&; $n-% d*d $n & * "and ! ,*%*-&$ n , #,&$& &*, a- n *n$*n-*!*!#*" + & * ( ( %a&$ n and & $, !a' %*ad & , !* #$a, a, & *

- a"a-&*"$,&$-( ,,*,,*d #' & * , #,&$& &*d n$& !a' #* d$++*"*n& +" ! & ,* ( ,,*,,*d #'& *

"$)$na% n$& $n-% d*d $n ,a!(%$n).

$. "aulty demarcation of sampling units

%. &ariability of the population. Sa!(%$n) *"" " !a' a%, d*(*nd & *a"$a#$%$&' "*&*" )*n*$&' + & * ( ( %a&$ n +" ! 6 $- & * ,a!(%*, a"* & #* d"a6n.

4.4. Non-Sampling $rrors

on-sampling errors " B$a, a & !a&$-a%%' -"**( $n d * & !an+a-& ", 6 $-a%6a', a"' +" ! n* $n *,&$)a& " & an & *". B$a, !a' a"$,* $n & * + %% 6$n)d$++*"*n&6a',.

?$@ D * & n*)%$)*n-* and -a"*%*,,n*,, n & * (a"& + & * $n *,&$)a& "

?$$@ D * & +a %&' (%ann$n) + ,a!(%$n)

?$$$@ D * & & * +a %&' ,*%*-&$ n + ,a!(%* n$&,

?$ @ D * & $n- !(%*&* $n *,&$)a&$ n and ,a!(%* , " *'

? @ D * & +"a!$n) + a 6" n) < *,&$ nna$"*

? $@ D * & n*)%$)*n-* and n n/"*,( n,* n & * (a"& + & * "*,( nd*n&,



11/61


? $$@ D * & , #,&$& &*, + ,*%*-&*d n$& #' an & *"

? $$$@ D * & *"" " $n - !($%a&$ n

?$9@ D * & a((%'$n) 6" n) ,&a&$,&$-a% !*a, "*.

4.9 T$STIN+ O& :;!OT:$SIS

T * &*,&$n) + '( & *,$, $, a (" -*d "* & a& *%(, , & a,-*"&a$n & *%$:*%$ d + '( & *,$>*d ( ( %a&$ n (a"a!*&*" #*$n) - ""*-& #' !a:$n) ,*

+ & * ,a!(%* ,&a&$,&$-. In &*,&$n) + '( & *,$, a ,&a&$,&$- $, - !( &*d +" ! a,a!(%* d"a6n +" ! & * (a"*n& ( ( %a&$ n and n & * #a,$, + & $, ,&a&$,&$-; $&$, #,*" *d 6 *& *" & * ,a!(%* , d"a6n a, - !* +" ! & * ( ( %a&$ n 6$&-*"&a$n ,(*-$+$*d - a"a-&*"$,&$-.

4.9.# Sampling %istribution

C n,$d*" a%% ( ,,$#%* ,a!(%*, + ,$>* n 6 $- -an #* d"a6n +" ! a)$ *n ( ( %a&$ n. F " *a- ,a!(%* 6* -an - !( &* a ,&a&$,&$- , - a, !*an;,&anda"d d* $a&$ n; *&-. 6 $- 6$%% a"' +" ! ,a!(%* & ,a!(%*. T *a))"*)a&* + a"$ , a% *, + & * ,&a&$,&$- nd*" - n,$d*"a&$ n !a' #*)" (*d $n& a +"*< *n-' d$,&"$# &$ n. T $, d$,&"$# &$ n $, :n 6n a, ,a!(%$n)d$,&"$# &$ n + & * ,&a&$,&$-. T , & * (" #a#$%$&' d$,&"$# &$ n + a%% &( ,,$#%* a% *, & a& a ,a!(%* ,&a&$,&$- -an &a:* $, -a%%*d & * ,a!(%$n)

d$,&"$# &$ n + & * ,&a&$,&$-.Sa!(%$n) !*an and ,a!(%* (" ( "&$ n #a,*d n "and ! ,a!(%* a"*

*9a!(%* + ,a!(%* ,&a&$,&$-

Sa!(%$n) d$,&"$# &$ n + & * M*an +" ! n "!a% ( ( %a&$ n

I+ 91; 9 2; 9 3; .. 9 n a"* n $nd*(*nd*n& "and ! ,a!(%*, d"a6n +" ! an "!a% ( ( %a&$ n 6$& !*an ! and ,&anda"d d* $a&$ n ,; & *n & * ,a!(%$n)d$,&"$# &$ n +9 ?& * ,a!(%* !*an@ + %% 6, a n "!a% d$,&"$# &$ n 6$& !*an! and ,&anda"d d* $a&$ n n .

I& !a' #* n &*d & a&

?$@ & * ,a!(%* !*an 9 $ 1n9$n 91 92 9nn T , 9 $, a "and ! a"$a#%* and 6$%% #* d$++*"*n& * *"' &$!* 6 *n an*6 ,a!(%* + n #,*" a&$ n, a"* &a:*n.



12/61


?$$@ 9 $, an n#$a,*d *,&$!a& " + & * ( ( %a&$ n !*an !. $.*. E ? 9 @ ; d*n &*d #' 9 .

?$$$@ T * ,&anda"d d* $a&$ n + & * ,a!(%* !*an 9 $, )$ *n #' 9 n .

Sampling distribution of proportions

S (( ,* & a& a ( ( %a&$ n $, $n+$n$&* and & a& & * (" #a#$%$&' +-- ""*n-* + an * *n; ,a' , --*,, $, P. L*& 1/P d*n &*, & * (" #a#$%$&' +

+a$% "*.

C n,$d*" a%% ( ,,$#%* ,a!(%*, + ,$>* n d"a6n +" ! & $, ( ( %a&$ n. F " *a-,a!(%*; d*&*"!$n* & * (" ( "&$ n ( + , --*,,*,. A((%'$n) -*n&"a% %$!$&& * "*!; $+ & * ,a!(%* + ,$>* n $, %a")*; & * d$,&"$# &$ n + & * ,a!(%*(" ( "&$ n ( + %% 6, a n "!a% d$,&"$# &$ n 6$& !*an !( P and S.D

( P n .

4.9. Standard $rror

T * ,&anda"d d* $a&$ n + & * ,a!(%$n) d$,&"$# &$ n + a ,&a&$,&$- $,-a%%*d & * ,&anda"d *"" " + & * ,&a&$,&$-. T * ,&anda"d d* $a&$ n + & *d$,&"$# &$ n + & * ,a!(%* !*an $, -a%%*d & * ,&anda"d *"" " + & * !*an.L$:*6$,*; & * ,&anda"d d* $a&$ n + & * d$,&"$# &$ n + & * ,a!(%* (" ( "&$ n$, -a%%*d & * ,&anda"d *"" " + & * (" ( "&$ n.

T * ,&anda"d *"" " $, ( ( %a"%' :n 6n a, ,a!(%$n) *"" ". Sa!(%$n)*"" " & " 6, %$) & n & * ("*-$,$ n and a-- "a-' + & * *,&$!a&*. T *,&anda"d *"" " $, $n *",*%' (" ( "&$ na% & & * ,a!(%* ,$>* $.*. & * %a")*" & *,a!(%* ,$>* & * ,!a%%*" & * ,&anda"d *"" ".

T * ,&anda"d *"" " !*a, "*, & * d$,(*",$ n + a%% ( ,,$#%* a% *, +& * ,&a&$,&$- $n "*(*a&*d ,a!(%*, + a +$9*d ,$>* +" ! a )$ *n ( ( %a&$ n. I& $,

,*d & ,*& ( - n+$d*n-* %$!$&, + " ( ( %a&$ n (a"a!*&*", $n &*,&, +,$)n$+$-an-*. T , & * ,&anda"d *"" ", + ,a!(%* !*an 9 and ,a!(%*(" ( "&$ n ( a"* ,*d & +$nd - n+$d*n-* %$!$&, + " & * ( ( %a&$ n !*an !and & * ( ( %a&$ n (" ( "&$ n P "*,(*-&$ *%'.

Statistic Standard$rror

'emar"

Sa!(%* !*an 9 n P ( %a&$ n ,$>* $, $n+$n$&* " ,a!(%*6$& "*(%a-*!*n&

nN/nN/1 P ( %a&$ n ,$>* N +$n$&* " ,a!(%*



13/61


6$& & "*(%a-*!*n&

Sa!(%* P" ( "&$ n?(@

P n P ( %a&$ n ,$>* $, $n+$n$&* " ,a!(%*6$& "*(%a-*!*n&

P n N/nN/1 P ( %a&$ n ,$>* N +$n$&* " ,a!(%*6$& & "*(%a-*!*n&

4.9.2 Null and (lternative : pot esis

Null : pot esis6 T * ,&a&$,&$-a% '( & *,$, & a& $, ,*& ( + " &*,&$n) a'( & *,$, $, :n 6n a, N %% H'( & *,$,. T * n %% '( & *,$, $, ,*& ( $n

&*,&$n) a ,&a&$,&$-a% '( & *,$, n%' & d*-$d* 6 *& *" & a--*(& " "*=*-& & *n %% '( & *,$,. I& a,,*"&, & a& & *"* $, n d$++*"*n-* #*&6**n & * ,a!(%*

,&a&$,&$- and ( ( %a&$ n (a"a!*&*" and 6 a&* *" d$++*"*n-* $, & *"*; $,a&&"$# &a#%* & ,a!(%$n) *"" ",. N %% H'( & *,$, , a%%' d*n &*d #' H 0 .

(lternative : pot esis6 T * n*)a&$ n + N %% H'( & *,$, $, -a%%*d & *A%&*"na&$ * '( & *,$,. In & *" 6 "d,; an' '( & *,$, 6 $- $, n & a n %%

'( & *,$, $, -a%%*d a%&*"na&$ * '( & *,$,. I& $, a%6a', d*n &*d #' H 1 " H a .I& $, ,*& $n , - a 6a' & a& "*=*-&$ n + n %% '( & *,$, $!(%$*, & * a--*(&an-*

+ a%&*"na&$ * '( & *,$,.

4.9.4 $rror in testing of pot esis

F " &*,&$n) & * '( & *,$,; 6* &a:* a ,a!(%* +" ! & * ( ( %a&$ n; ann & * #a,$, + & * ,a!(%* "*, %& #&a$n*d; 6* d*-$d* 6 *& *" & a--*(& "

"*=*-& & * '( & *,$,. H*"*; &6 &'(* + *"" ", a"* ( ,,$#%*. A n %% '( & *,$,- %d #* "*=*-&*d 6 *n $& $, &" *. T $, $, -a%%*d T'(* I *"" " and & *(" #a#$%$&' + - !!$&&$n) &'(* I *"" " $, d*n &*d #' K. A%&*"na&$ *%'; an *"" "- %d "*, %& #' a--*(&$n) a n %% '( & *,$, 6 *n $& $, +a%,*; & $, $, :n 6n a,

T'(* II *"" " and & * (" #a#$%$&' + - !!$&&$n) &'(* II *"" " $, d*n &*d #' .

T $, $, $%% ,&"a&*d $n & * + %% 6$n) &a#%*

Statistical %ecision of t eTest

TrueSituation

: 0 is True : 0 is &alse

H0 $, T" * C ""*-& T'(* I E"" "



14/61


D*-$,$ n

H0 $, Fa%,* T'(* II E"" " C ""*-&D*-$,$ n

4.9.9. 1ritical 'egion

A "*)$ n $n & * ,a!(%* ,(a-* 6 $- a! n&, & "*=*-&$ n + n %%'( & *,$, ?H 0@ $, -a%%*d & * -"$&$-a% "*)$ n. A+&*" + "! %a&$n) & * n %% and

a%&*"na&$ * '( & *,$, a# & a ( ( %a&$ n (a"a!*&*"; 6* &a:* a ,a!(%* +" !& * ( ( %a&$ n and -a%- %a&* & * a% * + & * "*%* an& ,&a&$,&$-; and - !(a"*$& 6$& & * '( & *,$>*d ( ( %a&$ n (a"a!*&*". A+&*" d $n) & $,; 6* a * &d*-$d* & * -"$&*"$a + " a--*(&$n) " "*=*-&$n) & * n %% '( & *,$,. T *,*-"$&*"$a a"* )$ *n a, a "an)* + a% *, $n & * + "! + an $n&*" a%; ,a' ?a;#@; ,& a& $+ & * ,&a&$,&$- a% * +a%%, &,$d* & * "an)*; 6*"* "*=*-& & * n %%

'( & *,$,. I+ & * ,&a&$,&$- a% * +a%%, 6$& $n & * $n&*" a% ?a;#@ & *n 6* a--*(&H0. T $, -"$&*"$ n a, & #* d*-$d*d n & * #a,$, + & * %* *% + ,$)n$+$-an-*.A& %* *% + ,$)n$+$-an-* !*an, & a& + & * ,&a&$,&$-a% a% * a""$ *d a&+" ! & * ,a!(%*, 6$%% +a%% &,$d* & $, "an)* ?a;#@and + & * a% *, 6$%%#* 6$& $n & * "an)* ?a;#@. T , & * %* *% + ,$)n$+$-an-* $, & * (" #a#$%$&' +

T'(* I *"" ". T * %* *%, + ,$)n$+$-an-* , a%%' *!(% '*d $n &*,&$n) +'( & *,$, a"* and 1 . A $) ,$)n$+$-an-* %* *% - ,* + " &*,&$n) a'( & *,$, 6 %d $!(%' & a& $) *" $, & * (" #a#$%$&' + "*=*-&$n) a n %%

'( & *,$, $+ $& $, &" *.Table of critical value < = of 9> #0>

T/o tailed test K2. 8

K 1. 5 K1.54

One tailed test K2.33

K1.54

K 1.28

4.9.? T/o tailed and one tailed test6



15/61


T * (" #a#$%$&' - " * + & * ,a!(%$n) d$,&"$# &$ n + & * &*,& ,&a&$,&$- $,a n "!a% - " *. In an' &*,&; & * -"$&$-a% "*)$ n $, "*("*,*n&*d #' a ( "&$ n +& * a"*a nd*" & $, n "!a% - " *. T $, - " * a, &6 ,$d*, ? " *nd,@ :n 6na, &6 &a$%,. T * "*=*-&$ n "*)$ n !a' #* "*("*,*n&*d #' a ( "&$ n + a"*a n*a- + & * &6 ,$d*, " #' n%' n & * ,$d* + & * n "!a% - " * and- ""*,( nd$n)%' & * &*,& $, :n 6n a, &6 &a$%*d &*,& ? " &6 ,$d*d &*,&@ " n*&a$%*d &*,& ? " n* ,$d*d &*,&@.

*n & * &*,& + '( & *,$, $, !ad* n & * #a,$, + "*=*-&$ n "*)$ n"*("*,*n&*d #' # & ,$d*, + & * ,&anda"d n "!a% - " *; $& $, -a%%*d a &6&a$%*d &*,& " &6 ,$d*d &*,&.

*n & * &*,& + '( & *,$, $, !ad* n & * #a,$, + "*=*-&$ n "*)$ n"*("*,*n&*d #' an' + & * ,$d*, + & * ,&anda"d n "!a% - " *; $& $, -a%%*d a

n* &a$%*d &*,& " n* ,$d*d &*,&.

4.9.@ *arge and small sample test

T * &*,& + ,$)n$+$-an-* $, ?a@ T*,& + ,$)n$+$-an-* + " %a")* ,a!(%* and?#@ T*,& + ,$)n$+$-an-* + " ,!a%% ,a!(%*,. F " %a")*" ,a!(%* ,$>* ?.30@; a%% & *d$,&"$# &$ n, %$:* B$n !$a%; P $,, n *&-.; a"* a((" 9$!a&*d #' n "!a%d$,&"$# &$ n. T , n "!a% (" #a#$%$&' - " * -an #* ,*d + " &*,&$n) +

'( & *,$,.

4.? !'O1$%U'$ &O' T$STIN+ O& :;!OT:$SIS6

S&*(, + " &*,&$n) '( & *,$, $, )$ *n #*% 6 ? + " # & %a")* ,a!(%* and ,!a%%,a!(%* &*,&,@

Step #6 Null pot esis6 S*& ( n %% '( & *,$, H 0

Step 6 (lternative : pot esis6 S*& ( a%&*"na&$ * H'( & *,$, H 1; 6 $-$, - !(%*!*n&a"' & H 0 6 $- 6$%% $nd$-a&* 6 *& *" n* &a$%*d ?"$) & " %*+&&a$%*d@ " &6 &a$%*d &*,& $, & #* a((%$*d

Step 26 *evel of significance6 C ,* an a((" ("$a&* %* *% + ,$)n$+$-an-*

[email protected] 46 Test Statistic ? " &*,& + -"$&*"$ n@

Ca%- %a&* & * a% * + & * &*,& ,&a&$,&$-; &/E?&@S.E.?&@

Und*" & * n %% '( & *,$,; 6 *"* & $, & * ,a!(%* ,&a&$,&$-



16/61


Step 96 1ritical Value6 F$nd & * -"$&$-a% a% * a + a& & * %* *% +,$)n$+$-an-*; +" ! & * &a#%* a"*a, nd*" & * n "!a% - " * a Q a% *, $n-a,* + %a")* ,a!(%*,; " a"*a, nd*" &/&a#%*; F/&a#%*; C $/,< a"* &a#%* $n-a,* + ,!a%% ,a!(%*,.

Step ?6 Inference6 * - !(a"* & * - !( &*d a% * + ?$n a#, % &*a% *@ 6$& & * ,$)n$+$-an& a% * ?-"$&$-a% a% *@ K 2 ? " a@. I+ a; 6*

"*=*-& & * n %% '( & *,$, H0 a& a %* *% + ,$)n$+$-an-* and $+ a ; 6*a--*(& H 0 a& a %* *% + ,$)n$+$-an-*.

4.@. *('+$ S()!*$ T$STS6

4.7.1 T*,& + " ,$n)%* !*an

Step #6 Setting up of a Null pot esis. T *"* $, n ,$)n$+$-an-*d$++*"*n-* #*&6**n & * ,a!(%* and & * ( ( %a&$ n !*an " & * ,a!(%* a,#**n d"a6n +" ! & * (a"*n& ( ( %a&$ n. H 0 9

Step 6 Setting up of an (lternative pot esis6 T *"* $, a ,$)n$+$-an-*d$++*"*n-* #*&6**n & * ,a!(%* !*an and & * ( ( %a&$ n !*an. H 1 9

Step 26 &i5ing of level of significance6 a ?n "!a%%' $& $, @

S&*( 4 C !( &a&$ n + &*,& S&a&$,&$-

< cal A 9/ n

Step 96 1ritical Value6 F$nd & * -"$&$-a% a% * a a& a %* *% + ,$)n$+$-an-*;+" ! & * &a#%* a"*a, nd*" & * n "!a% - " *, a Q a% *,.

Step ?6 Interference6 I+ & * ! d % , + & * -a%- %a&*d a% * -a%W K ;#&a$n*d $n ,&*( ; 6* a--*(& & * n %% '( & *,$, a& a %* *% + ,$)n$+$-an-*.

O& *"6$,* 6* "*=*-& & * n %% '( & *,$, a& a %* *% + ,$)n$+$-an-*.

N 6; 6* d$,- ,, & * a# * 6$& an *9a!(%*.

$5ample 4.# A Sa!(%* + 400 !a%* ,& d*n&, + a - %%*)* $, + nd & a * a!*an *$) & + 171.38-!. Can $& #* "*)a"d*d a, a ,a!(%* +" ! a %a")*( ( %a&$ n 6$& !*an *$) & 171.17-! and ,&anda"d d* $a&$ n 4.40-!X

Solution6



17/61


G$ *n n 400 ?La")* Sa!(%*@

171.17-! 9 171.38 -! 3.30 -!

Null : pot esis ,: 0 6 Sa!(%* !*an a, #**n d"a6n +" ! a %a")*

( ( %a&$ n 6$& !*an *$) & + 171.17 -!. $.*.; H 0 171.17 -!(lternative : pot esis ,: # 6 Sa!(%* !*an a, n & #**n d"a6n +" ! a%a")* ( ( %a&$ n 6$& !*an 171.17-! $.*.; H 1 171.17-!.

*evel of significance ?K@

Test Statistic6

< cal A 9/ n

< cal A 171.38/171.174.40400

< cal A 0.210.22 A 0.B94?

1ritical value6 A& %* *%; 0.0 1. 5

In&*"+*"*n-* S$n-* & * -a%- %a&*d a% * + $, %*,, & an & * -"$&$-a% a% * + a& %* *%; *n-* 6* a--*(& & * n %% '( & *,$, and - n-% d* & a&; & *

,a!(%* !*an a, #**n d"a6n +" ! a %a")* ( ( %a&$ n 6$& !*an *$) & +171.17-!.

$5ample 4. T * !*an %$+*&$!* + 100 +% "*,-*n& %$) & # %#, (" d -*d #' a- !(an' $, - !( &*d & #* 1 70 ", 6$& ,&anda"d d* $a&$ n + 120

",. I+ ! $, & * !*an %$+*&$!* + a%% & * # %#, (" d -*d #' & * - !(an';&*,& & * '( & *,$, 1500 ", a)a$n,& & * a%&*"na&$ * '( & *,$, ! 1500 ", ,$n) a %* *% + ,$)n$+$-an-*.

Solution6* a"* )$ *n

Z 1 70 ",. 1500 ",; , 120 ",; n 100

Null : pot esis ,: 0 6 ! 1500. $.*.; T *"* $, n ,$)n$+$-an& d$++*"*n-*#*&6**n & * ,a!(%* !*an and ( ( %a&$ n !*an.



18/61


(lternative : pot esis ,: # 6 M1 1500 ?& 6 &a$%*d A%&*"na&$ *@ T *"* $, a,$)n$+$-an& d$++*"*n-* #*&6**n & * ,a!(%* !*an and ( ( %a&$ n !*an.

*evel of Significance ,a 6

Test Statistic6

-a% 9/ n

-a% 1 70/1500120100 /2.

-a% 2.

1ritical value6 A& %* *%;

In&*"+*"*n-* S$n-* & * -a%- %a&*d a% * $, )"*a&*" & an & * -"$&$-a% a% * + a& %* *%; *n-* 6* "*=*-& & * n %% '( & *,$, and - n-% d* & a& ; & *"* $,a ,$)n$+$-an& d$++*"*n-* #*&6**n & * ,a!(%* !*an and ( ( %a&$ n !*an.

Self C (ssessment 8uestion

1. A "and ! ,a!(%* + 00 !*!#*", a, a !*an 3.4 -! and S.D 2.51-!. I, & * ,a!(%* +" ! a %a")* ( ( %a&$ n + !*an 3.2 -! and S.D2.51 -!X

n 00; 9 3.4; 3.2 ; 2.51; -a% 1.724

[H$n& H0 $, a--*(&*d a& %* *%\

2. A "and ! ,a!(%* , ,$>* 400 d"a6n and & * ,a!(%* !*an 6a, + nd& #* . T*,& 6 *& *" & * ,a!(%* - %d a * - !* +" ! a n "!a%( ( %a&$ n 6$& !*an 100 and ,&anda"d d* $a&$ n 8 a& %* *%.n 400; 9 ; 100; 8; -a% 2.[H$n& H0 $, "*=*-&*d a& %* *%\

4.@. Test for difference of mean6

Dor"ing 'ule6



19/61


Step #6 Setting up of a Null : pot esis6 T * &6 ,a!(%*, a *#**n d"a6n +" ! d$++*"*n& +" ! d$++*"*n& ( ( %a&$ n, a $n) & * ,a!*!*an, and *< a% ,&anda"d d* $a&$ nH0 1 2

Step 6 Setting up of an (lternative : pot esis. T * &6 ,a!(%*,a * #**n n & d"a6n +" ! d$++*"n* &+" ! d$++*"*n& ( ( %a&$ n,.

H0 1 2 ?T6 &a$%*d &*,&@; " H1 1 2 ?On* &a$%*d &*,&@; "

H1 1 2?On* &a$%*d &*,&@.

Step 26 &i5ing t e level of Significance6 K ?n "!a%%' $& $, @

Step 46 1omputation of test Statistic6

-a% 91/ 92 12n1 22n2 $+ & * ( ( %a&$ n ,.d , a"* :n 6n-a% 91/ 92,12n1 ,22n2 $+ & * ( ( %a&$ n ,.d , a"* n & :n 6n.

Step 96 1ritical Value6 F$nd & * -"$&$-a% a% * a a& K %* *% +,$)n$+$-an-*; +" ! & * &a#%* a"*a, nd*" & * n "!a% - " * a / a% *,

Step ?6 Inference6 I+ & * ! d % , + & * -a%- %a&*d a% *; #&a$n*d $n,&*( ; 6* a--*(& & * n %% '( & *,$, a& K %* *% + ,$)n$+$-an-*;O& *"6$,* 6* "*=*-& & * n %% '( & *,$, a& a %* *% + ,$)n$+$-an-*.

$5ample 4.26 A - %%*)* - nd -&, # & da' and * *n$n) -%a,,*, $n&*nd*d * $d*n&$-a%. A ,a!(%* + 100 da' ,& d*n&, '$*%d, *9a!$na&$ n "*, %&, a,

nd*" 91 72 and 1 14.8. A ,a!(%* + 200 * *n$n) ,& d*n&, '$*%d,*9a!$na&$ n "*, %& nd*" 92 73. and 2 17. . A"* & * &6 !*an,&a&$,&$-a%%' *< a% a& %* *%X

Solution6

* a"* )$ *nn 100; 91 72 and 1 14.8; n 200; 92 73. and 2 17.

Null : pot esis ,: 0 6 H0 1 2. .*; & * &6 !*an, a"* ,&a&$,&$-a%%' *< a%.

(lternative : pot esis ,: # 6 1 2 ?T6 &a$%*d &*,&@ $.*.; & * &6 !*an,a"* n & ,&a&$,&$-a%%' *< a%.



20/61


*evel of Significance ?K@

Test Statistics

-a% 91/ 92 12n1 22n2 72.4/73. ?14.8@2100 ?17. @2200 /1. 3.7 2/1. 1. 47 /0.77

>-a% 0.77

1ritical value6 (t 9> level ; 0.0 1. 5

Inference6 S$n-* & * -a%- %a&*d a% * + -a% $, %*,, & an & * -"$&$-a% a% * + a& %* *%; *n-* 6* a--*(& & * n %% '( & *,$, and - n-% d* & a&; & *

&6 !*an, a"* ,&a&$,&$-a%%' *< a%.

$5ample 4.4 A "and ! ,a!(%* + 1000 6 ":*", +" ! S & Ind$a , 6, & a&& *$" !*an 6a)*, a"* R, 47 (*" 6**: 6$& a ,&anda"d d* $a&$ n + R,. 28. A"and ! ,a!(%* + 1 00 6 ":*", +" ! N "& Ind$a )$ *, a !*an a)* + R,. 4(*" 6**: 6$& a ,&anda"d d* $a&$ n + R,. 40. I, & *"* an' ,$)n$+$-an&d$++*"*n-* #*&6**n & *$" !*an %* *%, + 6a)*,X

Solution

* a"* )$ *n; n 1 1000; 91 47 and , 1 28; n 2 1 00; 92 73. and

S2 17.

Null : pot esis ,: 0 6 H0 1 2 $.*.; & *"* $, n ,$)n$+$-an& d$++*"*n-*#*&6**n & *$" !*an %* *% + 6a)*,

(lternative : pot esis ,: # 6 H 1 1 2 ?T6 &a$%*d &*,&@ $.*.; & *"* $, a,$)n$+$-an& d$++*"*n-* #*&6**n & *$" !*an %* *% + 6a)*,.

*evel of significance ,= 6

Test Statistics

-a% 91/ 92,12n1 ,22n2 47/ 4 ?28@21000 ?40@21 00 /21.8 07 /21.3504 /1.47

-a% 1.47



21/61


1ritical value6 A& %* *%

Inference6 S$n-* & * -a%- %a&*d a% * + -a% $, %*,, & an & * -"$&$-a% a% * + a& %* *%; *n-* 6* a--*(& & * n %% '( & *,$, and - n-% d* & a& & *

&6 !*an, a"* ,&a&$,&$-a%%' *< a%.

Self-(ssessment 8uestion

1. In a , " *' $+ # '$n) a#$&,; 400 6 !*n , ((*", a"* - ,*n a&"and ! $n , (*" !a":*& A % -a&*d $n -*"&a$n ,*-&$ n + & * -$&'. T *$"a *"a)* 6**:%' + d *9(*nd$& "* $, R,. 2 0 6$& a ,&anda"d d* $a&$ n

+ R,. 40. F " 400 6 !*n , ((*", - ,*n a& "and ! $n , (*" !a":*&B $n an & *" ,*-&$ n + & * -$&'; & * a *"a)* 6**:%' + d *9(*nd$& "*

$, R,. 220 6$& ,&anda"d d* $a&$ n + R,. . T*,& a& %* *% +,$)n$+$-an-* 6 *& *" & * a *"a)* 6**:%' + d *9(*nd$& "* + & * &6

( ( %a&$ n, + , ((*", a"* *< a%.[H$n& n1 400; 91 2 0; , 1 40; n 2 400; 92 220 and , 2 ; -a%

8.82\2. Rand ! ,a!(%*, d"a6n +" ! &6 - n&"$*, )a * & * + %% 6$n) da&a

"*%a&$n) & & * *$) &, + ad %& !a%*,2.

1ountr(

1ountrE

M*an *$) & $n$n- *,

57.42 57.2

S&anda"dd* $a&$ n

2. 8 2. 0

N !#*" + ,a!(%* 1000 1200

I, & * d$++*"*n-* #*&6**n & * !*an, ,$)n$+$-an&X

[H$n& n1 1000; 91 57.42; , 1 2. 8; n 2 1200; 92 57.2 and , 2 2. 0; -a%1. 5\

[email protected] Test for single proportion

Dor"ing 'ule6



22/61


23/61


Inference6 S$n-* & * -a%- %a&*d a% * + -a% $, %*,, & an & * -"$&$-a% a% * + a& 1 %* *%; *n-* 6* a--*(& & * n %% '( & *,$, and - n-% d* & a&; & *

,a!(%* a, #**n d"a6n +" ! a ( ( %a&$ n 6$& (" ( "&$ n P; $.*.; H 0 P 0. .

Self C assessment 8uestion

1. In a "and ! ,a!(%* + 400 (*", n, +" ! a %a")* ( ( %a&$ n; 120 a"*+*!a%* -an $& ,a$d & a& !a%*, and +*!a%*, a"* $n & * "a&$ n 3 $n & *( ( %a&$ n. U,* %* *% + ,$)n$+$-an-*.[H$n& n 400; ( 120400 0.3 P 38 0.37 1/P 1/0.37 0.52 2. 8\

-a% 3.12

[email protected] Test for t/o proportions6

Dor"ing 'ule6

Step #6 Setting up of a Null : pot esis. T * &6 ,a!(%*, a * #**nd"a6n +" ! ,a!* ( ( %a&$ n; $.*.; H 0 P1 P 2 .

Step 6 Setting up of an (lternative pot esis. T * &6 ,a!(%*, a *n & #**n d"a6n +" ! ,a!* ( ( %a&$ n; $.*.; H 1 P1 P 2

Step 26 &i5ing of level of significance6 K ?n "!a%%' $& $, @

Step 46 1omputation of test statistics6

-a% (1/(2P ?1n1 1n2@ *"* P n1(1 n2(2n1 n2 and1/P

Step 961ritical value6 F$nd & * -"$&$-a% a% * a a& a %* *% + ,$)n$+$-an-*;+" ! & * &a#%* a"*a, nd*" & * n "!a% - " * aQ a% *,.

$5ample 4.? In a ,a!(%* + 500 !*n +" ! a -*"&a$n -$&'; 4 0 a"* + nd &

#* ,! :*",. In a ,a!(%* + 00 +" ! an & *" -$&' 4 0 a"* + nd & #*,! :*",. D & * da&a $nd$-a&* & a& & * &6 -$&$*, a"* ,$)n$+$-an&%' d$++*"*n&6$& "*,(*-& & ("* a%*n-* + ,! :$n) a#$&, a! n) !*nX

Solution6

G$ *n n 1 500 n 2 00 ( 1 4 0500 0.7 ( 2 4 0 00 0.



24/61


P n1(1 n2(2n1 n2 5000.7 00?0. @500 00 0.5

1/P 1/0.5 0.4

Null : pot esis ,: 0 6 T * &6 ,a!(%*, a * #**n d"a6n +" ! ,a!*( ( %a&$ n; $.*.; H0 P1 P 2

(lternative : pot esis ,: # 6 T * &6 ,a!(%*, a * n & #**n d"a6n +" !& * ,a!* ( ( %a&$ n; $.*.; H 1 P1 P 2.


T*,& S&a&$,&$-

-a% (1/(2P ?1n1 1n2@ 0.7 /0. 0.5?0.4@?1500 1 00@ 0.2 0.02 8 .7

-a% .7

1ritical Value6 A& %* *%; 0.0 1. 5

Inference6 S$n-* & * -a%- %a&*d a% * + -a% $, )"*a&*" & an & * -"$&$-a%a% * + a& %* *%; *n-* 6* "*=*-& & * n %% '( & *,$, and - n-% d*

& a&;& * &6 ,a!(%*, a * n & #**n d"a6n +" ! & * ,a!* ( ( %a&$ n.

$5ample 4.@ A !a- $n* ( &, & 15 $!(*"+*-& a"&$-%*, $n a ,a!(%* + 00.A+&*" & * !a- $n* $, *" a %*d; $& ( &, & 3 $!(*"+*-& a"&$-%*, $n a #a&- +100. Ha, & * !a- $n* $!(" *dX

Solution

G$ *n n 1 00 n 2 100 ( 1 15 00 0.032 ( 2 3100 0.03

P n1(1 n2(2n1 n2 000.032 100?0.03@ 00 100 0.0315

1/P 1/0.0315 0. 58

Null : pot esis,: 0 6 P 1 P 2 .

(lternative : pot esis ,: # 6 P 1 P 2 ? n* &a$%*d &*,&@


Test Statistic6

-a% (1/(2P ?1n1 1n2@ 0.032/0.030.0315?0. 58@?1 00 1100@ 0.0020.010 0.1



25/61


-a% 0.1

1ritical Value6 A& %* *%; 0.0 1.54

Inference6 S$n-* & * -a%- %a&*d a% * + $, %*,, & an & * -"$&$-a% a% * + a&

%* *% *n-* 6* a--*(& & * n %% '( & *,$, and - n-% d* & a&; & *"* $, n$!(" *!*n& a+&*" *" a %$n).

Self (ssessment 8uestion

1. In a "and ! ,a!(%*, + 500 and 1000 !*n +" ! &6 -$&$*, 400 and500 !*n a"* + nd & #* %$&*"a&*. D & * da&a $nd$-a&* a& %* *% +,$)n$+$-an-* & a& & * ( ( %a&$ n a"* ,$)n$+$-an&%' d$++*"*n& $n & *(*"-*n&a)* + %$&*"a-'X[H$n& n1 500; n 2 1000; ( 1 400500 0.57; ( 2 5001000 0.5P 0.52 0.37 ;\

-a% 2.572. B*+ "* an $n-"*a,* $n *9-$,* d &' n &*a 400 (* (%* & + a ,a!(%* +

00 (*", n, 6*"* + nd & #* &*a d"$n:*",. A+&*" an $n-"*a,* $n & *d &'; 400 (*", n, 6*"* :n 6n & & * &*a d"$n:*", $n ,a!(%* + 500(* (%*. D ' & $n: & a& & *"* a, #**n a ,$)n$+$-an& d*-"*a,* $n & *- n, !(&$ n + &*a a+&*" & * $n-"*a,* $n & * *9-$,* d &'X

3. [H$n& n1 00; n 2 500; ( 1 400 00 0.80; ( 2 400500 0.57P 0.73 0.27;\

4. -a% 4.81 H 0 P1 P 2 H1 P1 P 2 ? n* &a$%*d &*,&@.

4.F(N(*;SIS O& V('I(N1$6

In !an' ,&a&$,&$-a% ,& d$*, a a"$a#%* + $n&*"*,&; -a%%*d & * "*,( n,*a"$a#%* ? " d*(*nd*n& a"$a#%*@; $, $d*n&$+$*d. T *n & * da&a a"* - %%*-&*d

& a& &*%% , a# & 6 n* " ! "* +a-& ", ? " $nd*(*nd*n& a"$a#%*,@$n+% *n-* & * a"$a#%* + $n&*"*,&. I+ 6* -ann & - n&" % & * +a-& "?,@ #*$n),& d$*d; 6* ,a' & a& & * da&a #&a$n*d a"* #,*" a&$ na%. F " *9a!(%*;, (( ,* & a& $n "d*" & ,& d' 6 & * ,$>* + a !* "*%a&*, & & * ,a%*("$-* + & * !*; a "*a% *,&a&* a)*n& "and !%' ,*%*-&, 0 "*-*n&%' , %d

!*, and "*- "d, & * ,< a"* + &a)*, and ,a%* ("$-*, + & *,* !*,.B*-a ,* & * "*a% *,&a&* a)*n& -ann & - n&" % & * ,$>*, + & * "and !%',*%*-&*d !*,; 6* ,a' & a& da&a a"* #,*" a&$ na%.



26/61


27/61


L*& , , (( ,* & a& N #,*" a&$ n,; $ 1; 2; .: = 1;2 .@ + a"and ! a"$a#%* Z a"* )" (*d n , !* #a,$,; $n& : -%a,,*, + ,$>*, n 1;n 2 ; n : "*,(*-&$ *%'

?N $ 1:n$ @ a, *9 $#$&*d #*% 6

M*an T &a%

Z11 912 . . . . . . 9 1n1 91 T1

Z21 922 . . . . . . 9 2n2 92 T2

Z$1 9 $2 . . . . . . 9 $n$ 9$ T$

Z :1 9 :2 . . . . . . 9 :n: 9: T :

+rand Total+

T * & &a% a"$a&$ n $n & * #,*" a&$ n 9$= -an #* ,(%$& $n& & * + %% 6$n) &6- !( n*n&,

?$@ T * a"$a&$ n #*&6**n * -%a,,*, " & * a"$a&$ n d * &d$++*"*n& #a,*, + -%a,,$+$-a&$ n; - !! n%' :n 6n a,&"*a&!*n&,.

?$$@ T * a"$a&$ n 6$& $n & * -%a,,*, $.*.; & * $n *"*n& a"$a&$ n +& * "and ! a"$a#%* 6$& $n & * #,*" a&$ n, + a -%a,,.

T * +$",& &'(* + a"$a&$ n $, d * & a,,$)na#%* -a ,*, 6 $- -an #* d*&*-&*dand - n&" %%*d #' !an *nd*a " and & * ,*- nd &'(* + a"$a&$ n d * &- an-* -a ,*, 6 $- a"* #*' nd & * - n&" % + !an and.

In Pa"&$- %a"; %*& , - n,$d*" & * *++*-& + : d$++*n& "a&$ n, n & * '$*%d $n !$%:+ N - 6, ? + & * ,a!* #"**d and ,& -:@ d$ $d*d $n& : -%a,,*, + ,$>*, n 1; n 2;

..n : .

R*,(*-&$ *%' ?N $ 1:n$ @ H*n-* & * , "-*, + a"$a&$ n a"*

?$@ E++*-& + "a&$ n,?$$@ E"" " d * & - an-* -a ,*, (" d -*d #' n !*" , -a ,*, & a& & *'

a"* n & d*&*-&*d and $d*n&$+$*d.

Test !rocedure6



28/61


T * ,&*(, $n % *d $n -a""'$n) & & * ana%',$, a"*

1@ Null : pot esis ,: 0 6 T * +$",& ,&*( $, & ,*& ( + a n %% '( & *,$,H0 1 2 :

2@ (lternative : pot esis ,: # 6 a%% 1 , a"* n & *< a% ?$ 1;2; :@3@ *evel of significance L*& K 0.04 Test statistic6

Va"$ , , !, + ,< a"*, a"* #&a$n*d a, + %% 6,

a@ F$nd & * , ! + a% *, + a%% & * ?N@ $&*!, + & * )$ *n da&a. L*& & $,)"and & &a% "*("*,*n&*d #' G .

T *n - ""*-&$ n Fa-& " ?C.F@ G2N#@ F$nd & * , ! + ,< a"*, + a%% & * $nd$ $d a% $&*!, ?9$=@ and & *n & *

T &a% , ! + ,< a"*, ?TSS@ ^^ 9$=2/C.F.-@ F$nd & * , ! + ,< a"*, + a%% & * -%a,, & &a%, ? " *a- &"*a&!*n&

& &a%@ T$ ?$ 1;2; .:@ and & *n & * , ! + ,< a"*, #*&6**n & *-%a,,*, " #*&6**n & * &"*a&!*n&, ?SST@ $, SST $ 1:T$2n= / C.F.6 *"* n$ ?$ 1;2; ..:@ $, & * n !#*" + #,*" a&$ n, $n & * $ & -%a,,

" n !#*" + #,*" a&$ n, "*-*$ *d #' $ & &"*a&!*n&.d@ F$nd & * , ! + ,< a"*, 6$& $n & * -%a,, " , ! + ,< a"*, d * &

*"" " ?SSE@ #' , #&"a-&$ n. SSE TSS/SST

1@ %egrees of freedom ,d.f 6 T * d*)"**, + +"**d ! + " & &a% , ! +,< a"*, +"**d ! + " SSE $, ?N/:@

2@ )ean sum of s3uares6 T * !*an , ! + ,< "*, + " &"*a&!*n&, $,SST:/1 and !*an , ! + ,< a"*, + " *"" $, SSEN/:

3@ (NOV( Table6 T * a# * , ! + ,< "*, & )*& *" 6$& & *$""*,(*-&$ * d*)"**, + +"**d ! and !*an , ! + ,< "*, 6$%% #*, !!a"$>*d $n & * + %% 6$n) &a#%*.

(NOV( Table for one-/a classification

Sources ofVariation

d.f. S.S.

).S.S &-ratio

Eet/eenTreatments

:/1 SST SST:/1MST

MSTMSE F 1

$rror N/: SSE SSEN/:



29/61


MSE

Total N/1

Ca%- %a&$ n + a"$an-* "a&$ n

Va"$an-* "a&$ + F $, & * "a&$ #*&6**n )"*a&*" a"$an-* and ,!a%%*"a"$an-*; & ,

F 1 Va"$a&$ n #*&6**n & * &"*a&!*n&,Va"$a&$ n 6$& $n & * &"*a&!*n&,

MSTMSE

I+ a"$an-* 6$& $n & * &"*a&!*n& $, ! "* & an & * a"$an-* #*&6**n & *&"*a&!*n&,; & *n n !*"a& " and d*n !$na& " , %d #* $n&*"- an)*d andd*)"**, + +"**d ! ad= ,&*d a-- "d$n)%'.

4 1ritical Value of & or Table value of &6 T * -"$&$-a% a% * + F " &a#%* a% * + F $, #&a$n*d +" ! F &a#%* + " ?:/1; N/:@ d.+. a& %* *% + ,$)n$+$-an-*.

9 Inference6I+ -a%- %a&*d F a% * $, %*,, & an &a#%* a% * + F; 6* !a' a--*(& "n %% '( & *,$, H0 and ,a' & a& & *"* $, n ,$)n$+$-an& d$++*"*n-*#*&6**n &"*a&!*n&,. I+ Ca%- %a&*d F a% * $, )"*a&*" & an &a#%* a% * +F; 6* "*=*-& " H 0 and ,a' & a& & * d$++*"*n-* #*&6**n &"*a&!*n&, $,

,$)n$+$-an&.

$5ample 4.@

T * + %% 6$n) &a#%* )$ *, & * '$*%d, n 1 ,a!(%* (% &, nd*" & "** a"$*&$*,+ ,**d

A 20 21 23 15 20B 18 20 17 1 2C 2 28 22 28 32

P"*(a"* an ana%',$, + a"$an-* &a#%*

S % &$ n

Null : pot esis ,: 0 6 1 2 3 ?$.*.; a"$ , a"$&$*, + ,**d, a"*! )*n* ,@



30/61


(lternative pot esis ,: # 6 1 2 3 ?$.*.; a"$ , a"$&$*, + ,**d,a"* n & ! )*n* ,@

*evel of Significance,= 6 0.0

Test Statistic6Variet Tota

lS3uares

( 20 21 23 15 20 100 10000

E 18 20 17 1 2 02

1 2 28 22 28 32 13 1822

+rand Total 220

S3uares6

Variet Total

( 400 441 2 2 5 400 2025

E 324 400 28 22 52 18531 52 784 484 784 102

43701

Total @9B0

1orrection &actor ,1.&. G2N 33021 7250

T &a% , ! + S< a"*, ?TSS@ $, TSS ^^ 9$=2/ C.F 7 0/7250 330

S ! + ,< a"*, #*&6**n & * -%a,,*, " #*&6**n & * &"*a&!*n&, ?SST@ $,

SST $ 1:T$2n$ / C.F.

SST ? 1002 2 13 2 @/7250



31/61


74 0/7250 1 0

S ! + ,< a"*, d * & *"" " ?SSE@ TSS QSST 330/1 0 140

(NOV( Table for one-/a classification

Source ofVariation

d.f. S.S ).S.S &Aratio

Eet/eentreatments

3/1 2 1 0 1 02 11.557 8.142

$rror 14/2 12 140 14012 11.557

Total 1 /1 14

Table Value6 Ta#%* a% * + F + " 2;12@ d.+.; a& %* *% + ,$)n$+$-an-* $,3.8 ?F" ! F/&a#%*@

Inference6 S$n-* -a%- %a&*d F $, )"*a&*" & an &a#%* a% * + F; 6* !a' "*=*-&" H 0 and ,a' & a& a"$ , a"$*&$*, + ,**d, a"* n & ! )*n* ,.


1. T "** (" -*,,*, A; B and C $, &*,&*d & ,** 6 *& *" & *$" &( &, a"*

*< $ a%*n&. T * + %% 6$n) #,*" a&$ n + &( & a"* !ad*A 10 12 13 11 10 14 1 13B 11 10 12 13C 11 10 1 14 12 13

Ca""' & & * ana%',$, + a"$an-* and ,&a&* ' " - n-% ,$ n

:int6

S "-*, +Va"$a&$ n

d.+. S.S. M.S.S F/"a&$

B*&6**n&"*a&!*n&,

3/1 2 7 7 2 3. 3. 3.1 1.0 7

E"" " 18/2 15 1 1 15 3.



32/61


1

T &a% 1 /1 18

Ta#%* a% * + F + " ?2;15@ d.+. a& %* *% + ,$)n$+$-an-* $, 3.53

1 apter Summar

T $, - a(&*" a, *9(%a$n*d d$++*"*n& &'(*, + ,a!(%$n). F$",& 6*d$,- ,,*d & * n**d an *%*!*n&, + ,a!(%$n) (%an. * - n&$n *d #'d$,- ,,$n) ,a!(%$n) and n n/,a!(%$n) *"" ", and T*,&$n) + '( & *,$,. *,a6 & a& # & %a")* ,a!(%* and ,!a%% ,a!(%* &*,& a"* $n+*"*n-*, -an #*!ad*. * %*a"n*d & a& (" -*d "* + " &*,&$n) + '( & *,$, + " &*,&$n) ,$n)%*!*an; d$++*"*n-* + &6 !*an,; ,$n)%' (" ( "&$ n and d$++*"*n-* + &6

(" ( "&$ n, nd*" %a")* ,a!(%* &*,&,. T - n-% d* & $, - a(&*"; 6* *9(%a$n*d6 & &*,& ! )*n*$&' ,$n) n* 6a' ANOVA.



33/61


UNIT Q V

CORRELATION AND REGRESSION ANALYSIS

.0 OBJECTIVES

.1 MEANING OF CORRELATION

.2 TYPES OF CORRELATION.2.1 P ,$&$ * and N*)a&$ * C ""*%a&$ n.2.2 L$n*a" and N n/%$n*a" C ""*%a&$ n

.3 MEASUREMENT TECHNI UES OF CORRELATION COEFFICEINT

.3.1 S-a&&*" D$a)"a!.3.2 ]a"% P*a", n , C *++$-$*n& + C ""*%a&$ n

.3.3 S(*a"!an , Ran: C ""*%a&$ n Ran:, a"* )$ *n d$"*-&%'

N n /"*(*a&*d "an:,R*(*a&*d "an:,

.4 PROPERTIES OF CORRELATION COEFFICEINT

. MEANING OF REGRESSION

.5 TYPES OF REGRESSION LINES.5.1 R*)"*,,$ n %$n*, + Z n Y.5.2 R*)"*,,$ n %$n* + Y n Z

.7 CONSTUCTION OF REGRESSION E UATIONS

.8 PROPERTIES OF REGRESSION COEFFICENTS

. DIFFERENCES BET EEN CORRELATION AND REGRESSION

.10 APPLICATIONS OF REGRESSION ANALYSIS



34/61


INT'O%U1TION

In & $, n$& ' 6$%% #* a#%* & %*a"n & * - n-*(& + - ""*%a&$ n and

"*)"*,,$ n. A%, +" ! & $, n$& ' 6$%% #* a#%* & %*a"n & * a"$ , !*& d,+ #&a$n$n) & * - ""*%a&$ n - *++$-$*n&,; "an: - ""*%a&$ n - *++$-$*n&;

"*)"*,,$ n *< a&$ n, *&-. T $, n$& *9(%a$n, & * d$++*"*n-*, #*&6**n & *- ""*%a&$ n and "*)"*,,$ n. I& $, *a,' & nd*",&and & * &*- n$< *, & #*d$,- ,,*d $n & $, n$& #' !a:$n) ,* + -a%- %a&$ n. T"' & & * *9a!(%*(" #%*!, 6$& & * -a%- %a& ".

9.# )$(NIN+ O& 1O''$*(TION

* a"* +a!$%$a" & a&; & * - an)* $n n* +a-& "; ,a'; & * a! n& +"a$n+a%% a++*-&, & * - an)* $n & * & *" +a-& ",; ,a'; '$*%d + "$-*. T $, !*an,& a& & *"* *9$,&, , !* :$nd + "*%a&$ n, $( #*&6**n & * &6 +a-& ",. T ,- ""*%a&$ n $, "*%a&$ n, $( #*&6**n &6 +a-& ",.

In ,$!(%* 6 "d,; - ""*%a&$ n !*an, & * d*)"** + "*%a&$ n, $(#*&6**n &6 " ! "* +a-& ", . An *9a!(%* + & * "*%a&$ n, $( & a& *9$,*&6**n & * ("$-* and d*!and.

9. T;!$S O& 1O''$*(TION

T *"* a"* d$++*"*n& &'(*, + - ""*%a&$ n. T *' -an #* -%a,,$+$*d $n& & *+ %% 6$n) -a&*) "$*,.

a@ P ,$&$ * and N*)a&$ * d*)"** - ""*%a&$ n#@ L$n*a" and N n/%$n*a" - ""*%a&$ n

F$",& 6* 6$%% d$,- ,, ( ,$&$ * and N*)a&$ * d*)"** - ""*%a&$ n

9. .# !ositive and Negative degree correlation

I+ & * - an)*, $n & * +a-& ", a"* $n & * ,a!* d$"*-&$ n & *n & *- ""*%a&$ n $, ,a$d & #* P ,$&$ * d*)"** - ""*%a&$ n . R*%a&$ n, $( #*&6**n& * a! n& + "a$n+a%% and '$*%d + "$-* $, an *9a!(%* + ( ,$&$ * d*)"**- ""*%a&$ n. I+ & * "a$n+a%% %* *% $n-"*a,*, & *n & * '$*%d + "$-* a%, $n-"*a,*,and $-*/ *",a.

N 6; 6* 6$%% d$,- ,, & * %$n*a" and N n/%$n*a" - ""*%a&$ n.



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9. . *inear and Non-linear 1orrelation

I+ & * - an)*, $n & * +a-& ", a"* $n & * - n,&an& "a&$ & *n & *- ""*%a&$ n $, ,a$d & #* L$n*a" - ""*%a&$ n .

F " *9a!(%*

A! n& + "a$n+a%%?$n !!@

40 50 80 100 120

Y$*%d + "$-* 100 10

200

2 0 300

F" ! & * a# * *9a!(%*; $& -an #* #,*" *d & a& a! n& + "a$n+a%%$n-"*a,*, 6$& 20 !! a& *a- %* *% and '$*%d + "$-* $n-"*a,*, 6$& 0& nn*, a& *a- %* *%.

I+ & * - an)*, $n & * +a-& ", a"* n & $n & * - n,&an& "a&$ & *n & *- ""*%a&$ n $, ,a$d & #* N n/%$n*a" - ""*%a&$ n .

F " *9a!(%*

Fa-& "1

40 50 80 100 120

Fa-& "2

100 1 0 200 2 0 300

F" ! & * a# * *9a!(%*; $& -an #* #,*" *d & a&; & * - an)*, a&a"$ , %* *%, a"* d$++*"*n&

Self (ssessment 8uestion

S&a&* & * d$++*"*n& &'(*, + - ""*%a&$ n 6$& *9a!(%* $n & * ,(a-* )$ *n#*% 6. L$!$& ' " an,6*" $n a# & 80 6 "d,.



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_____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ ________________________________

9.2.# Scatter %iagram

I+ & * a% *, + a"$a#%*, " +a-& ",; ,a' Z and Y $, (% &&*d $n & * ZY Q(%an*; & * d$a)"a! + & * da&a #&a$n*d $, -a%%*d a, ,-a&&*" d$a)"a!. T *)"*a&*" & * ,-a&&*" + & * (% &&*d ( $n&, n & * d$a)"a!; & * %*,,*" $, & *"*%a&$ n, $( #*&6**n & * &6 a"$a#%*, " +a-& ",

1. I+ a%% & * ( $n&, %$* n a ,&"a$) & %$n* +a%%$n) +" ! & * % 6*" %*+&/ and- "n*" & & * ((*" "$) &/ and - "n*"; & * - ""*%a&$ n $, ,a$d & #*(*"+*-&$ * ( ,$&$ *?F$) .1@ $.*. & * - ""*%a&$ n - *++$-$*n& " 1

&igure 9.# rA G#

2. I+ a%% & * ( $n&, %$* n a ,&"a$) & %$n* +a%%$n) +" ! & * ((*" %*+&/ and - "n*"& & * ((*" "$) &/ and - "n*"; & * - ""*%a&$ n $, ,a$d & #* (*"+*-&$ *%'n*)a&$ * ?$.*. & * - ""*%a&$ n - *++$-$*n& " /1@ F$) .2.

&igure 9. r A -#

F$) "* I+ a%% & * ( $n&, %$* n a ,&"a$) & %$n* +a%% $n a na"" 6 #and and & *', 6 a "$,$n) &*nd*n-' +" ! & * % 6*" %*+&/ and - "n*" & & * ((*" "$) &/

and - "n*"; & *"* 6 %d #* $) d*)"** + ( ,$&$ * - ""*%a&$ n. F$) .3

9.2 r A #



37/61


I+ a%% & * ( $n&, %$* n a ,&"a$) & %$n* +a%% $n a na"" 6 #and and & *', 6 a d*-%$n$n) &*nd*n-' +" ! & * ((*" %*+& and - "n*" & & * % 6*""$) &/ and - "n*"; & *"* 6 %d #* $) d*)"** + n*)a&$ * - ""*%a&$ n.F$) .4.

F$) .4 " 1

I+ a a%% & * ( $n&, %$* n a ,&"a$) & %$n* +a%% $n a 6$d*%' #and and & *', 6 a "$,$n) &*nd*n-' +" ! ^& * % 6*" %*+&/ and - "n*" & & * ((*""$) &/ and - "n*"; & *"* 6 %d #* % 6 d*)"** + ( ,$&$ * - ""*%a&$ n.F$) . .

F$) . " 0

I+ a%% & * ( $n&, %$* n a ,&"a$) & %$n* +a%% $n a 6$d*%' #and and & *' , 6 ad*-%$n$n) &*nd*n-' +" ! & * ((*" %*+& and - "n*" & & * % 6*" "$) & and- "n*"; & *"* 6 %d #* % 6 d*)"** + n*)a&$ * - ""*%a&$ n. F$) .5

F$) .5 " 1

I+ & * (% &&*d ( $n&, %$* n a ,&"a$) & %$n* (a"a%%*% & 9/a9$, " $n

a( a>a"d !ann*" $& , 6, & * a#,*n-* + - ""*%a&$ n #*&6**n &6+a-& ",. F$) .7.

F$) .7 " 0



38/61


9.2. H('* !$('SON S 1O$&&I1$INT O& 1O''$*(TION

A, a !*a, "* + d*)"** + %$n*a" "*%a&$ n, $( #*&6**n &6 a"$a#%*,;]a"% P*a", n d* *% (*d a + "! %a -a%%*d - ""*%a&$ n - *++$-$*n&. T *- ""*%a&$ n - *++$-$*n& #*&6**n &6 a"$a#%* , a%%' d*n &*d #' " 9'; $, a!*a, "* + "*%a&$ n, $( #*&6**n & *! $, d*+$n*d a,;

"9' C ?9;'@ 9 '

Z/ZY/YZ/Z2Y/Y2

9'92'2

*"* 9 Z/Z ' Y/Y

Dor"ing !rocedure

Step # D*n &* n* ,*"$*, #' Z and & *" #' Y

Step 6 Ca%- %a&*Z and Y + & * Z and Y ,*"$*, "*,(*-&$ *%'; ,$n) & *+ "! %a;

Z Zn and Y Yn

Step 26 Ta:* & * d* $a&$ n, + & * #,*" a&$ n, $n Z/,*"$*, and +" ! Z and6"$&* $& nd*" & * - % !n *ad*d #' 9 /Z . Ta:* & * d* $a&$ n + & *

#,*" a&$ n, $n Y ,*"$*, +" ! Y and 6"$&* $& nd*" & * - % !n ' Y/Y.

Step 46 M %&$(%' & * "*,(*-&$ * d* $a&$ n, and 6"$&* $& nd*" & * - % !n*ad*d #' 9'.

Step 96 S< a"* & * d* $a&$ n, #&a$n*d $n ,&*( 4 + " Z and Y ,*"$*, and 6"$&*$& nd*" & * - % !n *ad*d #' 9 2 and ' 2 .

Step ?6 A((%' & * + %% 6$n) + "! %a & -a%- %a&* & * - ""*%a&$ n - *++$-$*n&?"@.

"9' 9'92'2

$5ample 9.# F$nd & * - *++$-$*n& + - ""*%a&$ n #*&6**n *$) & + #" & *",and ,$,&*", +" ! & * + %% 6$n) da&a



39/61


:eig t of Erot ers,in cm

5 55

57

58 5 70 71

:eig t of Sisters,in cm

57

58

55

5 72 72 5

Solution6 L*& & * *$) &, + B" & *", #* d*n &*d #' Z and & a& + S$,&*", #' Y. L*& , ("*(a"* & * + %% 6$n) &a#%*

J 5 A Z/Z ; A ZA ^ZnA 4757A ?F

Y/Y

5 J ;

5 /3 57 /2 5 4

55 /2 58 /1 2 4 1

57 /1 55 /3 3 1

58 0 5 0 0 0 0

5 1 72 3 3 1

70 2 72 3 5 4

71 3 5 0 0 0

4@?

- 4F2

- 0F

2

F" ! & * a# * &a#%*.

N 7 ^Z 475 ^Y 483 ^9' 20 ^9 2 28 ^' 2 32

Z A ^Zn A 4757 A ?F

Y A ^Yn A 4837 A ?B

]a"% P*a", n , C *++$-$*n& + C ""*%a&$ n $, n 6 -a%- %a&*d a, + %% 6,

" 9'92'2



40/61


2028 32

20 .2 1 ? .5 5 202 . 33

" 0.05581


Ca%- %a&* & * - ""*%a&$ n - *++$-$*n& #*&6**n & * *$) & + ,$,&*" and *$) &+ & * #" & *", +" ! & * )$ *n da&a

:eig t of Sisters,in cm

54

5 55

57

58

5 70

:eig t of Erot ers,in cm

55

57

5 58

70

58 72

[H$n& Z 57; Y 58; ^Z 2 28; ^Y 2 34; ^9' 2 ; " 0.81\

S ort 1ut )et od

T * a# * d$"*-& !*& d + " -a%- %a&$n) " $, n & - n *n$*n& 6 *n ?$@& * &*"!, + & * S*"$*, Z and Y a"* %a")*" and & * -a%- %a&$ n + Z and Y#*- !* d$++$- %& ? "@ ?$$@ & * !*an + Z " Y a"* n & $n&*)*",. In & *,* -a,*,6* a((%' & * + %% 6$n) + "! %a + a,, !*d !*an

"9' n^d9d'/?^d9@?^d'@n^d92/?^d9@2n^d'2/?^d'@2

6 *"*;

d9 Z/A; A $, & * a,, !*d !*an + Z Q ,*"$*,

d' Y/B; B $, & * a,, !*d !*an + Y Q ,*"$*,

n $, n !#*" + #,*" a&$ n + Z and Y

":$n) P" -*d "*

Step #6 D*n &* n* ,*"$*, #' Z and & * & *" #' Y.

Step 6 Ta:* an' &*"! A a, a,, !*d !*an + Z ,*"$*, and B a, a,, !*d!*an + Y ,*"$*, ?("*+*"a#%' & * !$dd%* n*@.



41/61


Step26 Ta:* & * d* $a&$ n, + & * #,*" a&$ n, $n Z Q ,*"$*, +" ! A and 6"$&$& nd*" & * - % !n *ad*d #' d9 Z/A. Ta:* & * d* $a&$ n, + & *

#,*" a&$ n, $n Y ,*"$*, +" ! B and 6"$&* $& nd*" & * - % !n *ad*d #' d' Y/B.

Step 46 M %&$(%' & * "*,(*-&$ * d* $a&$ n, and 6"$&* $& nd*" & * - % !n*ad*d #' d9 d'.

Step 96 S< a"* & * d* $a&$ n, #&a$n*d $n ,&*( 4 +" Z and Y ,*"$*, and 6"$&*$& nd*" & * - % !n *ad*d #' d9 2 and d' 2 .

Step ?6 A((%' & * + %% 6$n) + "! %a & -a%- %a&* & * - ""*%a&$ n - *++$-$*n&?"@.

"9' n^d9d'/?^d9@?^d'@n^d92/?^d9@2n^d'2/?^d'@2

E9a!(%* .2 Ca%- %a&* & * - *++$-$*n& + - ""*%a&$ n + " & * + %% 6$n) (a$",+ a% *, + Z and Y.

Z 17 1 21 25 20 28 25 2

Y 23 27 2 25 27 2 30 33

S % &$ n

L*& & * a,, !*d !*an, + " Z and Y #* 23 and 27 "*,(*-&$ *%'; , & a& d9 Z/23; d' Y/27;

* a * & * + %% 6$n) &a#%*

J ; d 5 A J-2

d A ;-@

d 5d d 5 d

17 23 /5 /4 24 35 15

1 27 /4 0 0 15 0

21 2 /2 /2 4 4 4

25 25 3 /1 /3 1

20 27 /3 0 0 0

28 2 /2 /10 2 4

25 30 3 3



42/61


2 33 5 5 35 35 35

#F? #? 0 ?0 #44 @0

N &* & a&; *"*Z ^Zn 1858 23.2 ; 6 $- $, n & an $n&*)*"; 6* ,*, "&/- & !*& d;

H*"* n 8; ^d9 2; ^d' 0; ^d9d' 50; ^d9 2 144; ^d' 2 70;

"9' n^d9d'/?^d9@?^d'@n^d92/?^d9@2n^d'2/?^d'@2

"9' 850/2?0@8?144@/?2@28?70@/?0@2

"9' 4801148 50

"9' 48033.8821?23.5543@

"9' 480801.7 52

"9' 0. 87


C !( &* & * - *++$-$*n& + - ""*%a&$ n + " & * + %% 6$n) da&a

Z 10 2 13 2 22 11 12 2 21 20

Y 12 22 15 1 18 18 17 23 24 17

[An, " 9' 0. 3\

9.2.2 6 Spearman s 'an" 1orrelation 1oefficient

T * - *++$-$*n& + "an: - ""*%a&$ n $, #a,*d n & * a"$ , a% *, + & *a"$*&$*, and $, d*n &*d #'

` 1/ 5^D2n3/n

6 *"*; D Q $, & * d$++*"*n-* + - ""*,( nd$n) "an:, and n Q $, & * n !#*" "(a$", + #,*" a&$ n,.



43/61


T;!$ I6 '(NHS ('$ +IV$N %I'$1T*;

Dor"ing !rocedure

Step #6 D*n &* "an: + Z ,*"$*, #' R 1 and "an: + Y ,*"$*, #' R 2 .

Step 6 Ca%- %a&* & * d$++*"*n-* " R1 and R 2 and 6"$&* $& nd*" & * - % !n*ad*d #' D

Step 26 S< a"* & * d$++*"*n-* D and 6"$&* $& nd*" & * - % !n *ad*d #' D 2 .

Step 46 A((%' & * + "! %a

` 1/ 5^D2n3/n

T $, !*& d $, d*,-"$#*d 6$& + %% 6$n) *9a!(%*

$5ample 9.26 T6 = d)*, $n a #*a &' - n&*,& "an: & * 12 *n&"$*, a, + %% 6,.

J d)*Z

1 2 3 4 5 7 8 10 11 12

J d)* Y

12

5 10 3 4 7 8 2 11 1

Ca%- %a&* & * "an: - ""*%a&$ n - *++$-$*n& #*&6**n & * &6 = d)*, Z and Y.

7udge J,' #

7udge ;,'

%A' # -'

%

1 12 /11 121

2 /7 4

3 5 /3

4 10 /5 35

3 2 45 1 1

7 4 3

8 7 1 1

8 1 1



44/61


10 2 8 54

11 11 0 0

12 1 11 121

Total 4#?

H*"* n 12 ^D 2 415

N 6;

` 1/ 5^D2n3/n

` 1/ 5?415@123/12

` 1/ 24 51728/12

` 1/ 24 51715 1/ 1.4 4

` /0.4 4

$5ample 9.46 T*n - !(*&$& ", $n a #*a &' - n&*,& 6*"* "an:*d #' & "** = d)*, $n & * + %% 6$n) "d*"

7udge#

1 5 10 3 2 4 7 8

7udge 3 8 4 7 10 2 1 5

7udge2

5 4 8 1 2 3 10 7

U,* & * "an: - ""*%a&$ n - *++$-$*n& & d*&*"!$n* 6 $- (a$" + = d)*, a, & *n*a"*,& a((" a- & - !! n &a,&* $n #*a &'.

S % &$ n

L*& R1; R 2; R 3 "*,(*-&$ *%' #* & * "an:, )$ *n #' +$",&; ,*- nd and & $"d = d)*.

L*& `$ = #* & * "an: - ""*%a&$ n - *++$-$*n& #*&6**n & * "an:, )$ *n #' $ & and =& = d)*,; $ 1;2;3 = 1;2;3.



45/61


L*& D$= R$ Q R=; #* & * d$++*"*n-* + "an:, + an $nd$ $d a% )$ * #' $ & and J &

J d)*.

7udge#

' #

7udge

'

7udge2

' 2

%# A' # -'

%# % 2 A' -' 2

% 2 %#2 A' # -' 2

%#2

1 3 5 /2 4 /3 / 2

5 4 1 1 1 1 2 4

8 /3 /1 1 4 15

10 4 8 5 35 /4 15 2 4

3 7 1 /4 15 5 35 2 4

2 10 2 /8 54 8 54 0 0

4 2 3 2 4 /1 1 1 1

1 10 8 54 / 81 /1 1

7 5 /1 1 1 1 2 4

8 7 /1 1 2 4 1 1 T &a% 0

0#

4?0

H*"* n 10 ^D 12 2 200; ^D 23 2 214 ^D 13 2 50

&irst and Second 7udges

` 12 1 / 5^D122n3/n 1/ 5?200@103/10 1/ 1200 0 1/ 1.2121 0.2121Second and T ird 7udges

` 23 1 / 5^D232n3/n 1/ 5?214@103/10 1/ 1284 0 1/ 1.2 5 0.2 5

&irst and T ird 7udges



46/61


` 13 1 / 5^D132n3/n 1/ 5?50@103/10 1/ 350 0 1/ 0.3535 0.5354

S$n-* ` 13 $, !a9$! !; & , & * (a$" + & * +$",& and & $"d = d)*, a, & *n*a"*,& a((" a- & - !! n &a,&* $n #*a &'.

Self Cassessment 8uestion

1. T6 = d)*, $n a ! ,$-a% - n&*,& "an: & * 10 *n&"$*, a, + %% 6, J d)*Z

3 8 4 7 10 2 1 5

J d)* Y

5 4 8 1 2 3 10 7

[H$n& n 10 ^D2

14 ` 0.84 \

2. T*n C !(*&$& ", $n a #*a &' - n&*,& 6*"* "an:*d #' & "** = d)*, $n& * + %% 6$n) "d*"

1 ,&

J d)*1 4 8 5 10 7 3 2

2 nd

J d)*4 8 7 5 10 3 2 1

3 "d

J d)*5 7 8 1 10 2 3 4

U,* ,(*a"!an , - *++$-$*n& + "an: - ""*%a&$ n & d*&*"!$n* 6 $- (a$" + = d)*, a, & * n*a"*,& a((" a- & - !! n &a,&* $n #*a &'

[H$n& n 10 ^D 12 2 74; ^D 23 2 44; ^D 13 2 1 5; ` 12 0 1 ; ` 23 0.7333` 13 0.0 4 \

T;!$ II6 '(NHS ('$ NOT +IV$N C NON C '$!$(T$% '(NHS

In & $, -a,* 6* a"* )$ *n n%' & * da&a. * a,,$)n & * "an:, & # && * ,*"$*, + Z and Y #' )$ $n) & * "an:, $n a,-*nd$n) "d*" + " # & ,*"$*,? " d*,-*nd$n) "d*"@.

Dor"ing 'ule



47/61


Step #6 A,,$)n "an:, & *a- $&*!, + # & ,*"$*, $n a,-*nd$n) "d*,-*nd$n) "d*".

Step 6 Ca%- %a&* & * d$++*"*n-* + "an:, and 6"$&* $& nd*" & * - % !n*ad*d #' D.

Step 26 S< a"* & * d$++*"*n-* D and 6"$&* $& nd*" & * *ad$n) D 2.

Step 46 A((%' & * + "! %a;

` 1 / 5^D2n3/n

T $, !*& d $, *9(%a$n*d 6$& & * *%( + & * + %% 6$n) *9a!(%*.

$5ample 9.9

F " & * + %% 6$n) da&a -a%- %a&* & * - *++$-$*n& + "an: - ""*%a&$ n.SeriesJ

80 1 71 51 81 70

Series ;

123 13 1 4 110 10 134 121 105

Solution6

SeriesJ

Series ;

'an" J

,' #

'an" ;

,'

% %

80 123 0 0

1 13 7 7 0 0

1 4 8 8 0 0

71 110 4 3 1 1

51 10 2 1 1 1

81 134 5 5 0 070 121 3 4 /1 1

105 1 2 /1 1

Total

4



48/61


H*"*; n 8 ^D 2 4

N 6;

` 1 / 5^D2n3/n 1 Q 5 ?4@83/8 1 / 24 04 1/0.0475 0.0 24


Ca%- %a&* & * "an: - ""*%a&$ n - *++$-$*n& + " & * + %% 6$n) da&a + &6 ,*"$*,

S*"$*,Z

2 8 87 85 83 77 71 53 3 0

S*"$*,

Y

85 83 1 77 58 8 2 82 37 7

[H$n& n 10 ^D 2 44 ` 0.733\

T;!$ III6 '(NHS ('$ NOT +IV$N C '$!$(T$% '(NHS

I+ &6 " ! "* $nd$ $d a%, a"* (%a-*d & )*& *" $n an' -%a,,$+$-a&$ n 6$&"*,(*-& & an a&&"$# &*; & *"* a"* ! "* & an n* $&*! 6$& & * ,a!* "an: $n*$& *" " # & & * ,*"$*,; & *n & * (" #%*! $, , % *d #' a,,$)n$n) a *"a)*"an: & *a- + & *$" $nd$ $d a%, 6 a"* ( & $n a &$*.

F " *9a!(%*; , (( ,* an $&*! $, "*(*a&*d a& "an: ; ?$.*.; & * & and5 & $&*! a"* a $n) ,a!* a% *,@; & *n & * - !! n "an: a,,$)n*d & & *and 5 & $, ? 5@ 2 . . T * n*9& "an: a,,$)n*d & "$-*; & *n & * - !! n "an:a,,$)n*d & & * a% * $, , ! + & * "an:, #' d$ $d*d #' 3. In "d*" & +$nd & *"an: - ""*%a&$ n - *++$-$*n& & * ad= ,&!*n& +a-& " $, add*d & & * + "! %a;6 $- $, )$ *n #'

Ad= ,&!*n& Fa-& " ?A.F@ 112 ?! 3/!@

*"* ! $, & * n !#*" + &$!*, an $&*! $, "*(*a&*d. T $,Ad= ,&!*n&, Fa-& " $, & #* add*d + " *a- "*(*a&*d a% * $n # & & * ,*"$*,.

T * ! d$+$*d + "! %a + " & * "an: - ""*%a&$ n - *++$-$*n& $, )$ *n #';

` 1 Q 5[^D2 112^?!3/!@\n3/n

T $, !*& d $, *9(%a$n*d 6$& & * + %% 6$n) *9a!(%*;



49/61


$5ample 9.? F" ! & * + %% 6$n) da&a "*%a&*d & & * ,*"$*, Z and Y; -a%- %a&*& * - *++$-$*n& + "an: - ""*%a&$ n.

S*"$*,Z

48 33 40 15 15 5 24 15 7

S*"$*, Y

13 13 24 5 1 4 20 5 1

Solution

SeriesJ

Series ;

'an" J,' #

'an" ;,'

%A' # -'

%

48 13 8 . 2. 5.2

33 13 5 . 0. 0.2

40 24 7 10 /3 .00

5 1 2. /1. 2.2

15 1 3 7 /4 15.00

15 4 3 1 2 4.00

5 20 10 1 1.0024 4 1 1.00

15 5 3 2. 0. 0.2

7 1 8 1 1.00

Total 4#.00

H*"* n 10; ^D 2 41

[R*!a": In & * Z ,*"$*,; 6* ,** & a& & * a% * 15 $, "*(*a&*d & "$-*; & *- !! n "an: $, )$ *n & & * Z a% * $, 3; 6 $- $, & * a *"a)* + 2.3 and 4.$.*.; ?2 3 4@ 3 3\

N 6; Ad= ,&*!*n& Fa-& "



50/61


F " Z ,*"$*,; AF 1 112 ?3 3/3@ 2

F " Y ,*"$*,; ;

AF2 112 ?2 3/2@ 0.

AF3 112 ?2 3/2@ 0.

T * - *++$-$*n& + "an: - ""*%a&$ n $,;

` 1 Q 5 [ ^D2 112^!3/!\n3/n

1 Q 5[41 2 0. 0. \103/10

1 Q 544 0

1 Q 254 0

1/ 0.2557

` 0.7333


O#&a$n & * "an: - ""*%a&$ n - *++$-$*n& + " & * + %% 6$n) da&aS*"$*,Z

58 54 7 0 54 80 7 40 54

S*"$*, Y

52 8 58 4 81 50 58 48 0 70

[H$n& n 10 ^D 2 72 ` 0. 4 \

9.4 properties of 1orrelation 1oefficient T * a% * + " d *, n & d*(*nd n 6 $- + & * &6 a"$a#%*, nd*"

,& d' $, %a#*%*d Z and 6 $- $, %a#*%*d Y. T * - ""*%a&$ n - *++$-$*n& %$*, #*&6**n /1 and 1 $.*.; /1W"W 1 T * - ""*%a&$ n - *++$-$*n& $, $nd*(*nd*n& + - an)* + "$)$n and

,-a%*.



51/61


" 1; $+ a%% ?Z$; Y =@ (a$", %$* n a ,&"a$) & %$n* 6$& ( ,$&$ * ,% (* and

" /1; $+ a%% ?Z$; Y =@ (a$", %$* n a ,&"a$) & %$n* 6$& n*)a&$ * ,% (*.

9.9 '$+'$SSION (N(*;SIS

Mana)*", +&*n !a:* d*-$,$ n, #' ,& d'$n) & * "*%a&$ n, $( #*&6**na"$a#%*, and (" -*,, $!(" *!*n&, -an +&*n #* !ad* #' nd*",&and$n)

6 - an)*, $n n* " ! "* a"$a#%*, a++*-& & * (" -*,, &( &. R*)"*,,$ nAna%',$, $, a ,&a&$,&$-a% &*- n$< * $n 6 $- 6* #,*" * da&a & "*%a&* a

a"$a#%* + $n&*"*,&; 6 $- $, -a%%*d & * d*(*nd*n& ? " "*,( n,*@ a"$a#%*; &n* " ! "* $nd*(*nd*n& ? " ("*d$-a& "@ a"$a#%*. T * #=*-&$ * $, & # $%d a

"*)"*,,$ n ! d*%; " ("*d$-&$ n *< a&$ n; & a& -an #* ,*d & d*,-"$#* ;("*d$-& and - n&" % & * d*(*nd*n& a"$a#%* n & * #a,$, + & * $nd*(*nd*n&

a"$a#%*,. F " *9a!(%*; a - !(an' !$) & 6$, & $!(" * $&, !a":*&$n)

(" -*,,. A+&*" - %%*-&$n) da&a - n-*"n$n) & * d*!and + " a (" d -&; & *(" d -& , ("$-*; and & * ad *"&$,$n) *9(*nd$& "*, !ad* & (" ! &* & *(" d -&; & * - !(an' !$) & ,* "*)"*,,$ n ana%',$, & d* *% ( an *< a&$ n& ("*d$-& d*!and n & * #a,$, + ("$-* and ad *"&$,$n) *9(*nd$& "*.P"*d$-&$ n, + d*!and + " a"$ , ("$-*/ad *"&$,$n) *9(*nd$& "*- !#$na&$ n, -an & *n #* ,*d & * a% a&* ( &*n&$a% - an)*, $n & *- !(an' , !a":*&$n) ,&"a&*)$*,.

In & * 6 "d, + M.M. B%a$";

R*)"*,,$ n ana%',$, $, a !a& *!a&$-a% !*a, "* + a *"a)*"*%a&$ n, $( #*&6**n &6 " ! "* a"$a#%*, $n &*"!, + & * "$)$na% n$& +& * da&a .

9.9. T pes of 'egression *ines

A %$n* + "*)"*,,$ n $, & * %$n*; 6 $- )$ *, & * #*,& *,&$!a&* + n* a"$a#%*Z; + " an' )$ *n a% * + & * & *" a"$a#%*. * a * &6 &'(*, + "*)"*,,$ n%$n*,; na!*%';

R*)"*,,$ n %$n* + Z n Y R*)"*,,$ n %$n* + Y n Z.

F$",& 6* 6$%% )$ * & * "*)"*,,$ n %$n* + Z n Y.

I& $, & * %$n*; 6 $- )$ *, & * #*,& *,&$!a&* + " & * a% *, + Z + " a ,(*-$+$*da% * + Y.



52/61


I& $, )$ *n #'

Z / Z # 9' ?Y / Y@

*"* # 9' $, & * "*)"*,,$ n - *++$-$*n& + Z n Y; 6 $- -an #* -a%- %a&*d

,$n) an' + & + "! %a nd*" & * na& "*, + & * da&a# 9' ^9'^'2

6 *"*; 9 Z / Z and ' Y / Y

"

# 9' n^d9d'/?^d9@?^d'@n^d92/?^d'@2

6 *"*; d 9 Z Q A; d ' Y Q B and A; B a"* a,, !*d !*an

"

# 9' " 9 '

6 *"* " $, & * - ""*%a&$ n - *++$-$*n&; 9 a n d ' a"* & * ,&anda"dd* $a&$ n, + " Z and Y ,*"$*,.

N 6 6* )$ * & * "*)"*,,$ n %$n* + Y n Z.

I& $, & * %$n*; 6 $- )$ *, & * #*,& *,&$!a&* + " & * a% * + Y + " a,(*-$+$*d a% * + Z.

I& $, )$ *n #'

Y / Y # '9 ?Z / Z@

*"* # '9 $, & * "*)"*,,$ n %$n* + Y n Z; 6 $- -an #* -a%- %a&*d ,$n)an' n* + & * + %% 6$n) + "! %a d*(*nd$n) ( n & * na& "* + da&a.

# '9 ^9'^92 6 *"* 9 Z / Z and ' Y / Y

"

# 9' n^d9d'/?^d9@?^d'@n^d92/?^d'@2 6 *"*; d 9 Z/ A d ' Y/B and A; B a"*a,, !*d !*an.

"



53/61


# '9 " ' 9 6 *"* " $, & * - ""*%a&$ n - *++$-$*n& 9; ' a"* & * ,&anda"dd* $a&$ n, + Z and Y ,*"$*,.

9.@ 1ONST'U1TION O& '$+'$SSION $8U(TION

$5ample 9.@ T * *$) & + a ,a!(%* + 10 +a& *", and & *$" *%d*,& , n, a"*)$ *n #*% 6 & & * n*a"*,& -!@.

H*$) & + Fa& *"?Z@

170 157 152 153 157 155 15 171 155 15

H*$) & + S n?Y@

155 157 154 155 155 154 158 170 153 155

?$@ O#&a$n & * &6 "*)"*,,$ n *< a&$ n,?$$@ E,&$!a&* & * %$:*%' *$) & + Fa& *" 6 *n & * *$) & + S n $, 1 0

-!.?$$$@ E,&$!a&* & * %$:*%' *$) &

22536177 mathematics and statistics unit iv v

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