biostatistic for medical

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LECTURE NOTES

For Health Science Students

Biostatistics

Getu DeguFasil Tessema

University of Gondar

In collaboration with the Ethiopia Public Health Training Initiative, The Carter Center,the Ethiopia Ministry of Health, and the Ethiopia Ministry of Education

January 2005



Funded under USAID Cooperative Agreement No. 663-A-00-00-0358-00.

Produced in collaboration with the Ethiopia Public Health Training Initiative, The Carter

Center, the Ethiopia Ministry of Health, and the Ethiopia Ministry of Education.

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publication for educational, not-for-profit use by health care workers, students or

faculty. All copies must retain all author credits and copyright notices included in theoriginal document. Under no circumstances is it permissible to sell or distribute on acommercial basis, or to claim authorship of, copies of material reproduced from thispublication.

©2005 by Getu Degu and Fasil Tessema

All rights reserved. Except as expressly provided above, no part of this publication maybe reproduced or transmitted in any form or by any means, electronic or mechanical,including photocopying, recording, or by any information storage and retrieval system,without written permission of the author or authors.

This material is intended for educational use only by practicing health care workers or

students and faculty in a health care field.



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PREFACE

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Acknowledgments

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Table of Contents

Preface i

Acknowledgements iii

Table of contents iv

Chapter One : Introduction to Statistics

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Chapter Two: Methods of Data collection,

Organization and presentation

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Chapter Three : Summarizing data

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Chapter Four : Demographic Methods and

Health Services Statistics

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Chapter Five : Elementary Probability

and probability distribution

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Chapter Six : Sampling methods

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Chapter seven : Estimation

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Chapter Eight : Hypothesis Testing

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Chapter Nine: Correlation and Regression

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Appendix : Statistical tables RUU

References RYS



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CHAPTER ONE

Introduction to Statistics

1.1. Learning objectives

After completing this chapter, the student will be able to:

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%,'*/#+#=29 :$ %*"$ /,7 5.-= #. 0.#',5-., <,%%,. %*&/ %*, #/,

'#11#/+2 "/ -$,a 4#7 1-'* <,%%,.a D*&%8 "3 &/28 &., %*, ."$C$

#3 $"5, ,33,'%$ &$$#'"&%,5 7"%* "%$ -$,a :/ %,$%"/= & /,7 5.-= *#7

1&/2 0&%",/%$ 1-$% <, %.,&%,58 &/5 "/ 7*&% 1&//,.8 "/ #.5,. %#

5,1#/$%.&%, "%$ 7#.%*a D*&% "$ %*, /#.1&+ ;&."&%"#/ "/ $#1,

'+"/"'&+ 1,&$-.,1,/%a 4#7 .,+"&<+, &/5 ;&+"5 "$ %*,



!"#$%&%"$%"'$

Y

1,&$-.,1,/%a D*&% "$ %*, 1&=/"%-5, &/5 ,33,'% #3 +&<#.&%#.2

&/5 %,'*/"'&+ ,..#.a 4#7 5#,$ #/, "/%,.0.,% &</#.1&+ ;&+-,$a

• E%&%"$%"'$ 0,.;&5,$ %*, 1,5"'&+ +"%,.&%-.,9 H$ & '#/$,>-,/', #3

%*, "/'.,&$"/=+2 >-&/%"%&%";, /&%-., #3 0-<+"' *,&+%* &/5 1,5"'"/,

&/5 "%$ .,+"&/', #/ $%&%"$%"'&+ 1,%*#5#+#=28 %*, 1,5"'&+ +"%,.&%-.,

"$ .,0+,%, 7"%* .,0#.%$ "/ 7*"'* $%&%"$%"'&+ %,'*/">-,$ &., -$,5

,B%,/$";,+29

"It is the interpretation of data in the presence of such variability

that lays at the heart of statistics."

Limitations of statistics:

:% 5,&+$ 7"%* #/+2 %*#$, $-<J,'%$ #3 "/>-".2 %*&% &., '&0&<+, #3 <,"/=

>-&/%"%&%";,+2 1,&$-.,5 &/5 /-1,."'&++2 ,B0.,$$,59

O9 :% 5,&+$ #/ &==.,=&%,$ #3 3&'%$ &/5 /# "10#.%&/', "$ &%%&'*,5 %#

"/5";"5-&+ "%,1$f$-"%,5 #/+2 "3 %*,". =.#-0 '*&.&'%,."$%"'$ &., 5,$".,5

%# <, $%-5",59

R9 E%&%"$%"'&+ 5&%& &., #/+2 &00.#B"1&%,+2 &/5 /#% 1&%*,1&%"'&++2

'#..,'%9



!"#$%&%"$%"'$

W

1.4 Scales of measurement

H/2 &$0,'% #3 &/ "/5";"5-&+ %*&% "$ 1,&$-.,5 &/5 %&C, &/2 ;&+-, 3#.

5"33,.,/% "/5";"5-&+$ #. '&$,$8 +"C, <+##5 0.,$$-.,8 #. .,'#.5$8 +"C, &=,8

$,B "$ '&++,5 & variable.

:% "$ *,+03-+ %# 5";"5, ;&."&<+,$ "/%# 5"33,.,/% %20,$8 &$ 5"33,.,/%

$%&%"$%"'&+ 1,%*#5$ &., &00+"'&<+, %# ,&'*9 )*, 1&"/ 5";"$"#/ "$ "/%#

>-&+"%&%";, L#. '&%,=#."'&+N #. >-&/%"%&%";, L#. /-1,."'&+ ;&."&<+,$N9

Qualitative variable: & ;&."&<+, #. '*&.&'%,."$%"' 7*"'* '&//#% <,

1,&$-.,5 "/ >-&/%"%&%";, 3#.1 <-% '&/ #/+2 <, "5,/%"3",5 <2 /&1, #.

'&%,=#.",$8 3#. "/$%&/', 0+&', #3 <".%*8 ,%*/"' =.#-08 %20, #3 5.-=8

$%&=,$ #3 <.,&$% '&/',. L:8 ::8 :::8 #. :ZN8 5,=.,, #3 0&"/ L1"/"1&+8

1#5,.&%,8 $,;,., #. -/<,&.&<+,N9

Quantitative variable: H >-&/%"%&%";, ;&."&<+, "$ #/, %*&% '&/ <,

1,&$-.,5 &/5 ,B0.,$$,5 /-1,."'&++2 &/5 %*,2 '&/ <, #3 %7# %20,$

L5"$'.,%, #. '#/%"/-#-$N9 )*, ;&+-,$ #3 & 5"$'.,%, ;&."&<+, &., -$-&++2

7*#+, /-1<,.$8 $-'* &$ %*, /-1<,. #3 ,0"$#5,$ #3 5"&..*#,& "/ %*,

3".$% 3";, 2,&.$ #3 +"3,9 H '#/%"/-#-$ ;&."&<+, "$ & 1,&$-.,1,/% #/ &

'#/%"/-#-$ $'&+,9 ?B&10+,$ "/'+-5, 7,"=*%8 *,"=*%8 <+##5 0.,$$-.,8

&=,8 ,%'9

H+%*#-=* %*, %20,$ #3 ;&."&<+,$ '#-+5 <, <.#&5+2 5";"5,5 "/%#

'&%,=#."'&+ L>-&+"%&%";,N &/5 >-&/%"%&%";, 8 "% *&$ <,,/ & '#11#/

0.&'%"', %# $,, 3#-. <&$"' %20,$ #3 5&%& L$'&+,$ #3 1,&$-.,1,/%N9



!"#$%&%"$%"'$

]

Nominal data:- K&%& %*&% .,0.,$,/% '&%,=#.",$ #. /&1,$9 )*,., "$ /#

"10+",5 #.5,. %# %*, '&%,=#.",$ #3 /#1"/&+ 5&%&9 :/ %*,$, %20,$ #3 5&%&8

"/5";"5-&+$ &., $"10+2 0+&',5 "/ %*, 0.#0,. '&%,=#.2 #. =.#-08 &/5 %*,

/-1<,. "/ ,&'* '&%,=#.2 "$ '#-/%,59 ?&'* "%,1 1-$% 3"% "/%# ,B&'%+2 #/,

'&%,=#.29

)*, $"10+,$% 5&%& '#/$"$% #3 -/#.5,.,58 5"'*#%#1#-$8 #. h,"%*,. ` #.h

%20,$ #3 #<$,.;&%"#/$8 "9,98 ,"%*,. %*, 0&%",/% +";,$ #. %*, 0&%",/% 5",$8

,"%*,. *, *&$ $#1, 0&.%"'-+&. &%%."<-%, #. *, 5#,$ /#%9

,=9 _#1"/&+ $'&+, 5&%&c $-.;";&+ $%&%-$ #3 0.#0&/#+#+ ` %.,&%,5 &/5

'#/%.#+ 0&%",/%$ 7"%* 12#'&.5"&+ "/3&.'%"#/

E%&%-$ R] 5&2$

&3%,. *#$0"%&+

&51"$$"#/

^.#0.&/#+#+

`%.,&%,5 0&%",/%

G#/%.#+

^&%",/%$

K,&5 W OW

H+";, S] RX

)#%&+

E-.;";&+ .&%,

VU

]Vi

VY

YSi

E#-.',c $/#78 ,33,'% #3 0.#0.&/#+#+ "/ 6: j)*, P&/',%8 OXYU9

)*, &<#;, %&<+, 0.,$,/%$ 5&%& 3.#1 & '+"/"'&+ %."&+ #3 %*, 5.-=0.#0.&/#+#+ "/ %*, %.,&%1,/% #3 12#'&.5"&+ "/3&.'%"#/9 )*,., 7,., %7#

=.#-0 #3 12#'&.5"&+ "/3&.'%"#/9 )*,., 7,., %7# =.#-0 #3 0&%",/%$ 7"%*

6:9 Q/, =.#-0 .,',";,5 0.#0.&/#+#+j %*, #%*,. 5"5 /#% &/5 7&$ %*,

'#/%.#+9 \#. ,&'* 0&%",/% %*, .,$0#/$, 7&$ 5"'*#%#1#-$j ,"%*,. *,



!"#$%&%"$%"'$

X

$-.;";,5 %*, 3".$% R] 5&2$ &3%,. *#$0"%&+ &51"$$"#/ #. *, $-''-1<,5

L5",5N $#1,%"1, 7"%*"/ %*"$ %"1, 0,."#59

D"%* /#1"/&+ $'&+, 5&%& %*, #<;"#-$ &/5 "/%-"%";, 5,$'."0%";, $-11&.2

1,&$-., "$ %*, 0.#0#.%"#/ #. 0,.',/%&=, #3 $-<J,'%$ 7*# ,B*"<"% %*,

&%%."<-%,9 )*-$8 7, '&/ $,, 3.#1 %*, &<#;, %&<+, %*&% ]V 0,.',/% #3 %*,

0&%",/%$ %.,&%,5 7"%* 0.#0.&/#+#+ $-.;";,58 "/ '#/%.&$% 7"%* #/+2 YSi #3

%*, '#/%.#+ =.#-09

Some other examples of nominal datac

?2, '#+#. ` <.#7/8 <+&'C8 ,%'9

T,+"="#/ ` G*."$%"&/"%28 :$+&18 4"/5-"$18 ,%'

E,B ` 1&+,8 3,1&+,

Ordinal Data:- *&;, #.5,. &1#/= %*, .,$0#/$, '+&$$"3"'&%"#/$

L'&%,=#.",$N9 )*, $0&',$ #. "/%,.;&+$ <,%7,,/ %*, '&%,=#.",$ &., /#%

/,',$$&."+2 ,>-&+9

Examplec

O9 $%.#/=+2 &=.,,

R9 &=.,,

S9 /# #0"/"#/

V9 5"$&=.,,

U9 $%.#/=+2 5"$&=.,,

:/ %*, &<#;, $"%-&%"#/8 7, #/+2 C/#7 %*&% %*, 5&%& &., #.5,.,59



!"#$%&%"$%"'$

O[

Interval Data:- :/ "/%,.;&+ 5&%& %*, "/%,.;&+$ <,%7,,/ ;&+-,$ &., %*,

$&1,9 \#. ,B&10+,8 "/ %*, \&*.,/*,"% %,10,.&%-., $'&+,8 %*, 5"33,.,/',

<,%7,,/ W[ 5,=.,,$ &/5 WO 5,=.,,$ "$ %*, $&1, &$ %*, 5"33,.,/',

<,%7,,/ SR &/5 SS 5,=.,,$9 !-% %*, $'&+, "$ /#% & TH):Q E'&+,9 V[

5,=.,,$ \&*.,/*,"% "$ /#% %7"', &$ 1-'* &$ R[ 5,=.,,$ \&*.,/*,"%9

Ratio Data:- )*, 5&%& ;&+-,$ "/ .&%"# 5&%& 5# *&;, 1,&/"/=3-+ .&%"#$8

3#. ,B&10+,8 &=, "$ & .&%"# 5&%&8 $#1, #/, 7*# "$ V[ "$ %7"', &$ #+5 &$

$#1,#/, 7*# "$ R[9

!#%* "/%,.;&+ &/5 .&%"# 5&%& "/;#+;, 1,&$-.,1,/%9 6#$% 5&%& &/&+2$"$

%,'*/">-,$ %*&% &00+2 %# .&%"# 5&%& &+$# &00+2 %# "/%,.;&+ 5&%&9 )*,.,3#.,8

"/ 1#$% 0.&'%"'&+ &$0,'%$8 %*,$, %20,$ #3 5&%& L"/%,.;&+ &/5 .&%"#N &.,

=.#-0,5 -/5,. 1,%."' 5&%&9 :/ $#1, #%*,. "/$%&/',$8 %*,$, %20, #3 5&%&

&., &+$# C/#7/ &$ /-1,."'&+ 5"$'.,%, &/5 /-1,."'&+ '#/%"/-#-$9

Numerical discrete

_-1,."'&+ 5"$'.,%, 5&%& #''-. 7*,/ %*, #<$,.;&%"#/$ &., "/%,=,.$ %*&%

'#..,$0#/5 7"%* & '#-/% #3 $#1, $#.%9 E#1, '#11#/ ,B&10+,$ &.,c

%*, /-1<,. #3 <&'%,."& '#+#/",$ #/ & 0+&%,8 %*, /-1<,. #3 ',++$ 7"%*"/ &

0.,$'."<,5 &.,& -0#/ 1"'.#$'#0"' ,B&1"/&%"#/8 %*, /-1<,. #3 *,&.%<,&%$ 7"%*"/ & $0,'"3",5 %"1, "/%,.;&+8 & 1#%*,.b$ *"$%#.2 #3 /-1<,. #3

<".%*$ L 0&."%2N &/5 0.,=/&/'",$ L=.&;"5"%2N8 %*, /-1<,. #3 ,0"$#5,$ #3

"++/,$$ & 0&%",/% ,B0,.",/',$ 5-."/= $#1, %"1, 0,."#58 ,%'9



!"#$%&%"$%"'$

OO

Numerical continuous

)*, $'&+, 7"%* %*, =.,&%,$% 5,=.,, #3 >-&/%"3"'&%"#/ "$ & /-1,."'&+

'#/%"/-#-$ $'&+,9 ?&'* #<$,.;&%"#/ %*,#.,%"'&++2 3&++$ $#1,7*,., &+#/=

& '#/%"/--19 Q/, "$ /#% .,$%."'%,58 "/ 0."/'"0+,8 %# 0&.%"'-+&. ;&+-,$ $-'*

&$ %*, "/%,=,.$ #3 %*, 5"$'.,%, $'&+,9 )*, .,$%."'%"/= 3&'%#. "$ %*, 5,=.,,#3 &''-.&'2 #3 %*, 1,&$-."/= "/$%.-1,/% 1#$% '+"/"'&+ 1,&$-.,1,/%$8

$-'* &$ <+##5 0.,$$-.,8 $,.-1 '*#+,$%,.#+ +,;,+8 *,"=*%8 7,"=*%8 &=,

,%'9 &., #/ & /-1,."'&+ '#/%"/-#-$ $'&+,9

1.5 Exercises

:5,/%"32 %*, %20, #3 5&%& L/#1"/&+8 #.5"/&+8 "/%,.;&+ &/5 .&%"#N .,0.,$,/%,5

<2 ,&'* #3 %*, 3#++#7"/=9 G#/3".1 2#-. &/$7,.$ <2 =";"/= 2#-. #7/

,B&10+,$9

O9 !+##5 =.#-0

R9 ),10,.&%-., LG,+$"-$N

S9 ?%*/"' =.#-0

V9 M#< $&%"$3&'%"#/ "/5,B LO`UN

U9 _-1<,. #3 *,&.% &%%&'C$

Y9 G&+,/5&. 2,&.

W9 E,.-1 -."' &'"5 L1=@O[[1+N

]9 _-1<,. #3 &''"5,/%$ "/ S ` 2,&. 0,."#5X9 _-1<,. #3 '&$,$ #3 ,&'* .,0#.%&<+, 5"$,&$, .,0#.%,5 <2 & *,&+%*

7#.C,.

O[9 )*, &;,.&=, 7,"=*% =&"/ #3 Y O`2,&. #+5 5#=$ L7"%* & $0,'"&+ 5",%

$-00+,1,/%N 7&$ XU[=.&1$ +&$% 1#/%*9



!"#$%&%"$%"'$

OR

CHAPTER TWO

Methods Of Data Collection, Organization And

Presentation

2.1. Learning Objectives

H% %*, ,/5 #3 %*"$ '*&0%,.8 %*, $%-5,/%$ 7"++ <, &<+, %#c

O9 :5,/%"32 %*, 5"33,.,/% 1,%*#5$ #3 5&%& #.=&/"F&%"#/ &/5

0.,$,/%&%"#/

R9 A/5,.$%&/5 %*, '."%,."#/ 3#. %*, $,+,'%"#/ #3 & 1,%*#5 %#

#.=&/"F, &/5 0.,$,/% 5&%&

S9 :5,/%"32 %*, 5"33,.,/% 1,%*#5$ #3 5&%& '#++,'%"#/ &/5 '."%,."#/

%*&% 7, -$, %# $,+,'% & 1,%*#5 #3 5&%& '#++,'%"#/

V9 K,3"/, & >-,$%"#//&".,8 "5,/%"32 %*, 5"33,.,/% 0&.%$ #3 &

>-,$%"#//&"., &/5 "/5"'&%, %*, 0.#',5-.,$ %# 0.,0&., &

>-,$%"#//&".,

2.2. Introduction

!,3#., &/2 $%&%"$%"'&+ 7#.C '&/ <, 5#/, 5&%& 1-$% <, '#++,'%,59

K,0,/5"/= #/ %*, %20, #3 ;&."&<+, &/5 %*, #<J,'%";, #3 %*, $%-52

5"33,.,/% 5&%& '#++,'%"#/ 1,%*#5$ '&/ <, ,10+#2,59



!"#$%&%"$%"'$

OS

2.3. Data Collection Methods

K&%& '#++,'%"#/ %,'*/">-,$ &++#7 -$ %# $2$%,1&%"'&++2 '#++,'% 5&%&

&<#-% #-. #<J,'%$ #3 $%-52 L0,#0+,8 #<J,'%$8 &/5 0*,/#1,/&N &/5

&<#-% %*, $,%%"/= "/ 7*"'* %*,2 #''-.9 :/ %*, '#++,'%"#/ #3 5&%& 7,

*&;, %# <, $2$%,1&%"'9 :3 5&%& &., '#++,'%,5 *&0*&F&.5+28 "% 7"++ <,

5"33"'-+% %# &/$7,. #-. .,$,&.'* >-,$%"#/$ "/ & '#/'+-$";, 7&29

Various data collection techniques can be used such as:

• Q<$,.;&%"#/

• \&',`%#`3&', &/5 $,+3`&51"/"$%,.,5 "/%,.;",7$

• ^#$%&+ #. 1&"+ 1,%*#5 &/5 %,+,0*#/, "/%,.;",7$

• A$"/= &;&"+&<+, "/3#.1&%"#/

• \#'-$ =.#-0 5"$'-$$"#/$ L\IKN

• Q%*,. 5&%& '#++,'%"#/ %,'*/">-,$ f T&0"5 &00.&"$&+ %,'*/">-,$8 SP

%,'*/">-,8 _#1"/&+ =.#-0 %,'*/">-,$8 K,+0*" %,'*/">-,$8 +"3,

*"$%#.",$8 '&$, $%-5",$8 ,%'9

1. Observation – Q<$,.;&%"#/ "$ & %,'*/">-, %*&% "/;#+;,$

$2$%,1&%"'&++2 $,+,'%"/=8 7&%'*"/= &/5 .,'#5"/= <,*&;"#.$ #3 0,#0+,

#. #%*,. 0*,/#1,/& &/5 &$0,'%$ #3 %*, $,%%"/= "/ 7*"'* %*,2 #''-.8

3#. %*, 0-.0#$, #3 =,%%"/= L=&"/"/=N $0,'"3",5 "/3#.1&%"#/9 :% "/'+-5,$

&++ 1,%*#5$ 3.#1 $"10+, ;"$-&+ #<$,.;&%"#/$ %# %*, -$, #3 *"=* +,;,+

1&'*"/,$ &/5 1,&$-.,1,/%$8 $#0*"$%"'&%,5 ,>-"01,/% #. 3&'"+"%",$8



!"#$%&%"$%"'$

OV

$-'* &$ .&5"#=.&0*"'8 <"#'*,1"'&+8 k`.&2 1&'*"/,$8 1"'.#$'#0,8

'+"/"'&+ ,B&1"/&%"#/$8 &/5 1"'.#<"#+#="'&+ ,B&1"/&%"#/$9

Q-%+"/, %*, =-"5,+"/,$ 3#. %*, #<$,.;&%"#/$ 0."#. %# &'%-&+ 5&%&

'#++,'%"#/9

Advantages: I";,$ .,+&%";,+2 1#., &''-.&%, 5&%& #/ <,*&;"#. &/5

&'%";"%",$

Disadvantagesc :/;,$%"=&%#.$ #. #<$,.;,.b$ #7/ <"&$,$8 0.,J-5"',8

5,$".,$8 &/5 ,%'9 &/5 /,,5$ 1#., .,$#-.',$ &/5 $C"++,5 *-1&/ 0#7,.

5-."/= %*, -$, #3 *"=* +,;,+ 1&'*"/,$9

2. :/%,.;",7$ &/5 $,+3`&51"/"$%,.,5 >-,$%"#//&".,

:/%,.;",7$ &/5 $,+3`&51"/"$%,.,5 >-,$%"#//&".,$ &., 0.#<&<+2 %*,

1#$% '#11#/+2 -$,5 .,$,&.'* 5&%& '#++,'%"#/ %,'*/">-,$9 )*,.,3#.,8

5,$"=/"/= =##5 lquestioning toolsm 3#.1$ &/ "10#.%&/% &/5 %"1,

'#/$-1"/= 0*&$, "/ %*, 5,;,+#01,/% #3 1#$% .,$,&.'* 0.#0#$&+$9

Q/', %*, 5,'"$"#/ *&$ <,,/ 1&5, %# -$, %*,$, %,'*/">-,$8 %*,

3#++#7"/= >-,$%"#/$ $*#-+5 <, '#/$"5,.,5 <,3#., 5,$"=/"/= #-. %##+$c

• D*&% ,B&'%+2 5# 7, 7&/% %# C/#78 &''#.5"/= %# %*, #<J,'%";,$

&/5 ;&."&<+,$ 7, "5,/%"3",5 ,&.+",.a :$ >-,$%"#/"/= %*, ."=*%

%,'*/">-, %# #<%&"/ &++ &/$7,.$8 #. 5# 7, /,,5 &55"%"#/&+

%,'*/">-,$8 $-'* &$ #<$,.;&%"#/$ #. &/&+2$"$ #3 .,'#.5$a



!"#$%&%"$%"'$

OU

• Q3 7*#1 7"++ 7, &$C >-,$%"#/$ &/5 7*&% %,'*/">-,$ 7"++ 7,

-$,a K# 7, -/5,.$%&/5 %*, %#0"' $-33"'",/%+2 %# 5,$"=/ &

>-,$%"#//&".,8 #. 5# 7, /,,5 $#1, +##$,+2 $%.-'%-.,5

"/%,.;",7$ 7"%* C,2 "/3#.1&/%$ #. & 3#'-$ =.#-0 5"$'-$$"#/ 3".$%

%# #.",/% #-.$,+;,$a

• H., #-. "/3#.1&/%$ 1&"/+2 +"%,.&%, #. "++"%,.&%,a :3 "++"%,.&%,8 %*,

-$, #3 $,+3`&51"/"$%,.,5 >-,$%"#//&".,$ "$ /#% &/ #0%"#/9

• 4#7 +&.=, "$ %*, $&10+, %*&% 7"++ <, "/%,.;",7,5a E%-5",$ 7"%*

1&/2 .,$0#/5,/%$ #3%,/ -$, $*#.%,.8 *"=*+2 $%.-'%-.,5

>-,$%"#//&".,$8 7*,.,&$ $1&++,. $%-5",$ &++#7 1#., 3+,B"<"+"%2

&/5 1&2 -$, >-,$%"#//&".,$ 7"%* & /-1<,. #3 #0,/`,/5,5

>-,$%"#/$9

Q/', %*, 5,'"$"#/ *&$ <,,/ 1&5, :/%,.;",7$ 1&2 <, +,$$ #. 1#.,

$%.-'%-.,59 A/$%.-'%-.,5 "/%,.;",7 "$ 3+,B"<+,8 %*, '#/%,/% 7#.5"/= &/5

#.5,. #3 %*, >-,$%"#/$ ;&.2 3.#1 "/%,.;",7 %# "/%,.;",79 )*,

"/;,$%"=&%#.$ #/+2 *&;, "5,& #3 7*&% %*,2 7&/% %# +,&./ <-% 5# /#%

5,'"5, "/ &5;&/', ,B&'%+2 7*&% >-,$%"#/$ 7"++ <, &$C,58 #. "/ 7*&%

#.5,.9

:/ #%*,. $"%-&%"#/$8 & 1#., $%&/5&.5"F,5 %,'*/">-, 1&2 <, -$,58 %*,

7#.5"/= &/5 #.5,. #3 %*, >-,$%"#/$ <,"/= 5,'"5,5 "/ &5;&/',9 )*"$

1&2 %&C, %*, 3#.1 #3 & *"=*+2 $%.-'%-.,5 "/%,.;",78 "/ 7*"'* %*,

>-,$%"#/$ &., &$C,5 #.5,.+28 #. & $,+3 &51"/"$%,.,5 >-,$%"#//&".,8 "/

7*"'* '&$, %*, .,$0#/5,/% .,&5$ %*, >-,$%"#/$ &/5 3"++ "/ %*, &/$7,.$



!"#$%&%"$%"'$

OY

<2 *"1$,+3 L$#1,%"1,$ "/ %*, 0.,$,/', #3 &/ "/%,.;",7,. 7*# g$%&/5$

<2b %# =";, &$$"$%&/', "3 /,',$$&.2N9

E%&/5&.5"F,5 1,%*#5$ #3 &$C"/= >-,$%"#/$ &., -$-&++2 0.,3,..,5 "/

'#11-/"%2 1,5"'"/, .,$,&.'*8 $"/', %*,2 0.#;"5, 1#., &$$-.&/',

%*&% %*, 5&%& 7"++ <, .,0.#5-'"<+,9 P,$$ $%.-'%-.,5 "/%,.;",7$ 1&2 <,

-$,3-+ "/ & 0.,+"1"/&.2 $-.;,28 7*,., %*, 0-.0#$, "$ %# #<%&"/

"/3#.1&%"#/ %# *,+0 "/ %*, $-<$,>-,/% 0+&//"/= #3 & $%-52 .&%*,. %*&/

3&'%#.$ 3#. &/&+2$"$8 &/5 "/ "/%,/$";, $%-5",$ #3 0,.',0%"#/$8 &%%"%-5,$8

1#%";&%"#/ &/5 &33,'%";, .,&'%"#/$9 A/$%.-'%-.,5 "/%,.;",7$ &.,

'*&.&'%,."$%"' #3 >-&+"%&%";, L/#/`>-&/%"%&%";,N .,$,&.'*9

)*, -$, #3 $,+3`&51"/"$%,.,5 >-,$%"#//&".,$ "$ $"10+,. &/5 '*,&0,.j

$-'* >-,$%"#//&".,$ '&/ <, &51"/"$%,.,5 %# 1&/2 0,.$#/$

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2.6 Steps in Designing a Questionnaire

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>-,$%"#//&".,8 2#- $*#-+5 .,'#/$"5,. %*, ;&."&<+,$ 2#- *&;, '*#$,/8

&/58 "3 /,',$$&.28 &558 5.#0 #. '*&/=, $#1,9 o#- 1&2 ,;,/ '*&/=,

$#1, #3 2#-. #<J,'%";,$ &% %*"$ $%&=,9

Step 2: FORMULATING QUESTIONS

Formulate one or more questions that will provide the

information needed for each variable.

)&C, '&., %*&% >-,$%"#/$ &., $0,'"3"' &/5 0.,'"$, ,/#-=* %*&%

5"33,.,/% .,$0#/5,/%$ 5# /#% "/%,.0.,% %*,1 5"33,.,/%+29 \#. ,B&10+,8

& >-,$%"#/ $-'* &$c lD*,., 5# '#11-/"%2 1,1<,.$ -$-&++2 $,,C

%.,&%1,/% 7*,/ %*,2 &., $"'Cam '&//#% <, &$C,5 "/ $-'* & =,/,.&+

7&2 <,'&-$, ,&'* .,$0#/5,/% 1&2 *&;, $#1,%*"/= 5"33,.,/% "/ 1"/5

7*,/ &/$7,."/= %*, >-,$%"#/c

Q/, "/3#.1&/% 1&2 %*"/C #3 1,&$+,$ 7"%* '#10+"'&%"#/$ &/5

$&2 *, =#,$ %# %*, *#$0"%&+8 &/#%*,. #3 '#-=* &/5 $&2 =#,$ %#

%*, 0.";&%, 0*&.1&'2j



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?;,/ "3 <#%* %*"/C #3 %*, $&1, 5"$,&$,8 %*,2 1&2 *&;,

5"33,.,/% 5,=.,,$ #3 $,."#-$/,$$ "/ 1"/5 &/5 %*-$ &/$7,.

5"33,.,/%+2j

:/ &++ '&$,$8 $,+3`'&., 1&2 <, #;,.+##C,59

)*, >-,$%"#/8 %*,.,3#.,8 &$ .-+, *&$ %# <, <.#C,/ -0 "/%# 5"33,.,/%

0&.%$ &/5 1&5, $# $0,'"3"' %*&% &++ "/3#.1&/%$ 3#'-$ #/ %*, $&1,

%*"/=9 \#. ,B&10+,8 #/, '#-+5c

G#/',/%.&%, #/ "++/,$$ %*&% *&$ #''-..,5 "/ %*, 3&1"+2 #;,. %*,

0&$% OV 5&2$ &/5 &$C 7*&% *&$ <,,/ 5#/, %# %.,&% "3 3.#1 %*,

#/$,%j #.

G#/',/%.&%, #/ & /-1<,. #3 5"$,&$,$8 &$C 7*,%*,. %*,2 *&;,

#''-..,5 "/ %*, 3&1"+2 #;,. %*, 0&$% k 1#/%*$ L'*.#/"' #.

$,."#-$ 5"$,&$,$ *&;, & +#/=,. .,'&++ 0,."#5 %*&/ 1"/#.

&"+1,/%$N &/5 7*&% *&$ <,,/ 5#/, %# %.,&% ,&'* #3 %*,1 3.#1

%*, #/$,%9

Check whether each question measures one thing at a time.

\#. ,B&10+,8 %*, >-,$%"#/8 ee4#7 +&.=, &/ "/%,.;&+ 7#-+5 2#- &/5 2#-.

*-$<&/5 0.,3,. <,%7,,/ %7# $-'',$$";, <".%*$aee 7#-+5 <,%%,. <,

5";"5,5 "/%# %7# >-,$%"#/$ <,'&-$, *-$<&/5 &/5 7"3, 1&2 *&;,

5"33,.,/% #0"/"#/$ #/ %*, 0.,3,..,5 "/%,.;&+9



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Avoid leading questions.

H >-,$%"#/ "$ +,&5"/= "3 "% $-==,$%$ & ',.%&"/ &/$7,.9 \#. ,B&10+,8

%*, >-,$%"#/8 eeK# 2#- &=.,, %*&% %*, 5"$%."'% *,&+%* %,&1 $*#-+5 ;"$"%

,&'* *,&+%* ',/%,. 1#/%*+2aee *&.5+2 +,&;,$ .##1 3#. l/#m #. 3#. #%*,.

#0%"#/$9 !,%%,. 7#-+5 <,c lK# 2#- %*"/= %*&% 5"$%."'% *,&+%* %,&1$

$*#-+5 ;"$"% ,&'* *,&+%* ',/%,.a :3 2,$8 *#7 #3%,/am

E#1,%"1,$8 & >-,$%"#/ "$ +,&5"/= <,'&-$, "% 0.,$-00#$,$ & ',.%&"/

'#/5"%"#/9 \#. ,B&10+,c lD*&% &'%"#/ 5"5 2#- %&C, 7*,/ 2#-. '*"+5

*&5 5"&..*#,& %*, +&$% %"1,am 0.,$-00#$,$ %*, '*"+5 *&$ *&5

5"&..*#,&9 H <,%%,. $,% #3 >-,$%"#/$ 7#-+5 <,c l4&$ 2#-. '*"+5 *&5

5"&..*#,&a :3 2,$8 7*,/ 7&$ %*, +&$% %"1,am lK"5 2#- 5# &/2%*"/= %#

%.,&% "%a :3 2,$8 7*&%am

Step 3: SEQUENCING OF QUESTIONS

Design your interview schedule or questionnaire to be

“consumer friendly.”

)*, $,>-,/', #3 >-,$%"#/$ 1-$% <, +#="'&+ 3#. %*, .,$0#/5,/%

&/5 &++#7 &$ 1-'* &$ 0#$$"<+, 3#. & l/&%-.&+m 5"$'-$$"#/8 ,;,/

"/ 1#., $%.-'%-.,5 "/%,.;",7$9

H% %*, <,="//"/= #3 %*, "/%,.;",78 C,,0 >-,$%"#/$ '#/',./"/=

l<&'C=.#-/5 ;&."&<+,$m L,9=98 &=,8 .,+"="#/8 ,5-'&%"#/8 1&."%&+

$%&%-$8 #. #''-0&%"#/N %# & 1"/"1-19 :3 0#$$"<+,8 0#$, 1#$% #.

&++ #3 %*,$, >-,$%"#/$ +&%,. "/ %*, "/%,.;",79 LT,$0#/5,/%$



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SO

1&2 <, .,+-'%&/% %# 0.#;"5, l0,.$#/&+m "/3#.1&%"#/ ,&.+2 "/ &/

"/%,.;",7N

E%&.% 7"%* &/ "/%,.,$%"/= <-% /#/`'#/%.#;,.$"&+ >-,$%"#/

L0.,3,.&<+2 #0,/N %*&% "$ 5".,'%+2 .,+&%,5 %# %*, $-<J,'% #3 %*,

$%-529 )*"$ %20, #3 <,="//"/= $*#-+5 *,+0 %# .&"$, %*,

"/3#.1&/%$b "/%,.,$% &/5 +,$$,/ $-$0"'"#/$ '#/',./"/= %*,

0-.0#$, #3 %*, "/%,.;",7 L,9=98 %*&% "% 7"++ <, -$,5 %# 0.#;"5,

"/3#.1&%"#/ %# -$, "/ +,;2"/= %&B,$N9

^#$, 1#., $,/$"%";, >-,$%"#/$ &$ +&%, &$ 0#$$"<+, "/ %*,

"/%,.;",7 L,9=98 >-,$%"#/$ 0,.%&"/"/= %# "/'#1,8 $,B-&+

<,*&;"#.8 #. 5"$,&$,$ 7"%* $%"=1& &%%&'*,5 %# %*,18 ,%'9

A$, $"10+, ,;,.25&2 +&/=-&=,9

6&C, %*, >-,$%"#//&"., &$ $*#.% &$ 0#$$"<+,9 G#/5-'% %*, "/%,.;",7

"/ %7# 0&.%$ "3 %*, /&%-., #3 %*, %#0"' .,>-".,$ & +#/= >-,$%"#//&".,

L1#., %*&/ O *#-.N9

Step 4: FORMATTING THE QUESTIONNAIRE

When you finalize your questionnaire, be sure that:

?&'* >-,$%"#//&"., *&$ & *,&5"/= &/5 $0&', %# "/$,.% %*,

/-1<,.8 5&%& &/5 +#'&%"#/ #3 %*, "/%,.;",78 &/58 "3 .,>-".,5 %*,



!"#$%&%"$%"'$

SR

/&1, #3 %*, "/3#.1&/%9 o#- 1&2 &55 %*, /&1, #3 %*,

"/%,.;",7,. %# 3&'"+"%&%, >-&+"%2 '#/%.#+9

P&2#-% "$ $-'* %*&% >-,$%"#/$ <,+#/="/= %#=,%*,. &00,&.

%#=,%*,. ;"$-&++29 :3 %*, >-,$%"#//&"., "$ +#/=8 2#- 1&2 -$,

$-<*,&5"/=$ 3#. =.#-0$ #3 >-,$%"#/$9

E-33"'",/% $0&', "$ 0.#;"5,5 3#. &/$7,.$ %# #0,/`,/5,5

>-,$%"#/$9

!#B,$ 3#. 0.,`'&%,=#."F,5 &/$7,.$ &., 0+&',5 "/ & '#/$"$%,/%

1&//,. *&+3 #3 %*, 0&=,9

Your questionnaire should not only be consumer but also user

friendly!

Step 5: TRANSLATION

:3 "/%,.;",7 7"++ <, '#/5-'%,5 "/ #/, #. 1#., +#'&+ +&/=-&=,$8 %*,

>-,$%"#//&"., *&$ %# <, %.&/$+&%,5 %# $%&/5&.5"F, %*, 7&2 >-,$%"#/$

7"++ <, &$C,59 H3%,. *&;"/= "% %.&/$+&%,5 2#- $*#-+5 *&;, "%

.,%.&/$+&%,5 "/%# %*, #."="/&+ +&/=-&=,9 o#- '&/ %*,/ '#10&., %*,

%7# ;,.$"#/$ 3#. 5"33,.,/',$ &/5 1&C, & 5,'"$"#/ '#/',./"/= %*, 3"/&+

0*.&$"/= #3 5"33"'-+% '#/',0%$9

2.7 Methods of data organization and presentation

)*, 5&%& '#++,'%,5 "/ & $-.;,2 "$ '&++,5 raw data9 :/ 1#$% '&$,$8

-$,3-+ "/3#.1&%"#/ "$ /#% "11,5"&%,+2 ,;"5,/% 3.#1 %*, 1&$$ #3

-/$#.%,5 5&%&9 G#++,'%,5 5&%& /,,5 %# <, #.=&/"F,5 "/ $-'* & 7&2 &$



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SS

%# '#/5,/$, %*, "/3#.1&%"#/ %*,2 '#/%&"/ "/ & 7&2 %*&% 7"++ $*#7

0&%%,./$ #3 ;&."&%"#/ '+,&.+29 ^.,'"$, 1,%*#5$ #3 &/&+2$"$ '&/ <,

5,'"5,5 -0 #/ #/+2 7*,/ %*, '*&.&'%,."$%"'$ #3 %*, 5&%& &.,

-/5,.$%##59 \#. %*, 0."1&.2 #<J,'%";, #3 %*"$ 5"33,.,/% %,'*/">-,$ #3

5&%& #.=&/"F&%"#/ &/5 0.,$,/%&%"#/ +"C, #.5,. &..&28 %&<+,$ &/5

5"&=.&1$ &., -$,59

2.7.1 Frequency Distributions

\#. 5&%& %# <, 1#., ,&$"+2 &00.,'"&%,5 &/5 %# 5.&7 >-"'C '#10&."$#/$8

"% "$ #3%,/ -$,3-+ %# &..&/=, %*, 5&%& "/ %*, 3#.1 #3 & %&<+,8 #. "/ #/, #3 &

/-1<,. #3 5"33,.,/% =.&0*"'&+ 3#.1$9

D*,/ &/&+2$"/= ;#+-1"/#-$ 5&%& '#++,'%,5 3.#1 $&28 & *,&+%* ',/%,.e$

.,'#.5$8 "% "$ >-"%, -$,3-+ %# 0-% %*,1 "/%# '#10&'% %&<+,$9 p-"%, #3%,/8

%*, 0.,$,/%&%"#/ #3 5&%& "/ & 1,&/"/=3-+ 7&2 "$ 5#/, <2 0.,0&."/= &

3.,>-,/'2 5"$%."<-%"#/9 :3 %*"$ "$ /#% 5#/, %*, .&7 5&%& 7"++ /#% 0.,$,/%

&/2 1,&/"/= &/5 &/2 0&%%,./ "/ %*,1 L"3 &/2N 1&2 /#% <, 5,%,'%,59

Array (ordered array) "$ & $,."&+ &..&/=,1,/% #3 /-1,."'&+ 5&%& "/ &/

&$',/5"/= #. 5,$',/5"/= #.5,.9 )*"$ 7"++ ,/&<+, -$ %# C/#7 %*, .&/=,

#;,. 7*"'* %*, "%,1$ &., $0.,&5 &/5 7"++ &+$# =,% &/ "5,& #3 %*,".

=,/,.&+ 5"$%."<-%"#/9 Q.5,.,5 &..&2 "$ &/ &00.#0."&%, 7&2 #3

0.,$,/%&%"#/ 7*,/ %*, 5&%& &., $1&++ "/ $"F, L-$-&++2 +,$$ %*&/ R[N9



!"#$%&%"$%"'$

SV

H $%-52 "/ 7*"'* V[[ 0,.$#/$ 7,., &$C,5 *#7 1&/2 3-++`+,/=%*

1#;",$ %*,2 *&5 $,,/ #/ %,+,;"$"#/ 5-."/= %*, 0.,',5"/= 7,,C9 )*,

3#++#7"/= =";,$ %*, 5"$%."<-%"#/ #3 %*, 5&%& '#++,'%,59

Number of movies Number of persons Relative frequency (%)

[ WR O]9[

O O[Y RY9U

R OUS S]9S

S V[ O[9[

V O] V9U

U W O9]

Y S [9]

W [ [9[

] O [9S

)#%&+ V[[ O[[9[

:/ %*, &<#;, 5"$%."<-%"#/ _-1<,. #3 1#;",$ .,0.,$,/%$ %*, ;&."&<+,

-/5,. '#/$"5,.&%"#/8 _-1<,. #3 0,.$#/$ .,0.,$,/%$ %*, 3.,>-,/'28

&/5 %*, 7*#+, 5"$%."<-%"#/ "$ '&++,5 3.,>-,/'2 5"$%."<-%"#/ 0&.%"'-+&.+2

$"10+, 3.,>-,/'2 5"$%."<-%"#/9

A categorical distribution f /#/`/-1,."'&+ "/3#.1&%"#/ '&/ &+$# <,

.,0.,$,/%,5 "/ & 3.,>-,/'2 5"$%."<-%"#/9 E,/"#.$ #3 & *"=* $'*##+

7,., "/%,.;",7,5 #/ %*,". 0+&/ &3%,. '#10+,%"/= *"=* $'*##+9 )*,

3#++#7"/= 5&%& =";, 0+&/$ #3 UV] $,/"#.$ #3 & *"=* $'*##+9



!"#$%&%"$%"'$

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SSEENNIIOORRSS’’ PPLLAANN NNUUMMBBEERR OOFF SSEENNIIOORRSS

^+&/ %# &%%,/5 '#++,=, RV[

6&2 &%%,/5 '#++,=, OVY

^+&/ %# #. 1&2 &%%,/5 & ;#'&%"#/&+ $'*##+ UW

D"++ /#% &%%,/5 &/2 $'*##+ O[U

)#%&+ UV]

G#/$"5,. %*, 0.#<+,1 #3 & $#'"&+ $'",/%"$% 7*# 7&/%$ %# $%-52 %*, &=,

#3 0,.$#/$ &..,$%,5 "/ & '#-/%.29 :/ '#//,'%"#/ 7"%* +&.=, $,%$ #3

5&%&8 & =##5 #;,.&++ 0"'%-., &/5 $-33"'",/% "/3#.1&%"#/ '&/ #3%,/ <,

'#/;,2,5 <2 =.#-0"/= %*, 5&%& "/%# & /-1<,. #3 '+&$$ "/%,.;&+$ &$

$*#7/ <,+#79

Age (years) Number of persons

A/5,. O] O8WV]

O] f RV S8SRU

RU f SV S8OVX

SU f VV O8SRS

VU f UV UOR

UU &/5 #;,. SSU

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5"$%."<-%"#/9



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SY

\.,>-,/'2 5"$%."<-%"#/$ 0.,$,/% 5&%& "/ & .,+&%";,+2 '#10&'% 3#.18

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3#. 1&/2 0-.0#$,$8 <-% %*,., &., -$-&++2 $#1, %*"/=$ 7*"'* '&/ <,

5,%,.1"/,5 #/+2 3.#1 %*, #."="/&+ 5&%&9 \#. "/$%&/',8 %*, &<#;,

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0,.$#/$ &., OX 2,&.$ #+58 #. *#7 1&/2 &., #;,. YR9

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LON G*##$"/= %*, '+&$$,$8 LRN $#.%"/= L#. %&++2"/=N #3 %*, 5&%& "/%# %*,$,

'+&$$,$8 LSN '#-/%"/= %*, /-1<,. #3 "%,1$ "/ ,&'* '+&$$8 &/5 LVN

5"$0+&2"/= %*, .,$-+%$ "/ %*, 3#.1& #3 & '*&.% #. %&<+,

G*##$"/= $-"%&<+, '+&$$"3"'&%"#/ "/;#+;,$ '*##$"/= %*, /-1<,. #3

'+&$$,$ &/5 %*, .&/=, #3 ;&+-,$ ,&'* '+&$$ $*#-+5 '#;,.8 /&1,+28

3.#1 7*,., %# 7*,., ,&'* '+&$$ $*#-+5 =#9 !#%* #3 %*,$, '*#"',$ &.,

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&/5 "%$ &''-.&'28 &/5 #/ %*, 0-.0#$, %*, 5"$%."<-%"#/ "$ %# $,.;,9 )*,

3#++#7"/= &., $#1, .-+,$ %*&% &., =,/,.&++2 #<$,.;,5c

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K = 1 + 3.322×log(n), 7*,., / "$ %*, /-1<,. #3 #<$,.;&%"#/$

H/5 %*, +,/=%* #. 7"5%* #3 %*, '+&$$ "/%,.;&+ L7N '&/ <, '&+'-+&%,5 <2c

W = (Maximum value – Minimum value)/K = Range/K

RN D, &+7&2$ 1&C, $-., %*&% ,&'* "%,1 L1,&$-.,1,/% #.

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'+&$$,$ 5# /#% #;,.+&08 /&1,+28 %*&% $-'',$$";, '+&$$,$ *&;, /#

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Note %*&% %*, E%-.=,$ .-+, $*#-+5 /#% <, .,=&.5,5 &$ 3"/&+8 <-% $*#-+5

<, '#/$"5,.,5 &$ & =-"5, #/+29 )*, /-1<,. #3 '+&$$,$ $0,'"3",5 <2 %*,

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SN Determination of class limitsc L"N G+&$$ +"1"%$ $*#-+5 <, 5,3"/"%,

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3-.%*,. 5,$'."0%"#/$ %*&% 1&2 <, #3 "/%,.,$%9 )*,$, &., '+&$$,$ +"C,

+,$$ %*,/ O[8 =.,&%,. %*&/ YU8 &/5 $# #/9 L""N )*, $%&.%"/= 0#"/%8 "9,98 %*,



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+#7,. +"1"% #3 %*, 3".$% '+&$$ <, 5,%,.1"/,5 "/ $-'* & 1&//,. %*&%

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"/%,.;&+9 )*"$ "$ /,',$$&.2 <,'&-$, "/ %*, "/%,.0.,%&%"#/ #3 & 3.,>-,/'2

%&<+, &/5 "/ $-<$,>-,/% '&+'-+&%"#/ <&$,5 -0 #/ "%8 %*, 1"5`0#"/% #3

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3.,>-,/'2 #3 %*&% '+&$$9

:% "$ "10#.%&/% %# 7&%'* 7*,%*,. %*,2 &., =";,/ %# %*, /,&.,$% "/'* #.

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#-/', #. %# %*, /,&.,$% *-/5.,5%* #3 &/ #-/',8 &/5 $# 3#.%*9 \#.

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3".$% #3 %*, 3#++#7"/= %*.,, '+&$$"3"'&%"#/$ "3 %*, 7,"=*%$ &., =";,/ %# %*,

/,&.,$% C"+#=.&18 %*, $,'#/5 "3 %*, 7,"=*%$ &., =";,/ %# %*, /,&.,$%

%,/%* #3 & C"+#=.&18 &/5 %*, %*".5 "3 %*, 7,"=*%$ &., =";,/ %# %*,

/,&.,$% *-/5.,5%* #3 & C"+#=.&1c

Weight (kg) Weight (kg) Weight (kg)

O[ f OV O[9[ f OV9X O[9[[ f OV9XX

OU f OX OU9[ f OX9X OU9[[ f OX9XX

R[ f RV R[9[ f RV9X R[9[[ f RV9XX

RU f RX RU9[ f RX9X RU9[[ f RX9XX

S[ f SV S[9[ f SV9X S[9[[ f SV9XX



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Examplec G#/$%.-'% & =.#-0,5 3.,>-,/'2 5"$%."<-%"#/ #3 %*, 3#++#7"/=

5&%& #/ %*, &1#-/% #3 %"1, L"/ *#-.$N %*&% ][ '#++,=, $%-5,/%$

5,;#%,5 %# +,"$-., &'%";"%",$ 5-."/= & %20"'&+ $'*##+ 7,,Cc

RS RV O] OV R[ RV RV RY RS RO

OY OU OX R[ RR OV OS R[ OX RW

RX RR S] R] SV SR RS OX RO SO

OY R] OX O] OR RW OU RO RU OY

S[ OW RR RX RX O] RU R[ OY OO

OW OR OU RV RU RO RR OW O] OU

RO R[ RS O] OW OU OY RY RS RR

OO OY O] R[ RS OX OW OU R[ O[

A$"/= %*, &<#;, 3#.1-+&8 n q O r S9SRR × +#= L][N q W9SR ≈ W '+&$$,$6&B"1-1 ;&+-, q S] &/5 6"/"1-1 ;&+-, q O[ T&/=, q S] f O[ q

R] &/5 D q R]@W q V

A$"/= 7"5%* #3 U8 7, '&/ '#/$%.-'% =.#-0,5 3.,>-,/'2 5"$%."<-%"#/ 3#.

%*, &<#;, 5&%& &$c

)"1, $0,/% L*#-.$N )&++2 \.,>-,/'2 G-1-+&%";, 3.,>

O[ f OV &&&& @@@ ] ]

OU f OX &&&& &&&& &&&& &&&& &&&& @@@ R] SYR[ f RV &&&& &&&& &&&& &&&& &&&& @@ RW YS

RU f RX &&&& &&&& @@ OR WU

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Table 1: Overall immunization status of children in Adami Tullu

Woreda, Feb. 1995

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Source: Fikru T et al. EPI Coverage in Adami Tulu. Eth J Health Dev

1997;11(2): 109-113

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Source: Mikael A. et al Tetanus Toxoid immunization coverage

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CHAPTER THREE

Summarizing Data

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Consider the following data sets:

Mean

Set 1c Y[ V[ S[ U[ Y[ V[ W[ U[

Set 2: U[ VX VX UO V] U[ US U[

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#3 ;&."&%"#/9

1. Range

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T&/=, q B1&B ` B1"/



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2. Quantiles

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3.5 Exercises

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CHAPTER FOUR

DEMOGRAPHY AND HEALTH SERVICES

STATISTICS

44..11.. LLEEAARRNNIINNGG OOBBJJEECCTTIIVVEESS

H% %*, ,/5 #3 %*"$ '*&0%,.8 %*, $%-5,/%$ 7"++ <, &<+, %#c

O9 K,3"/, &/5 -/5,.$%&/5 %*, '#/',0%$ #3 5,1#=.&0*"' $%&%"$%"'$

R9 :5,/%"32 %*, 5"33,.,/% 1,%*#5$ #3 5&%& '#++,'%"#/ 3#.

5,1#=.&0*"'

$%-5",$

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V9 A/5,.$%&/5 3,.%"+"%2 &/5 1#.%&+"%2 1,&$-.,$

U9 A/5,.$%&/5 1,%*#5$ #3 0#0-+&%"#/ 0.#J,'%"#/ &/5 '#10-%&%"#/

#3 5#-<+"/= %"1,

Y9 A/5,.$%&/5 &/5 '#10-%, %*, 5"33,.,/% "/5"',$ .,+&%"/= %#

*#$0"%&+$

VV99RR99 ::__))TTQQKKAAGG))::QQ__

Definitionc K,1#=.&0*2 "$ & $'",/', %*&% $%-5",$ *-1&/ 0#0-+&%"#/

7"%* .,$0,'% %# $"F,8 5"$%."<-%"#/8 '#10#$"%"#/8 $#'"&+ 1#<"+"%2 &/5 "%$

;&."&%"#/ 7"%* .,$0,'% %# &++ %*, &<#;, 3,&%-.,$ &/5 %*, '&-$,$ #3 $-'*



!"#$%&%"$%"'$

XY

;&."&%"#/ &/5 %*, ,33,'% #3 &++ %*,$, #/ *,&+%*8 $#'"&+8 ,%*"'&+8 &/5

,'#/#1"' '#/5"%"#/$9

Sizec "$ %*, /-1<,. #3 0,.$#/$ "/ %*, 0#0-+&%"#/ &% & =";,/ %"1,9

?B&10+,c )*, $"F, #3 ?%*"#0"&/ 0#0-+&%"#/ "/ R[[R "$ &<#-% YU 1"++"#/9

Distributionc "$ %*, &..&/=,1,/% #3 %*, 0#0-+&%"#/ "/ %*, %,.."%#.2 #3 %*,

/&%"#/ "/ =,#=.&0*"'&+8 .,$"5,/%"&+ &.,&8 '+"1&%"' F#/,8 ,%'9

?B&10+,c K"$%."<-%"#/ #3 ?%*"#0"&/ 0#0-+&%"#/ <2 d#/,

Composition (Structure)c "$ %*, 5"$%."<-%"#/ #3 & 0#0-+&%"#/ "/%# "%$

;&."#-$ =.#-0"/=$ 1&"/+2 <2 &=, &/5 $,B9

?B&10+,c )*, &=, &/5 $,B 5"$%."<-%"#/ #3 %*, ?%*"#0"&/ 0#0-+&%"#/

.,3,.$ %# %*, /-1<,. LiN #3 %*, 0#0-+&%"#/ 3&++"/= "/ ,&'* &=, =.#-0 L&%

,&'* &=,N <2 $,B9

Changec .,3,.$ %# %*, "/'.,&$, #. 5,'+"/, #3 %*, %#%&+ 0#0-+&%"#/ #. "%$

'#10#/,/%$9 )*, '#10#/,/%$ #3 '*&/=, &., birth, death, and

migration9

)*,.,3#.,8 5,1#=.&0*"' $%&%"$%"'$ "$ %*, &00+"'&%"#/ #3 $%&%"$%"'$ %# %*,

$%-52 #3 *-1&/ 0#0-+&%"#/ "/ .,+&%"#/ %# %*, ,$$,/%"&+ 5,1#=.&0*"'

;&."&<+,$ &/5 %*, $#-.', #3 ;&."&%"#/$ #3 %*, 0#0-+&%"#/ 7"%* .,$0,'% %#

%*,$, ;&."&<+,$8 $-'* &$ 3,.%"+"%28 1#.%&+"%2 &/5 1"=.&%"#/9 :% "$

"/5"$0,/$&<+, %# $%-52 %*,$, ;&."&<+,$ %*.#-=* #%*,. $#'"#`,'#/#1"'



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;&."&<+,$ $-'* &$ 0+&', #3 <".%*8 .,+"="#/8 #''-0&%"#/8 ,5-'&%"#/8 1&."%&+

$%&%-$8 ,%'9

4.3 Sources of Demographic Data

K,1#=.&0*"' "/3#.1&%"#/ "$ &'>-".,5 %*.#-=* %7# 1&"/ 7&2$c <2

'#10+,%, ,/-1,.&%"#/$ L',/$-$N &/5 $&10+, $-.;,2$ &% & 0#"/% "/

%"1,8 &/5 %*.#-=* .,'#.5"/= ;"%&+ ,;,/%$ &$ %*,2 #''-. #;,. & 0,."#5

#3 %"1,9

G#10+,%, ,/-1,.&%"#/$ #. ',/$-$,$ &., %&C,/ <2 #<%&"/"/= "/3#.1&%"#/

'#/',./"/= ,;,.2 "/*&<"%&/% #3 %*, &.,&9 H+$# '#1"/= "/%# "/'.,&$"/= -$,&., $&10+, $-.;,2$8 '#/5-'%,5 <2 "/%,.;",7"/= & 0&.% #3 %*, 0#0-+&%"#/

%# .,0.,$,/% %*, 7*#+,9 Q/ %*, #%*,. *&/5 %*, "/3#.1&%"#/ #<%&"/,5 3.#1

%*, .,'#.5"/= #3 ;"%&+ ,;,/%$ L<".%*8 5,&%*8 1&.."&=,8 5";#.',8 ,%'N #/ &

'#/%"/-#-$ <&$"$ '#10+,%,$ %*, 5&%& '#++,'%,5 3.#1 0,."#5"' ',/$-$,$ |

$&10+, $-.;,2$9

I) THE CENSUS

:/ 1#5,./ -$&=,8 %*, %,.1 l',/$-$m .,3,.$ %# & /&%"#/`7"5, '#-/%"/= #3

0#0-+&%"#/9 :% "$ #<%&"/,5 <2 & 5".,'% '&/;&$$ #3 ,&'* 0,.$#/ #.

*#-$,*#+58 7*"'* "$ & +&.=, &/5 '#10+"'&%,5 -/5,.%&C"/=9 )*,., &., %7#

1&"/ 5"33,.,/% $'*,1,$ 3#. ,/-1,.&%"/= & 0#0-+&%"#/ "/ & ',/$-$9



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K, J-.,c` %*, ,/-1,.&%"#/ L #. '#-/%N "$ 5#/, &''#.5"/= %# %*, -$-&+ #.

+,=&+ 0+&', #3 .,$"5,/',

K, 3&'%#c` )*, ,/-1,.&%"#/ "$ 5#/, &''#.5"/= %# %*, &'%-&+ 0+&', #3

.,$"5,/', #/ %*, 5&2 #3 %*, ',/$-$9

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.,$"5,/%$ &/5 ;"$"%#.$8 <-% "/'+-5,$ 0,.1&/,/% .,$"5,/%$ 7*# &.,

%,10#.&."+2 &7&29 H 5, 3&'%# '#-/% "/'+-5,$ %,10#.&.2 .,$"5,/%$ &/5

;"$"%#.$8 <-% ,B'+-5,$ 0,.1&/,/% .,$"5,/%$ 7*# *&00,/ %# <, &7&2 #/

%*, 5&2 #3 %*, ',/$-$9

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7*,%*,. %*,". 0.,$,/', "/ %*&% 0+&', <, 0,.1&/,/% #. %,10#.&.2N "$

3&;#-.,5 <2 !."%&"/ 7*"+, %*, A/"%,5 E%&%,$ #3 H1,."'& *&$ %.&5"%"#/&++2

-$,5 %*, 5, J-., L0,.1&/,/% .,$"5,/',N $'*,1,$9

Advantages and Disadvantages of the two schemes

de jure:

&N H5;&/%&=,

• :% 2",+5$ "/3#.1&%"#/ .,+&%";,+2 -/&33,'%,5 <2 $,&$#/&+ &/5 #%*,.

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1&2 <, '#-/%,5 %7"',

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-$-&+ #. +,=&+ .,$"5,/',9 L:/ 0+&',$ 7*,., 1#<"+"%2 "$ *"=* &/5 /#

3"B,5 .,$"5,/', "$ "/5"'&%,5N

• :/3#.1&%"#/ '#++,'%,5 .,=&.5"/= 0,.$#/$ &7&2 3.#1 *#1, "$ #3%,/

"/'#10+,%, #. "/'#..,'%9

de facto:

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• #33,.$ +,$$ '*&/', #3 5#-<+, '#-/%"/=

• I";,$ +,$$ '*&/', 3#. %*, #1"$$"#/ #3 0,.$#/$ 3.#1 %*, '#-/%

<N 5"$&5;&/%&=,$c

• ^#0-+&%"#/ 3"=-.,$ 1&2 <, "/3+&%,5 #. 5,3+&%,5 <2 %#-."$%$8 %.&;,++"/=

$&+,$1,/8 &/5 #%*,. %.&/$",/%$9

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$%&%"$%"'$N "$ $-<J,'% L +"&<+,N %# 5"$%#.%"#/9

\#. 1#$% 0.&'%"'&+ 0-.0#$,$8 ;&."#-$ '#1<"/&%"#/$ #. 1#5"3"'&%"#/$ #3

%*, %7#`$'*,1,$ L"9,9 5, J-., | 5, 3&'%#N &., -$,5 5,0,/5"/= -0#/

/&%"#/&+ /,,5$ &/5 %*, ,/-1,.&%"#/ 0+&/ 3#++#7,5 9



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Information to be collected

E,B8 &=,8 1&."%&+ $%&%-$8 ,5-'&%"#/&+ '*&.&'%,."$%"'$8 ,'#/#1"'

'*&.&'%,."$%"'$8 0+&', #3 <".%*8 +&/=-&=,8 3,.%"+"%2 1#.%&+"%2 8 '"%"F,/$*"0 L

/&%"#/&+"%2N8 +";"/= '#/5"%"#/$ L,9=9 *#-$,`#7/,.$*"08 %20, #3 *#-$"/=&/5 %*, +"C,N8 .,+"="#/8 ,%'99

Essential features ( characteristics) of a census

ON E,0&.&%, ,/-1,.&%"#/ &/5 .,'#.5"/= #3 %*, '*&.&'%,."$%"'$ #3 ,&'*

"/5";"5-&+

RN :% $*#-+5 .,3,. %# 0,#0+, "/*&<"%"/= & 7,++`5,3"/,5 %,.."%#.2

SN )*, 0#0-+&%"#/ $*#-+5 <, ,/-1,.&%,5 7"%* .,$0,'% %# & 7,++ 5,3"/,50#"/% "/ %"1,

VN :% $*#-+5 <, %&C,/ &% .,=-+&. "/%,.;&+$ L -$-&++2 ,;,.2 %,/ 2,&.$N

UN :/ 1#$% '#-/%.",$ %*, 0,.$#/&+ 5&%& '#++,'%,5 "/ & ',/$-$ &., /#%

-$,5

3#. #%*,. %*&/ $%&%"$%"'&+ 0-.0#$,$9

YN )*, '#10"+&%"#/ &/5 0-<+"'&%"#/ #3 5&%& <2 =,#=.&0*"' &.,&$ &/5 <2

<&$"' 5,1#=.&0*"' ;&."&<+,$ "$ &/ "/%,=.&+ 0&.% #3 & ',/$-$9

:/ $*#.%8 %*, 1&"/ '*&.&'%,."$%"'$ #3 & ',/$-$ '#-+5 <, $-11&."F,5 &$3#++#7$c

:/5";"5-&+ ,/-1,.&%"#/8 -/";,.$&+"%2 7"%*"/ & 5,3"/,5 %,.."%#.28

$"1-+%&/,"%2 &/5 5,3"/,5 0,."#5"'"%2



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Census Operation

)*, ,/%"., ',/$-$ #0,.&%"#/ *&$ S 0&.%$ L$%&=,$N

ON 0.,`,/-1,.&%"#/ → 0+&//"/= &/5 0.,0&.&%#.2 7#.C

RN ,/-1,.&%"#/→

3",+5 7#.C L '#++,'%"#/ #3 %*, 5&%&NSN 0#$%`,/-1,.&%"#/ → ,5"%"/=8 '#5"/=8 '#10"+&%"#/8

%&<-+&%"#/8 &/&+2$"$8 &/5 0-<+"'&%"#/ #3 %*, .,$-+%$

Uses of a census

ON =";,$ '#10+,%, &/5 ;&+"5 0"'%-., #3 %*, 0#0-+&%"#/ '#10#$"%"#/ &/5

'*&.&'%,."$%"'$

RN $,.;,$ &$ & $&10+"/= 3.&1,SN 0.#;"5,$ 7"%* ;"%&+ $%&%"$%"'$ #3 %*, 0#0-+&%"#/ "/ %,.1$ #3 3,.%"+"%2 &/5

1#.%&+"%29

VN G,/$-$ 5&%& &., -%"+"F,5 "/ & /-1<,. #3 7&2$ 3#. 0+&//"/= %*,

7,+3&., #3 %*, 0,#0+,

?=9 )# &$',.%&"/ 3##5 .,>-".,1,/%$8 %# 0+&/ $#'"&+ 7,+3&., $'*,1,$ +"C,

$'*##+$8 *#$0"%&+$8 *#-$,$8 #.0*&/&=,$8 0,/$"#/$8 ,%'9

Common errors in census data

O9 Q1"$$"#/ &/5 #;,. ,/-1,.&%"#/9



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O[R

R9 6"$.,0#.%"/= #3 &=, 5-, %# 1,1#.2 +&0$,8 0.,3,.,/', #3 %,.1"/&+

5"="%$8 #;,.@-/5,. ,$%"1&%"#/9

S9 Q;,.$%&%"/= #3 %*, $%&%-$ 7"%*"/ %*, #''-0&%"#/9

V9 A/5,. .,0#.%"/= #3 <".%*$ 5-, %# 0.#<+,1 #3 .,3,.,/', 0,."#5 &/5

1,1#.2 +&0$,9

U9 A/5,. .,0#.%"/= #3 5,&%*$ 5-, %# 1,1#.2 +&0$, &/5 %,/5,/'2 /#% %#

.,0#.% #/ 5,&%*$8 0&.%"'-+&.+2 #/ "/3&/% 5,&%*$9

ii) Surveysc H $-.;,2 "$ & %,'*/">-, <&$,5 #/ $&10+"/= 1,%*#5$ <2

1,&/$ #3 7*"'* 7, %.2 %# #<%&"/ $0,'"3"' "/3#.1&%"#/ 3.#1 0&.% #3 %*,

0#0-+&%"#/ +"&<+, %# <, '#/$"5,.,5 &$ .,0.,$,/%&%";, #3 %*, 7*#+,9

E-.;,2$ &., 1&5, &% & =";,/ 1#1,/%8 "/ & $0,'"3"' %,.."%#.2j $0#.&5"'&++2

&/5 7"%*#-% 0,."#5"'"%2 3#. %*, 5,,0 $%-52 #3 & 0.#<+,19

iii) Registration of vital events (mainly births and deaths)

G*&/=,$ "/ 0#0-+&%"#/ /-1<,.$ &., %&C"/= 0+&', ,;,.2 5&29 H55"%"#/$

&., 1&5, <2 <".%*$ #. %*.#-=* /,7 &..";&+$ 3.#1 #-%$"5, %*, &.,&9

T,5-'%"#/$ %&C, 0+&', <,'&-$, #3 5,&%*$8 #. %*.#-=* 0,#0+, +,&;"/= %*,

&.,&9

)*, %&C"/= #3 & ',/$-$ "$ 1,.,+2 & $/&0$*#% #3 &/ ,;,/% 7*"+, %*,

'#-/%"/= #3 <".%*$ &/5 5,&%*$ L;"%&+ .,'#.5$N "$ & '#/%"/-#-$ 0.#',$$9

)*, .,="$%.&%"#/ #3 ;"%&+ ,;,/%$ L<".%*$8 5,&%*$8 1&.."&=,$8 ,%'N "$ &

$2$%,1 <2 7*"'* &++ <".%*$8 5,&%*$8 ,%'9 #''-.."/= /&%"#/7"5, &.,



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.,="$%,.,58 .,0#.%,5 %# & '#/%.#+ <#52 &/5 '#10"+,5 ',/%.&++29 H

',.%"3"'&%, "$ "$$-,5 3#. ,;,.2 5,&%* &/5 <".%*9 )*, 3#-. 1&"/

'*&.&'%,."$%"'$ #3 ;"%&+ .,="$%.&%"#/ &., '#10.,*,/$";,/,$$8 '#10-+$#.2

<2 +&78 '#10"+&%"#/ 5#/, ',/%.&++2 &/5 %*, .,="$%.&%"#/ "$ &/ #/=#"/=

L'#/%"/-#-$N 0.#',$$9

)*"$ $2$%,1 "$ /#% 7,++ 5,;,+#0,5 "/ ?%*"#0"& &/5 *,/', '&//#% $,.;,

&$ & .,+"&<+, 1,&/$ #3 =,%%"/= "/3#.1&%"#/9

4.4 Stages in Demographic Transition

K,1#=.&0*"' %.&/$"%"#/ "$ & %,.1 -$,5 %# 5,$'."<, %*, 1&J#.

5,1#=.&0*"' %.,/5$ #3 %*, 0&$% %7# ',/%-.",$9 )*, '*&/=, "/ 0#0-+&%"#/

<&$"'&++2 '#/$"$%$ #3 & $*"3% 3.#1 &/ ,>-"+"<."-1 '#/5"%"#/ #3 *"=* <".%*

&/5 5,&%* .&%,$8 '*&.&'%,."$%"'$ #3 &=.&."&/ $#'",%",$8 %# & /,7,.

,>-"+"<."-1 "/ 7*"'* <#%* <".%* &/5 5,&%* .&%,$ &., &% 1-'* +#7,. +,;,+$9

)*, ,/5 $%&=, L0,."#5 SN #3 %*"$ 5,1#=.&0*"' %.&/$"%"#/ "$ & $"%-&%"#/ "/

7*"'* <".%* &/5 5,&%* .&%,$ &., &=&"/ ,$$,/%"&++2 "/ <&+&/', <-% & +#7,.

+,;,+9

)*, %*,#.2 #3 5,1#=.&0*"' %.&/$"%"#/ &%%,10%$ %# ,B0+&"/ %*, '*&/=,$ "/

1#.%&+"%2 &/5 3,.%"+"%2 "/ %*.,, 5"33,.,/% $%&=,$9

ON 0.,`%.&/$"%"#/&+ c` '*&.&'%,."F,5 <2 *"=* 1#.%&+"%2 &/5 *"=* 3,.%"+"%28

7"%* +#7 L1#5,.&%,N 0#0-+&%"#/ =.#7%* L2#-/= 0#0-+&%"#/N9 )*"$ "$

$*#7/ <2 %*, %."&/=-+&.8 <.#&5 <&$,5 0&%%,./ .,3+,'%"/= %*, *"=* <".%*

.&%,$8 #;,. & +#/= 0,."#5 #3 %"1,9 :9,98 #/+2 & $1&++ 0.#0#.%"#/ #3 %*,



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O[V

0,.$#/$ *&;, $-.;";,5 "/ %# %*, #+5,. &=, =.#-09 )*"$ %20, #3 0#0-+&%"#/

"$ 3#-/5 "/ 0."1"%";, $#'",%",$ &/5 "$ $#1,%"1,$ C/#7/ &$ ,B0&/$";,

L%20, :N9

RN ).&/$"%"#/&+c` '*&.&'%,."F,5 <2 *"=* <".%* .&%, &/5 .,5-',5 5,&%*

.&%,8 7"%* *"=* L.&0"5N =.#7%* .&%, Ll2#-/= 0#0-+&%"#/mN9 )*, 5.#0 "/ %*,

5,&%* .&%, "$ -$-&++2 <.#-=*% &<#-% <2 "10.#;,5 1,5"'&+ '&., 3#++#7"/=

$#'"# f,'#/#1"' 5,;,+#01,/%9 )*, $*&0, #3 %*, 0#0-+&%"#/ 02.&1"5 "$

%."&/=-+&. '*&.&'%,."F"/= & 5,;,+#0"/= $#'",%2 L$#1,%"1,$ C/#7/ &$

,B0&/$";, L%20, ::N9

SN ^#$% f%.&/$"%"#/&+c` '*&.&'%,."F,5 <2 +#7 <".%* &/5 5,&%* .&%,$ 7"%*

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)*"$ %20"'&+ #3 &5;&/',5 #. 5,;,+#0,5 '#-/%.",$ &/5 "$ $#1,%"1,$

C/#7/ &$ $%&%"#/&.2 L )20, :::N9 P"3, ,B0,'%&/'2 "$ *"=*,. &/5 & *"=*

0.#0#.%"#/ #3 %*, 0#0-+&%"#/ $-.;";,$ "/ %# %*, #+5 &=, Ll #+5 0#0-+&%"#/mN9

Population Pyramid

)*, &=, f$,B 5"$%."<-%"#/ #3 & 0#0-+&%"#/ '&/ <, 1#$% '+,&.+2 0.,$,/%,5

"/ & =.&0*"'&+ 3#.1 C/#7/ &$ & lpopulation pyramidm9 :% "$ %*, 1#$%

7"5,+2 -$,5 #3 &++ =.&0*"' 5,;"',$ "/ 0#0-+&%"#/ $%-5",$9

^#0-+&%"#/ 02.&1"5$ 0.,$,/% %*, 0#0-+&%"#/ #3 &/ &.,& #. '#-/%.2 "/

%,.1$ #3 "%$ '#10#$"%"#/ <2 &=, &/5 $,B &% & 0#"/% "/ %"1,9 !2



!"#$%&%"$%"'$

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'#/;,/%"#/8 1&+,$ &., $*#7/ #/ %*, +,3% #3 %*, 02.&1"58 3,1&+,$ #/ %*,

."=*%8 2#-/= 0,.$#/$ &% %*, <#%%#18 &/5 %*, ,+5,.+2 &% %*, %#09

)*, 02.&1"5 '#/$"$%$ #3 & $,.",$ #3 <&.$8 ,&'* 5.&7/ 0.#0#.%"#/&%,+2 %#

.,0.,$,/% %*, 0,.',/%&=, '#/%."<-%"#/ #3 ,&'* &=,`$,B =.#-0 L#3%,/ "/

3";,`2,&. =.#-0"/=$N %# %*, %#%&+ 0#0-+&%"#/j %*&% "$8 %*, %#%&+ &.,& #3 %*,

<&.$ .,0.,$,/%$ O[[i #3 %*, 0#0-+&%"#/9

H 02.&1"5 '#/;,2$ &% & =+&/', %*, ,/%"., $*&0, #3 %*, &=, $%.-'%-.,9 :%

$*#7$ &/2 =.#$$ "..,=-+&."%",$ 5-, %# $0,'"&+ 0&$% ,;,/%$ L$-'* &$ &

7&.8 ,0"5,1"' #. &=,`$,+,'%";, 1"=.&%"#/N8 3+-'%-&%"#/$ #3 3,.%"+"%28 ,%'99

D, .,3,. %# & 0#0-+&%"#/ &$ l#+5m #. l2#-/=m8 &''#.5"/= %# %*, .,+&%";,

7,"=*% #3 #+5 &/5 2#-/= &=, =.#-0$ "/ %*, %#%&+9

^#0-+&%"#/ 02.&1"5 '&/ <, '#/$%.-'%,5 ,"%*,. -$"/= &<$#+-%, /-1<,. #.

0,.',/%&=,$9 D*,/ '#/$%.-'%"/= 0,.',/% 0#0-+&%"#/ 02.&1"5$8 %&C, %*,

0,.',/%&=, 3.#1 %*, =.#-0 %#%&+9

)*, 3#++#7"/= 02.&1"5 "$ 5.&7/ 3#. 5&%& #<%&"/,5 3.#1 ',/$-$

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Fig. 11: ^#0-+&%"#/ ^2.&1"5 #3 G*,*& K"$%."'%8 I-.&=, d#/,8 M-+2

R[[[



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VV99UU TT H H))::QQ88 ^TTQQ^QQTT))::QQ__ H H__KK TT H H))??

4. Ratio: H .&%"# >-&/%"3",$ %*, 1&=/"%-5, #3 #/,

#''-..,/', #. '#/5"%"#/ "/ .,+&%"#/ %# &/#%*,.9

O9 Sex Ratio LETNc $,B .&%"# "$ 5,3"/,5 &$ %*, %#%&+ /-1<,. #3 1&+,

0#0-+&%"#/ 0,. O[[ 3,1&+, 0#0-+&%"#/8

ET q F

M B O[[ 7*,., 6 &/5 \ &., %#%&+ /-1<,. #3 1&+, &/5

3,1&+, 0#0-+&%"#/$8 .,$0,'%";,+29

E,B .&%"#$ &., -$,5 3#. 0-.0#$,$ #3 '#10&."$#/9

&N )*, <&+&/', <,%7,,/ %*, %7# $,B,$

<N )*, ;&."&%"#/ "/ %*, $,B <&+&/', &% 5"33,.,/% &=,$

'N :% "$ &+$# -$,5 3#. 5,%,'%"/= ,..#.$ "/ 5,1#=.&0*"' 5&%&

2. Child-Woman-Ratio LGDTNc :% "$ 5,3"/,5 &$ %*, .&%"# #3 %*, /-1<,.

#3 '*"+5.,/ -/5,. U 2,&.$ #3 &=, %# %*, /-1<,. #3 7#1,/ "/ %*,

'*"+5<,&."/= &=, =.#-0 L-$-&++2 OU`VXN9

GDT q ^[`V @ ^3 OU`VX B O[[[ q _-1<,. #3 '*"+5.,/ -/5,. U 2,&.$ #3 &=,

0,. O[[[ 7#1,/ "/ %*, '*"+5 <,&."/= &=,9



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O[]

)*, '*"+5 7#1&/ .&%"# "$ &+$# C/#7/ &$ 1,&$-., #3 ,33,'%";, 3,.%"+"%2

<,'&-$, 7, &., '#/$"5,."/= $-.;";&+$ -0 %# %*, &=, #3 V /#% %*, 5,&5

#/,$9

3. Dependency Ratio (DR)c )*, 5,0,/5,/'2 .&%"# 5,$'."<,$ %*,

.,+&%"#/ <,%7,,/ %*, 0#%,/%"&++2 $,+3`$-00#.%"/= 0#.%"#/ #3 %*, 0#0-+&%"#/

&/5 %*, 5,0,/5,/% 0#.%"#/$ &% %*, ,B%.,1,$ #3 &=,9 :% "$ -$,3-+ "/

,'#/#1"' $%-5",$9

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4.5.2 Proportionc` "$ & %20, #3 .&%"# 7*"'* >-&/%"3",$ #''-..,/',$ "/

.,+&%"#/ %# %*, 0#0-+&%"#/ "/ 7*"'* %*,$, #''-..,/',$ %&C, 0+&',9 :9,98

%*, /-1,.&%#. "$ &+$# "/'+-5,5 "/ %*, 5,/#1"/&%#.9

Examplec )*, 0.#0#.%"#/ #3 1&+&."& '&$,$ &1#/= "/*&<"%&/%$ #3 &

',.%&"/ +#'&+"%29

4.5.3 Ratec H .&%, "$ & 0.#0#.%"#/ 7"%* & %"1, ,+,1,/%8 "9,98 "/

7*"'* #''-..,/',$ &., >-&/%"3",5 #;,. & 0,."#5 #3 %"1,9

)*, %,.1 .&%, &00.#0."&%,+2 .,3,.$ %# %*, .&%"# #3 5,1#=.&0*"' ,;,/%$ %#

%*, 0#0-+&%"#/ &% ."$C "/ & $0,'"3",5 0,."#59



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k ×=2(.] -*@/4<,-*(/=

(=*121.*/E 1L1=*.6?1)/J2-4F(/E ;<)D12\-*1

7*,., n "$ & '#/$%&/% 1&"/+2 & 1-+%"0+, #3 O[ LO[[8 O[[[8 O[[[[8 ,%'9N9

Population at risk: )*"$ '#-+5 <, %*, 1"5`2,&. 0#0-+&%"#/ L0#0-+&%"#/

&% %*, 3".$% #3 M-+2 ON8 0#0-+&%"#/ &% %*, <,="//"/= #3 %*, 2,&. #. & 1#.,

'#10+,B 5,3"/"%"#/9 ^,."#5 3#. & .&%, "$ -$-&++2 & 2,&.9

Rate could be crude or specific

:% "$ '#/$"5,.,5 &$ '.-5, 7*,/ "% $*#7$ %*, 3.,>-,/'2 #3 & '+&$$ #3

,;,/%$ %*.#-=* #-% %*, ,/%"., 0#0-+&%"#/ 7"%*#-% .,=&.5"/= %# &/2 #3 %*,

$1&++,. =.#-0"/=$9 G.-5, .&%,$ &., *"=*+2 $,/$"%";, %# %*, $%.-'%-.,

L&=,N #3 %*, 0#0-+&%"#/ &/5 &., /#% 5".,'%+2 -$,5 3#. '#10&."$#/

0-.0#$,$9 D*,., &$ & $0,'"3"' .&%, "10+",$ %*, ,;,/%$ "/ & 0&.%"'-+&.

'&%,=#.2 #3 &=,8 $,B8 .&',8 0&.%"'-+&. 5"$,&$,8 #. #%*,. '+&$$"3"'&%"#/

;&."&<+,$ &., -$,59

44..66 66?? H HEEAATT??EE QQ\\ \\??TT))::PP::))oo

O9 Crude Birth Rate LG!TNc "$ %*, /-1<,. #3 +";, <".%*$ "/ & 2,&. 0,.

O[[[ 1"5 2,&. 0#0-+&%"#/ "/ %*, $&1, 2,&.9

"777A1-2 .-)1(= *F1 4/4<,-*(/=A1-2ZH(?

A1-2 -(= D(2*F,(L1/E ;<)D12^/*-,+_\

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(*) Live Birth is the complete expulsion or extraction from its mother as

a product of conception irrespective of the duration of pregnancy, which

after such separation show evidence of life, (like breathing, pulsation of

the heart, etc.).

R9 General Fertility Rate LI\TNc "$ %*, /-1<,. #3 <".%*$ "/ & $0,'"3",5

0,."#5 0,. O[[[ 7#1,/ &=,5 OU`VX 2,&.j "9,9

"777A1-2.8RZ"G-J1? 4/4<,-*(/=E1)-,1A1-2H(?

A1-2 -?<2(=J D(2*F.,(L1/E =<)D12^/*-,aB\ ×=

I\T "$ =,/,.&+ <,'&-$, 7, &., '#/$"5,."/= &++ 3,1&+,$ "/ %*, &=,

=.#-0 OU`VX 7"%* #-% .,$%."'%"/= %# %*#$, 7*# *&;, '*"+5 L'*"+5.,/N9

S9 Age Specific Fertility Rate LHE\TNc "$ %*, /-1<,. #3 +";, <".%*$ "/ &

$0,'"3",5 0,."#5 0,. O[[[ 7#1,/ #3 & =";,/ &=, #. &=, =.#-09

"777

A1-2 .-)1*F1?<2(=JJ2/<4-J1

.-)1/= *F1 4/4<,-*(/=E1)-,1A1-2H(?

A1-2 -?<2(=JJ2/<4-J1J(L1=-/E

0/)1=*/21J(.*121? D(2*F.,(L1/E =<)D12^/*-,

PIB\ ×=

HE\T "$ -$,5 %# 1,&$-., %*, .,0.#5-'%";"%2 0,.3#.1&/', #3 & =";,/

&=,8 %*-$ $*#7"/= ;&."&%"#/ "/ 3,.%"+"%2 <2 &=,9

V9 Total Fertility Rate L)\TNc "$ %*, $-1 #3 &++ &=, $0,'"3"' 3,.%"+"%2

.&%,$ 3#. ,&'* 2,&. #3 &=, 3.#1 OU`VX 2,&.$9 :% "$ %*, &;,.&=, /-1<,. #3

'*"+5.,/ %*&% & $2/%*,%"' L&.%"3"'"&+N '#*#.% L& =.#-0 #3 0,.$#/$ 7*# $*&.,

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%#%&+ #3 SU 2,&.$9

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3#. U 2,&. &=, =.#-0 '+&$$"3"'&%"#/

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^"3 q 3,1&+, 0#0-+&%"#/ &% &=, L&=, "/%,.;&+N "9

)\T "$ -$,5 &$ & $%&/5&.5"F,5 "/5,B 3#. %*, #;,. &++ 3,.%"+"%2 +,;,+ &/5

1,&$-.,$ %*, %#%&+ /-1<,. #3 '*"+5.,/ & '#*#.% #3 7#1,/ 7"++ *&;, &%

%*, ,/5 #3 %*,". .,0.#5-'%";, &=,9

U9 Gross Reproduction Rate LITTNc "$ %*, %#%&+ 3,.%"+"%2 .&%, .,$%."'%,5

%# 3,1&+, <".%*$ #/+29 4,.,8 "/ #.5,. %# '#10-%, %*, .&%, 7, /,,5 %*,

HE\T$ .,$%."'%,5 %# 3,1&+, <".%*$ &% ,&'* $"/=+, 2,&. #3 &=,9

J2/<4(=J-J1A1-2.GE/2"777

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)*, &<#;, %7# 3#.1-+&, &., -$,5 "3 7, 5#/e% *&;, 3,1&+, <".%* &/5

3,1&+, 0#0-+&%"#/ <2 &=,9 !-% "3 7, *&;, 3,1&+, <".%* &/5 3,1&+,

0#0-+&%"#/ <2 &=,

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#3 7#1,/ 7#-+5 *&;, &% %*, ,/5 #3 .,0.#5-'%"#/8 "/ %*, &<$,/', #3

1#.%&+"%29

?B&10+,c :3 ITT q O[[[ q• %*, '-..,/% =,/,.&%"#/ #3 3,1&+,$ #3 '*"+5

<,&."/= &=, 7"++ 1&"/%&"/ "%$,+3 #/ %*, <&$"$ #3 '-..,/% 3,.%"+"%2 .&%, 7"%*

#-% 1#.%&+"%29

:3 ITT • O[[[ q• /# &1#-/% #3 .,5-'%"#/ #3 5,&%*$ 7"++ ,/&<+, "% %#

,$'&0, 5,'+"/, $##/,. #. +&%,. &/5 "3 ITT t O[[[ %*, .,;,.$, *&00,/$9

:/ %*, &<$,/', #3 <".%* 5&%& '.#$$ '+&$$"3",5 <2 &=, #3 1#%*,. &% <".%*

&/5 $,B #3 %*, /,7 <#./8 7, '&/ &00.#B"1&%, ITT 3.#1 )\T $"10+2 <2

1-+%"0+2"/= )\T <2 0.#0#.%"#/ #3 3,1&+, <".%*$ #/ %*, &$$-10%"#/ %*&%

$,B .&%"# &% <".%* "$ '#/$%&/%9 )*&% "$8 %*, .&%"# #3 %*, /-1<,. #3 1&+,

<".%*$ %# %*, /-1<,. #3 3,1&+, <".%*$ .,1&"/$ '#/$%&/% #;,. &++ &=,$ #3

1#%*,.$9



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OOS

"77

D(2*F-*2-*(/I1Q"

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D(2*F.E1)-,1^/*-,

D(2*F.)-,1^/*-,"

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_#%,c H .&%, "$ !".%* T&%, "3 %*, 5,/#1"/&%#. "$ 1"5 2,&. 0#0-+&%"#/

&/5 "% "$ 3,.%"+"%2 .&%, "3 %*, 5,/#1"/&%#. "$ .,$%."'%,5 %# 3,1&+,$ "/ %*,'*"+5 <,&."/= &=,9

6. Net reproduction rate (NRR)

)*, 1&"/ 5"$&5;&/%&=, #3 %*, =.#$$ .,0.#5-'%"#/ .&%, "$ %*&% "% 5#,$

/#% %&C, "/%# &''#-/% %*, 3&'% %*&% /#% &++ %*, 3,1&+,$ 7"++ +";, -/%"+ %*,

,/5 #3 %*, .,0.#5-'%";, 0,."#59 :/ '#10-%"/= %*, /,% .,0.#5-'%"#/

.&%,8 1#.%&+"%2 #3 %*, 3,1&+,$ "$ %&C,/ "/%# &''#-/%9 )*, /,%

.,0.#5-'%"#/ .&%, 1,&$-.,$ %*, ,B%,/% %# 7*"'* %*, 3,1&+,$ "/ %*,

'*"+5<,&."/= &=,`=.#-0$ &., .,0+&'"/= %*,1$,+;,$ "/ %*, /,B%

=,/,.&%"#/9 )*, /,% .,0.#5-'%"#/ .&%, "$ #/, "/ & $%&%"#/&.2

0#0-+&%"#/j & 0#0-+&%"#/ 7*"'* /,"%*,. "/'.,&$,$ /#. 5,'.,&$,$ L"9,9 .

q [ N9 :/ 1#$% '&$,$8 _TT "$ ,B0.,$$,5 0,. 7#1&/ "/$%,&5 #3 0,.

O[[[ 7#1,/9

_TT q O ⇒ $%&%"#/&.2 0#0-+&%"#/ L"9,98 O 5&-=*%,. 0,. 7#1&/N

_TT t O⇒ 5,'+"/"/= 0#0-+&%"#/

_TT • O⇒ =.#7"/= 0#0-+&%"#/



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OOV

44..77 66?? H HEEAATT??EE QQ\\ 66QQTT)) H HPP::))oo

O9 Crude Death Rate (CDR)c "$ 5,3"/,5 &$ %#%&+ /-1<,. #3 5,&%*$ 5-,

%# &++ '&-$,$ #''-.."/= "/ & 5,3"/,5 &.,& 5-."/= & 5,3"/,5 0,."#5 0,.

O[[[ 1"5 2,&. 0#0-+&%"#/ "/ %*, $&1, &.,& 5-."/= %*, $&1, 0,."#59

"777

A1-2 J(L1=(= *F1

-21-.-)1(= *F1 4/4<,-*(/=A1-2H(?

A1-2 J(L1=-(=-21--=-(=/66<22(=J

6-<.1.-,,*/?<1?1-*F./E =<)D12^/*-,

+K\ ×=

7*,., 1"5 2,&. 0#0-+&%"#/ "$ 0#0-+&%"#/ #3 %*, &.,& &$ #3 M-+2 O L1"55+,

#3 %*, 2,&.N9

GKT 1,&$-.,$ %*, .&%, &% 7*"'* 5,&%*$ &., %&C"/= 0+&', 3.#1 &++

'&-$,$ "/ & =";,/ 0#0-+&%"#/ 5-."/= & $0,'"3",5 2,&.9

R9 Age-Specific Death Rate LHEKTNc "$ 5,3"/,5 &$ %#%&+ /-1<,. #3

5,&%*$ #''-.."/= "/ & $0,'"3",5 &=, =.#-0 #3 %*, 0#0-+&%"#/ #3 & 5,3"/,5

&.,& 5-."/= & $0,'"3",5 0,."#5 0,. O[[[ 1"5 2,&. 0#0-+&%"#/ #3 %*, $&1,

&=, =.#-0 #3 %*, $&1, &.,& 5-."/= %*, $&1, 0,."#59

"777-J2/<4-J1/2-J1-* 4/4<,-*(/=A1-2H(?

-J2/<4-J1/2-J1-*?1-*F.^/*-,PIK\ - ×=



!"#$%&%"$%"'$

OOU

S9 Cause Specific Death Ratio and Rate: H '&-$, $0,'"3"' 5,&%* .&%"#

L0.#0#.%"#/&%, 1#.%&+"%2 .&%"#N .,0.,$,/%$ %*, 0,.',/% #3 &++ 5,&%*$ 5-, %#

& 0&.%"'-+&. '&-$, #. =.#-0 #3 '&-$,$9

GEK .&%"# 3#. '&-$, ' q "777

K

K

*

6 × 8 7*,., K' "$ %#%&+ 5,&%*$ 3.#1

'&-$, ' &/5 K% "$ %#%&+ 5,&%*$ 3.#1 &++ '&-$,$ "/ & $0,'"3",5 %"1, 0,."#59

G&-$, E0,'"3"' K,&%* T&%, LGEKTN "$ %*, /-1<,. #3 5,&%*$ 3#.1 '&-$,

' 5-."/= & 2,&. 0,. O[[[ #3 %*, 1"5 2,&. 0#0-+&%"#/8 "9,9

"7772(.]-*@/4<,-*(/=

66-<.1J(L1=-E2/)?1-*F.^/*-,+IK\ 6 ×=

V9 Infant Mortality Rate L:6TNc 1,&$-.,$ %*, ."$C #3 52"/= 5-."/="/3&/'2 L"9,9 %*, 3".$% &=, #3 +"3,N8 &/5 "$ 5,3"/,5 &$c

"777 D(2*F.,(L1^/*-,

-J1/E A1-2/=1<=?126F(,?21=/E K1-*F.CH\ ×=

:/3&/% 6#.%&+"%2 .&%,c %*, 0.#<&<"+"%2 #3 52"/= <,%7,,/ <".%* &/5 &=, #/,

2,&. 0,. O[[[ +";, <".%*$9

U9 Neonatal Mortality Rate L_6TNc 1,&$-.,$ %*, ."$C #3 52"/= 7"%*"/ R]

5&2$ #3 <".%*9 :% "$ 5,3"/,5 &$



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"777_(2*F.,(L1^/*-,

-J1/E ?-A.$:<=?126F(,?21=/E K1-*F. ;H\ ×=

U9 Post - Neonatal Mortality Rate L^_6TNc 6,&$-.,$ %*, ."$C #3 52"/=

5-."/= "/3&/'2 &3%,. %*, 3".$% V 7,,C$ #3 +"3,8 &/5 "$ 5,3"/,5 &$c

"77 D(2*F.,(L1^/*-,

A1-2 /=1<=?12*/?-A.$:-J1?6F(,?21=/E K1-*F.@;H\ ×=

Y9 Maternal Mortality Rate L66TNc "$ 5,3"/,5 &$ %*, /-1<,. #3 5,&%*$

#3 1#%*,.$ LK1N 5-, %# 1&%,./&+ '&-$,$8 "9,9 '#10+"'&%"#/$ #3

0.,=/&/'28 '*"+5 <".%*8 &/5 0-,.0,."-18 0,. O[[8[[[ +";, <".%*$ 5-."/= &

2,&.8 "9,9

777T"77A1-2 .-)1(= *F1 D(2*F.,(L1^/*-,

A1-2 -(=6-<.1.)-*12=-,

*/?<1H/*F12./E K1-*F.

HH\ ×=

66T 1,&$-.,$ %*, ."$C #3 52"/= #3 1#%*,.$ 3.#1 1&%,./&+ '&-$,$9

:5,&++2 %*, 5,/#1"/&%#. $*#-+5 "/'+-5, &++ 5,+";,.",$ &/5 &<#.%"#/$9



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OOW

44..88 ^QQ^AAPP H H))::QQ__ IITTQQDD))44 H H__KK ^TTQQMM??GG))::QQ__

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Rate of population growth

G.-5, !".%* T&%, ` G.-5, K,&%* T&%, q crude rate of natural

increase. )*"$ .&%, "$ <&$,5 #/ /&%-.&++2 #''-.."/= ,;,/%$ f <".%*$

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A more practical approach to calculate the population doubling

time is:

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4.10 Exercises

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ORY

CHAPTER FIVE

ELEMENTARY PROBABILITY AND

PROBABILITY DISTRIBUTIONS

5.1 Learning Objectives

H% %*, ,/5 #3 %*"$ '*&0%,.8 %*, $%-5,/% 7"++ <, &<+, %#c

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frequentist definition of Probability.



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The Classical Probability Concept

:3 %*,., &., / equally likely possibilities8 #3 7*"'* #/, 1-$% #''-.

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diea

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Definitions of some terms commonly encountered in probability

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5.3 Mutually exclusive events and the additive law

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H8 !8 Gu &., 1-%-&++2 ,B'+-$";, ,;,/%$8 Pr(A or B or C or ) = Pr (A)

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)*,.,3#.,8 7*,/ H &/5 ! &., /#% 1-%-&++2 ,B'+-$";,, ^.LH #. !N q ^. LHN



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OSO

r ^.L!N f ^.LH &/5 !N9 )*, 3#.1-+& '#/$"5,.,5 ,&.+",. 3#. 1-%-&++2

,B'+-$";, ,;,/%$ "$ & $0,'"&+ '&$, #3 %*"$8 $"/', 0.LH &/5 !N q [9

?=9 Q3 R[[ $,/"#.$ &% & ',.%&"/ '#++,=,8 X] &., 7#1,/8 SV &., 1&J#."/=

"/ !"#+#=28 &/5 R[ !"#+#=2 1&J#.$ &., 7#1,/9 :3 #/, $%-5,/% "$ '*#$,/

&% .&/5#1 3.#1 %*, $,/"#. '+&$$8 7*&% "$ %*, 0.#<&<"+"%2 %*&% %*, '*#"',

7"++ <, ,"%*,. & !"#+#=2 1&J#. #. & 7#1,/N9

^. L !"#+#=2 1&J#. #. 7#1&/ N q ^. L!"#+#=2 1&J#.N r .L7#1&/ N ` ^.

L!"#+#=2 1&J#. &/5 7#1&/N qSV@R[[ r X]@R[[ ` R[@R[[ q OOR@R[[ q

9UY

5.4 Conditional probabilities and the multiplicative law

E#1,%"1,$ %*, '*&/', & 0&.%"'-+&. ,;,/% *&00,/$ 5,0,/5$ #/ %*,

#-%'#1, #3 $#1, #%*,. ,;,/%9 )*"$ &00+",$ #<;"#-$+2 7"%* 1&/2

,;,/%$ %*&% &., $0.,&5 #-% "/ %"1,9

?=9 )*, '*&/', & 0&%",/% 7"%* $#1, 5"$,&$, $-.;";,$ %*, /,B% 2,&.

5,0,/5$ #/ *"$ *&;"/= $-.;";,5 %# %*, 0.,$,/% %"1,9 E-'* 0.#<&<"+"%",$

&., '&++,5 '#/5"%"#/&+9

)*, /#%&%"#/ "$ ^.L!@HN8 7*"'* "$ .,&5 &$ l%*, 0.#<&<"+"%2 ,;,/% ! #''-.$

=";,/ %*&% ,;,/% H *&$ &+.,&52 #''-..,5 9m



!"#$%&%"$%"'$

OSR

P,% H &/5 ! <, %7# ,;,/%$ #3 & $&10+, $0&', E9 )*, '#/5"%"#/&+

0.#<&<"+"%2 #3 &/ ,;,/% H8 =";,/ !8 5,/#%,5 <2 ^. L H@! Nq ^LH / !N @

^L!N 8 ^L!N ≠ [9

E"1"+&.+28 ^L!@HN q ^LH / !N @ ^LHN 8 ^LHN ≠ [9 )*"$ '&/ <, %&C,/ &$ &/

&+%,./&%";, 3#.1 #3 %*, 1-+%"0+"'&%";, +&79

?=9 E-00#$, "/ '#-/%.2 k %*, '*&/', %*&% &/ "/3&/% +";,$ %# &=, RU "$

9XU8 7*,.,&$ %*, '*&/', %*&% *, +";,$ %# &=, YU "$ 9YU9 \#. %*, +&%%,.8 "% "$

-/5,.$%##5 %*&% %# $-.;";, %# &=, YU 1,&/$ %# $-.;";, <#%* 3.#1 <".%* %#

&=, RU &/5 3.#1 &=, RU %# YU9 D*&% "$ %*, '*&/', %*&% & 0,.$#/ RU

2,&.$ #3 &=, $-.;";,$ %# &=, YUa

&

_#%&%"#/ ?;,/% ^.#<&<"+"%2

H E-.;";, <".%* %# &=, RU 9XU

H &/5 ! E-.;";, <#%* <".%* %# &=, RU &/5 &=, RU %# YU 9YU

!@H E-.;";, &=, RU %# YU =";,/ $-.;";&+ %# &=, RU a

)*,/8 ^.L!@HN q ^.LH / ! N @ ^.LHN q 9YU@9XU q 9Y]V 9 )*&% "$8 & 0,.$#/

&=,5 RU *&$ & Y]9V 0,.',/% '*&/', #3 +";"/= %# &=, YU9

Independent Events

Q3%,/ %*,., &., %7# ,;,/%$ $-'* %*&% %*, #''-..,/', #.

/#/#''-..,/', #3 #/, 5#,$ /#% "/ &/2 7&2 &33,'% %*, #''-..,/', #.



!"#$%&%"$%"'$

OSS

/#/#''-..,/', #3 %*, #%*,.9 )*"$ 5,3"/,$ "/5,0,/5,/% ,;,/%$9 )*-$8 "3

,;,/%$ H &/5 ! &., "/5,0,/5,/%8 ^.L!@HN q ^L!Nj ^.LH@!N q ^LHN9

?=9 ON H '+&$$"' ,B&10+, "$ / %#$$,$ #3 & '#"/ &/5 %*, '*&/',$ %*&% #/

,&'* %#$$ "% +&/5$ *,&5$9 )*,$, &., "/5,0,/5,/% ,;,/%$9 )*, '*&/', #3*,&5$ #/ &/2 #/, %#$$ "$ "/5,0,/5,/% #3 %*, /-1<,. #3 0.,;"#-$ *,&5$9

_# 1&%%,. *#7 1&/2 *,&5$ *&;, &+.,&52 <,,/ #<$,.;,58 %*, '*&/', #3

*,&5$ #/ %*, /,B% %#$$ "$ {9

?= RN H $"1"+&. $"%-&%"#/ 0.,;&"+$ 7"%* %*, $,B #3 #33$0."/=9 )*, '*&/', #3

& 1&+, "$ &00.#B"1&%,+2 {9 T,=&.5+,$$ #3 %*, $,B,$ #3 0.,;"#-$

#33$0."/=8 %*, '*&/', %*, /,B% '*"+5 "$ & 1&+, "$ $%"++ {9

7"%* "/5,0,/5,/% ,;,/%$8 %*, 1-+%"0+"'&%";, +&7 <,'#1,$ c

^.LH &/5 !N q ^.LHN ^.L!N

4,/',8 ^.LHN q ^.LH &/5 !N @ ^.L!N 8 7*,., ^.L!N ≠ [

^.L!N q ^.LH &/5 !N @ ^.LHN 8 7*,., ^.LHN ≠ [



!"#$%&%"$%"'$

OSV

EXERCISE

G#/$"5,. %*, 5.&7"/= #3 %7# '&.5$ #/, &3%,. %*, #%*,. 3.#1 & 5,'C #3 UR

'&.5$9 D*&% "$ %*, 0.#<&<"+"%2 %*&% <#%* '&.5$ 7"++ <, $0&5,$a

&N 7"%* .,0+&',1,/%

<N 7"%*#-% .,0+&',1,/%

Summary of basic Properties of probability

O9 ^.#<&<"+"%",$ &., .,&+ /-1<,.$ #/ %*, "/%,.;&+ 3.#1 [ %# Oj "9,98 [ ≤

^.LHN ≤ O

R9 :3 &/ ,;,/% "$ ',.%&"/ %# #''-.8 "%$ 0.#<&<"+"%2 "$ O8 &/5 "3 %*, ,;,/%

"$ ',.%&"/ /#% %# #''-.8 "%$ 0.#<&<"+"%2 "$ [9S9 :3 %7# ,;,/%$ &., 1-%-&++2 ,B'+-$";, L5"$J#"/%N8 %*, 0.#<&<"+"%2 %*&%

#/, #. %*, #%*,. 7"++ #''-. ,>-&+$ %*, $-1 #3 %*, 0.#<&<"+"%",$j ^.LH #.

!N q ^.LHN r ^.L!N9

V9 :3 H &/5 ! &., %7# ,;,/%$8 /#% /,',$$&."+2 5"$J#"/%8 %*,/ ^.L H #. !Nq

^. LHN r^. L!N f ^.L H &/5 !N9

U9 )*, $-1 #3 %*, 0.#<&<"+"%",$ %*&% &/ ,;,/% 7"++ #''-. &/5 %*&% "% 7"++

/#% #''-. "$ ,>-&+ %# Oj *,/',8 ^LHbN q O f ^LHN

Y9 :3 H &/5 ! &., %7# "/5,0,/5,/% ,;,/%$8 %*,/ ^. L H &/5 !N q ^.LHN ^. L!N



!"#$%&%"$%"'$

OSU

5.5 Random variables and probability distributions

A$-&++2 /-1<,.$ '&/ <, &$$#'"&%,5 7"%* %*, #-%'#1,$ #3 &/

,B0,."1,/%9 \#. ,B&10+,8 %*, /-1<,. #3 *,&5$ %*&% '#1, -0 7*,/ &

'#"/ "$ %#$$,5 3#-. %"1,$ "$ [8 O8R8S #. V9 E#1,%"1,$8 7, 1&2 3"/5 &$"%-&%"#/ 7*,., %*, ,+,1,/%$ #3 & $&10+, $0&', &., '&%,=#.",$9 :/

$-'* '&$,$8 7, '&/ &$$"=/ /-1<,.$ %# %*, '&%,=#.",$ 9

?=9 )*,., &., R8U[[ 1,/ &/5 R[[[ 7#1,/ "/ & $,/"#. '+&$$9 H$$-1, &

0,.$#/ "$ .&/5#1+2 $,+,'%,5 9

E&10+, $0&', _-1<,.

&$$"=/,5

6&/

D#1&/

OR

^. L6&/N q 9UY )*&% "$8 ^. LON q 9UY

^. LD#1,/N q 9VV )*&% "$8 ^. LRN q 9VV

Definitionc H .&/5#1 ;&."&<+, 3#. 7*"'* %*,., ,B"$%$ & 5"$'.,%,

5,3"/"%"#/ #3 ;&+-,$ 7"%* $0,'"3",5 0.#<&<"+"%",$ "$ & 5"$'.,%, .&/5#1

;&."&<+,9



!"#$%&%"$%"'$

OSY

Definitionc H .&/5#1 ;&."&<+, 7*#$, ;&+-,$ 3#.1 & '#/%"/--1 L"9,98

*&;, /# =&0$N $-'* %*&% .&/=,$ #3 ;&+-,$ #''-. 7"%* $0,'"3",5

0.#<&<"+"%",$ "$ & '#/%"/-#-$ .&/5#1 ;&."&<+,9

)*, ;&+-,$ %&C,/ <2 & 5"$'.,%, .&/5#1 ;&."&<+, &/5 "%$ &$$#'"&%,50.#<&<"+"%",$ '&/ <, ,B0.,$$,5 <2 & .-+,8 #. .,+&%"#/$*"0 %*&% "$ '&++,5

& probability mass (density) function.

Definitionc H probability distribution (mass function N"$ &

1&%*,1&%"'&+ .,+&%"#/$*"08 #. .-+,8 %*&% &$$"=/$ %# &/2 0#$$"<+, ;&+-,

#3 & 5"$'.,%, .&/5#1 ;&."&<+, k %*, 0.#<&<"+"%2 ^Lk q B"N9 )*"$

&$$"=/1,/% "$ 1&5, 3#. &++ ;&+-,$ B" %*&% *&;, 0#$"%";, 0.#<&<"+"%29)*, 0.#<&<"+"%2 5"$%."<-%"#/ '&/ <, 5"$0+&2,5 "/ %*, 3#.1 #3 & %&<+,

=";"/= %*, ;&+-,$ &/5 %*,". &$$#'"&%,5 0.#<&<"+"%",$ &/5@#. "% '&/ <,

,B0.,$$,5 &$ & 1&%*,1&%"'&+ 3#.1-+& =";"/= %*, 0.#<&<"+"%2 #3 &++

0#$$"<+, ;&+-,$9

General rules which apply to any probability distribution:

O9 E"/', %*, ;&+-,$ #3 & 0.#<&<"+"%2 5"$%."<-%"#/ &., 0.#<&<"+"%",$8

%*,2 1-$% <, /-1<,.$ "/ %*, "/%,.;&+ 3.#1 [ %# O9

R9 E"/', & .&/5#1 ;&."&<+, *&$ %# %&C, #/ #/, #3 "%$ ;&+-,$8 %*,

$-1 #3 &++ %*, ;&+-,$ #3 & 0.#<&<"+"%2 5"$%."<-%"#/ 1-$% <, ,>-&+

%# O9



!"#$%&%"$%"'$

OSW

?=9 )#$$ & '#"/ S %"1,$9 P,% B <, %*, /-1<,. #3 *,&5$ #<%&"/,59 \"/5

%*, 0.#<&<"+"%2 5"$%."<-%"#/ #3 B 9

3 LBN q ^. Lk q B"N 8 " q [8 O8 R8 S9

^. LB q [N q O@] uuuuuuuuuuu99 )))

^. LB q ON q S@] uuuuuuuuuuu9 4)) )4) ))4

^. LB q RN q S@] uuuuuuuuuuu9944) )44 4)4

^. LB q SN q O@] uuuuuuuuuuu9 444

^.#<&<"+"%2 5"$%."<-%"#/ #3 k9

k q B" [ O R S

^.LkqB"N O@] S@] S@] O@]

)*, .,>-".,5 '#/5"%"#/$ &., &+$# $&%"$3",59 "N 3LBN ≥ [ ""N ∑ 3 LB"N q O

UU99UU99OO ))44?? ??kk^??GG))??KK ZZ H HPPAA?? QQ\\ H H KK::EEGGTT??))?? TT H H__KKQQ66 ZZ H HTT:: H H!!PP??

)*, ,B0,'%,5 ;&+-,8 5,/#%,5 <2 ?LBN #. μ8 .,0.,$,/%$ %*, l&;,.&=,m

;&+-, #3 %*, .&/5#1 ;&."&<+,9 :% "$ #<%&"/,5 <2 1-+%"0+2"/= ,&'*

0#$$"<+, ;&+-, <2 "%$ .,$0,'%";, 0.#<&<"+"%2 &/5 $-11"/= #;,. &++ %*,

;&+-,$ %*&% *&;, 0#$"%";, 0.#<&<"+"%29

Definitionc )*, ,B0,'%,5 ;&+-, #3 & 5"$'.,%, .&/5#1 ;&."&<+, "$

5,3"/,5 &$



!"#$%&%"$%"'$

OS]

?LkN q μ q #Q@U5=

Q ("(

( =∑=

D*,., %*, B"b$ &., %*, ;&+-,$ %*, .&/5#1 ;&."&<+, &$$-1,$ 7"%*

0#$"%";, 0.#<&<"+"%2

Examplec G#/$"5,. %*, .&/5#1 ;&."&<+, .,0.,$,/%"/= %*, /-1<,. #3

,0"$#5,$ #3 5"&..*#,& "/ %*, 3".$% R 2,&.$ #3 +"3,9 E-00#$, %*"$ .&/5#1

;&."&<+, *&$ & 0.#<&<"+"%2 1&$$ 3-/'%"#/ &$ <,+#7

T [ O R S V U Y

^L

k q

.N

9ORX 9RYV 9RWO 9O]U 9[XU 9[SX 9[OW

D*&% "$ %*, ,B0,'%,5 /-1<,. #3 ,0"$#5,$ #3 5"&..*#,& "/ %*, 3".$% R2,&.$ #3 +"3,a

?LkN q [L9ORXNrOL9RYVNrRL9RWONrSL9O]UNrVL9[XUNrUL9[SXNrYL9[OWNq

R9[S]

)*-$8 #/ %*, &;,.&=, & '*"+5 7#-+5 <, ,B0,'%,5 %# *&;, R ,0"$#5,$

#3 5"&..*#,& "/ %*, 3".$% R 2,&.$ #3 +"3,9

""99 )

)44?? ZZ H HTT:: H H__GG?? QQ\\ H H KK::EEGGTT??))?? TT H H__KKQQ66 ZZ H HTT:: H H!!PP??

)*, ;&."&/', .,0.,$,/%$ %*, $0.,&5 #3 &++ ;&+-,$ %*&% *&;, 0#$"%";,

0.#<&<"+"%2 .,+&%";, %# %*, ,B0,'%,5 ;&+-,9 :/ 0&.%"'-+&.8 %*, ;&."&/', "$

#<%&"/,5 <2 1-+%"0+2"/= %*, $>-&.,5 5"$%&/', #3 ,&'* 0#$$"<+, ;&+-,



!"#$%&%"$%"'$

OSX

3.#1 %*, ,B0,'%,5 ;&+-, <2 "%$ .,$0,'%";, 0.#<&<"+"%2 &/5 $-11"/=

#;,.&++ %*, ;&+-,$ %*&% *&;, 0#$"%";, 0.#<&<"+"%29

Definitionc )*, ;&."&/', #3 & 5"$'.,%, .&/5#1 ;&."&<+, 5,/#%,5 <2 k

"$ 5,3"/,5 <2

ZLkN q ∑ ∑=

−===−==

]

"

] $(

$((

$(

$

"μ#Q@U5Q#Q@U5#μQUσ

i i

D*,., %*, k"b$ &., %*, ;&+-,$ 3#. 7*"'* %*, .&/5#1 ;&."&<+, %&C,$ #/

0#$"%";, 0.#<&<"+"%29 )*, EK #3 & .&/5#1 ;&."&<+, k8 5,/#%,5 <2

EKLkN #. σ "$ 5,3"/,5 <2 $>-&., .##% #3 "%$ ;&."&/',9

Examplec G#10-%, %*, ;&."&/', &/5 EK 3#. %*, .&/5#1 ;&."&<+,

.,0.,$,/%"/= /-1<,. #3 ,0"$#5,$ #3 5"&..*#,& "/ %*, 3".$% R 2,&.$ #3+"3,9

?LkN q μ q R9[V

#Q@U5=

Q ("(

( =∑=

q [RL9ORXN r ORL9RYVN r RRL9RWON r SRL9O]UN r

VRL9[XUN r URL9[SXN r YRL[9[OWN q Y9OR

)*-$8 ZLkN q Y9OR f LR9[VNR q O9XYW &/5 %*, EK #3 k "$

"S87$"SR9Oσ ==



!"#$%&%"$%"'$

OV[

""""99 ))44?? !!::__QQ66:: H HPP KK::EE))TT::!!AA))::QQ__

Binomial assumptions:

ON )*, $&1, ,B0,."1,/% "$ '&..",5 #-% / %"1,$ L / %."&+$ &., 1&5,N9

RN ?&'* %."&+ *&$ %7# 0#$$"<+, #-%'#1,$ L -$-&++2 %*,$, #-%'#1,$ &.,

'&++,5 l $-'',$$m &/5 l 3&"+-.,m9 _#%, %*&% & $-'',$$3-+ #-%'#1, 5#,$

/#% "10+2 & =##5 #/,8 /#. 3&"+-., & <&5 #-%'#1,9 :3 ^ "$ %*, 0.#<&<"+"%2

#3 $-'',$$ "/ #/, %."&+8 %*,/ 8 O`0 "$ %*, 0.#<&<"+"%2 #3 3&"+-.,9

SN )*, .,$-+% #3 ,&'* %."&+ "$ "/5,0,/5,/% #3 %*, .,$-+% #3 &/2 #%*,. %."&+9

Definitionc :3 %*, <"/#1"&+ &$$-10%"#/$ &., $&%"$3",58 %*, 0.#<&<"+"%2 #3

. $-'',$$,$ "/ / %."&+$ "$c

2 =2 @#U"@2

= 2#@U5 −−⎟⎟

⎞⎜⎜⎝

⎛ == 8 . q [8 O8 R8 u8 /

)*"$ 0.#<&<"+"%2 5"$%."<-%"#/ "$ '&++,5 %*, <"/#1"&+ 5"$%."<-%"#/9

⎟ ⎠ ⎞

⎜⎝ ⎛

2

= q / G. "$ %*, /-1<,. #3 7&2$ #3 '*##$"/= . "%,1$ 3.#1 /8 &/5 "$

& /-1<,. 7, *&;, %# '&+'-+&%,9 )*, =,/,.&+ 3#.1-+& 3#. %*,

'#,33"'",/%$ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ 2

= "$ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ 2

=q

2#dU=2d

=d

−



!"#$%&%"$%"'$

OVO

:3 %*, %.-, 0.#0#.%"#/ #3 ,;,/%$ #3 "/%,.,$% "$ ^8 %*,/ "/ & $&10+, #3

$"F, / %*, 1,&/ #3 %*, <"/#1"&+ 5"$%."<-%"#/ "$ n×p &/5 %*, $%&/5&.5

5,;"&%"#/ "$ 4#=4U"−

Example: H$$-1, %*&%8 7*,/ & '*"+5 "$ <#./8 %*, 0.#<&<"+"%2 "% "$ & =".+

"$ { &/5 %*&% %*, $,B #3 %*, '*"+5 5#,$ /#% 5,0,/5 #/ %*, $,B #3 &/

#+5,. $"<+"/=9

HN \"/5 %*, 0.#<&<"+"%2 5"$%."<-%"#/ 3#. %*, /-1<,. #3 =".+$ "/ & 3&1"+2

7"%* V '*"+5.,/9

!N \"/5 %*, 1,&/ &/5 %*, $%&/5&.5 5,;"&%"#/ #3 %*"$ 5"$%."<-%"#/9

3LBN q 0Lkq.N q V G. LO@RN. LO@RNV``. j kq [ 8O8 R8 S8 V9

HN ^.#<&<"+"%2 5"$%."<-%"#/

k [ O R S V

^Lkq.N O@OY V@OY Y@OY V@OY O@OY

!N 6,&/ q /^ q V B O@R q R

E%&/5&.5 5,;"&%"#/ q #"U P nP − q $&"$&"8 ×× q " q O



!"#$%&%"$%"'$

OVR

Exercisec E-00#$, %*&% "/ & ',.%&"/ 1&+&."#-$ &.,& 0&$% ,B0,.",/',

"/5"'&%,$ %*&% %*, 0.#<&<"+"%2 #3 & 0,.$#/ 7"%* & *"=* 3,;,. 7"++ <, 0#$"%";,

3#. 1&+&."& "$ [9W9 G#/$"5,. S .&/5#1+2 $,+,'%,5 0&%",/%$ L7"%* *"=*

3,;,.N "/ %*&% $&1, &.,&9

ON D*&% "$ %*, 0.#<&<"+"%2 %*&% /# 0&%",/% 7"++ <, 0#$"%";, 3#. 1&+&."&a

RN D*&% "$ %*, 0.#<&<"+"%2 %*&% ,B&'%+2 #/, 0&%",/% 7"++ <, 0#$"%";, 3#.

1&+&."&a

SN D*&% "$ %*, 0.#<&<"+"%2 %*&% ,B&'%+2 %7# #3 %*, 0&%",/%$ 7"++ <, 0#$"%";,

3#. 1&+&."&a

VN D*&% "$ %*, 0.#<&<"+"%2 %*&% &++ 0&%",/%$ 7"++ <, 0#$"%";, 3#. 1&+&."&a

UN \"/5 %*, 1,&/ &/5 %*, EK #3 %*, 0.#<&<"+"%2 5"$%."<-%"#/ =";,/

&<#;,9

H/$7,.c ON [9[RW RN [9O]X SN [9VVO VN [9SVS

UN μ q R9O &/5 σ q [9WXV

5.5.4 The Normal Distribution

)*, Normal Distribution "$ <2 3&. %*, 1#$% "10#.%&/% 0.#<&<"+"%25"$%."<-%"#/ "/ $%&%"$%"'$9 :% "$ &+$# $#1,%"1,$ C/#7/ &$ %*, I&-$$"&/

5"$%."<-%"#/8 &3%,. %*, 1&%*,1&%"'"&/ I&-$$9 )*, 5"$%."<-%"#/$ #3 1&/2

1,5"'&+ 1,&$-.,1,/%$ "/ 0#0-+&%"#/$ 3#++#7 & /#.1&+ 5"$%."<-%"#/ L,=9

E,.-1 -."' &'"5 +,;,+$8 '*#+,$%,.#+ +,;,+$8 <+##5 0.,$$-.,8 *,"=*% &/5



!"#$%&%"$%"'$

OVS

7,"=*%N9 )*, /#.1&+ 5"$%."<-%"#/ "$ & %*,#.,%"'&+8 '#/%"/-#-$ 0.#<&<"+"%2

5"$%."<-%"#/ 7*#$, ,>-&%"#/ "$c

$

$

"Z

1

$

"EUQ#

⎟ ⎠ ⎞

⎜⎝ ⎛ −

= σ

μ

σ π

x

3#. `∝ t B t r∝

)*, &.,& %*&% .,0.,$,/%$ %*, 0.#<&<"+"%2 <,%7,,/ %7# 0#"/%$ ' &/5 5

#/ &<$'"$$& "$ 5,3"/,5 <2c

^L' t k t 5N q ?Q

$

σ

μQ

$

"

σπ$

"1

⎟ ⎠ ⎞

⎜⎝ ⎛ −

∫d

c

The important characteristics of the Normal Distribution are:

ON :% "$ & 0.#<&<"+"%2 5"$%."<-%"#/ #3 & '#/%"/-#-$ ;&."&<+,9 :% ,B%,/5$

3.#1 1"/-$ "/3"/"%2L ∞N %# 0+-$ "/3"/"%2 Lr∞N9

RN :% "$ -/"1#5&+8 <,++`$*&0,5 &/5 $211,%."'&+ &<#-% B q -9

SN :% "$ 5,%,.1"/,5 <2 %7# >-&/%"%",$c "%$ 1,&/ ( μ N &/5 EK L σ N9

G*&/="/= μ &+#/, $*"3%$ %*, ,/%"., /#.1&+ '-.;, %# %*, +,3% #. ."=*%9

G*&/="/= σ &+#/, '*&/=,$ %*, 5,=.,, %# 7*"'* %*, 5"$%."<-%"#/ "$

$0.,&5 #-%9

V9 )*, *,"=*% #3 %*, 3.,>-,/'2 '-.;,8 7*"'* "$ '&++,5 %*, 0.#<&<"+"%2

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&/5σ

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'&+'-+&%"/= c

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Area under any Normal curve

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Z- Scores

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The number of standard deviation units is called a Z-score or Z-

value.

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*20,.%,/$";,a

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⎜⎝ ⎛ −

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q ^ Ld t O9RUN − ^Ld t [9]SN q [9]XVV − [9WXYW q [9[X]

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<,+#7 X[=@O[[[ '&+a

Solutionc P,% k <, '&.<#*25.&%, "/%&C, "/ ORÒV`2,&.`#+5 1&+,$ &/5

k ∼ _ LORV8 V[[N

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q O− ^Ld t [9]N q O− [9W]]O q [9ROOX

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q ^Ld • O9WN q O− ^Ld t O9WN q O− [9XUUV q [9[VVY

b. Exercises

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3. Answer the following questions by referring to the table of the

standard normal distribution.

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CHAPTER SIX

SAMPLING METHODS

66..11 LLEEAARRNNIINNGG OOBBJJEECCTTIIVVEESS

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$&10+"/= 1,%*#5$

66..22 IINNTTRROODDUUCCTTIIOONN

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Advantages of samples

• '#$% ` $&10+"/= $&;,$ %"1,8 +&<#-. &/5 1#/,2

• >-&+"%2 #3 5&%& ` 1#., %"1, &/5 ,33#.% '&/ <, $0,/% #/ =,%%"/=

.,+"&<+, 5&%& #/ ,&'* "/5";"5-&+ "/'+-5,5 "/ %*, $&10+,9

` K-, %# %*, -$, #3 <,%%,. %.&"/,5 0,.$#//,+8 1#., '&.,3-+

$-0,.;"$"#/ &/5 0.#',$$"/= & $&10+, '&/ &'%-&++2 0.#5-',

0.,'"$, .,$-+%$9

If we have to draw a sample, we will be confronted with the

following questionsc

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5.&7 & $&10+,a

<N 4#7 1&/2 0,#0+, 5# 7, /,,5 "/ #-. $&10+,a

'N 4#7 7"++ %*,$, 0,#0+, <, $,+,'%,5a

Apart from persons, a population may consist of mosquitoes,

villages, institutions, etc.

6.3 Common terms used in sampling

Reference population (also called source population or target

population) - %*, 0#0-+&%"#/ #3 "/%,.,$%8 %# 7*"'* %*, "/;,$%"=&%#.$



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Study or sample population ` %*, 0#0-+&%"#/ "/'+-5,5 "/ %*, $&10+,9

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Study unit ` %*, -/"% #/ 7*"'* "/3#.1&%"#/ "$ '#++,'%,59

` %*, $&10+"/= -/"% "$ /#% /,',$$&."+2 %*, $&1, &$ %*, $%-52 -/"%9

` "3 %*, #<J,'%";, "$ %# 5,%,.1"/, %*, &;&"+&<"+"%2 #3 +&%."/,8 %*,/ %*,

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0.,;&+,/', #3 %.&'*#1&8 %*,/ %*, $%-52 -/"% 7#-+5 <, %*, "/5";"5-&+9

Sampling frame ` %*, +"$% #3 &++ %*, -/"%$ "/ %*, .,3,.,/', 0#0-+&%"#/8

3.#1 7*"'* & $&10+, "$ %# <, 0"'C,59

Sampling fraction (Sampling interval) ` %*, .&%"# #3 %*, /-1<,. #3

-/"%$ "/ %*, $&10+, %# %*, /-1<,. #3 -/"%$ "/ %*, .,3,.,/', 0#0-+&%"#/

(n/N)



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6.4 Sampling methods (Two broad divisions)

6.4.1 Non-probability Sampling Methods

` A$,5 7*,/ & $&10+"/= 3.&1, 5#,$ /#% ,B"$%

` _# .&/5#1 $,+,'%"#/ L-/.,0.,$,/%&%";, #3 %*, =";,/

0#0-+&%"#/N

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3"/5"/=$ #<%&"/,5 3.#1 & $&10+, %# %*, 0#0-+&%"#/9

Two such non-probability sampling methods are:

A) Convenience samplingc "$ & 1,%*#5 "/ 7*"'* 3#. '#/;,/",/',

$&C, %*, $%-52 -/"%$ %*&% *&00,/ %# <, &;&"+&<+, &% %*, %"1, #3 5&%&

'#++,'%"#/ &., $,+,'%,59

B) Quota samplingc "$ & 1,%*#5 %*&% ,/$-.,$ %*&% & ',.%&"/ /-1<,. #3

$&10+, -/"%$ 3.#1 5"33,.,/% '&%,=#.",$ 7"%* $0,'"3"' '*&.&'%,."$%"'$

&., .,0.,$,/%,59 :/ %*"$ 1,%*#5 %*, "/;,$%"=&%#. "/%,.;",7$ &$ 1&/2

0,#0+, "/ ,&'* '&%,=#.2 #3 $%-52 -/"% &$ *, '&/ 3"/5 -/%"+ *, *&$

3"++,5 *"$ >-#%&9

Both the above methods do not claim to be representative of the

entire population9



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6.4.2 Probability Sampling methods

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` :/;#+;, .&/5#1 $,+,'%"#/ 0.#',5-.,$9 H++ -/"%$ #3 %*, 0#0-+&%"#/

$*#-+5 *&;, &/ ,>-&+ #. &% +,&$% & C/#7/ '*&/', #3 <,"/= "/'+-5,5

"/ %*, $&10+,9

` I,/,.&+"F&%"#/ "$ 0#$$"<+, L3.#1 $&10+, %# 0#0-+&%"#/N

A) Simple random sampling (SRS)

` )*"$ "$ %*, 1#$% <&$"' $'*,1, #3 .&/5#1 $&10+"/=9

` ?&'* -/"% "/ %*, $&10+"/= 3.&1, *&$ &/ ,>-&+ '*&/', #3 <,"/=

$,+,'%,5

` .,0.,$,/%&%";,/,$$ #3 %*, $&10+, "$ ,/$-.,59

4#7,;,.8 "% "$ '#$%+2 %# '#/5-'% ETE9 6#.,#;,.8 1"/#."%2 $-<=.#-0$ #3

"/%,.,$% "/ %*, 0#0-+&%"#/ 12 /#% <, 0.,$,/% "/ %*, $&10+, "/ $-33"'",/%

/-1<,.$ 3#. $%-529

To select a simple random sample you need to:

•

6&C, & /-1<,.,5 +"$% #3 &++ %*, -/"%$ "/ %*, 0#0-+&%"#/ 3.#1 7*"'*

2#- 7&/% %# 5.&7 & $&10+,9

• ?&'* -/"% #/ %*, +"$% $*#-+5 <, /-1<,.,5 "/ $,>-,/', 3.#1 O %#

_ L7*,., _ "$ %*, $"F, #3 %*, 0#0-+&%"#/N

• K,'"5, #/ %*, $"F, #3 %*, $&10+,



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• E,+,'% %*, .,>-".,5 /-1<,. #3 $%-52 -/"%$8 -$"/= & l+#%%,.2m

1,%*#5 #. & %&<+, #3 .&/5#1 /-1<,.$9

hLottery” methodc 3#. & $1&++ 0#0-+&%"#/ "% 1&2 <, 0#$$"<+, %# -$, %*,

l+#%%,.2m 1,%*#5c ,&'* -/"% "/ %*, 0#0-+&%"#/ "$ .,0.,$,/%,5 <2 & $+"0 #3

0&0,.8 %*,$, &., 0-% "/ & <#B &/5 1"B,58 &/5 & $&10+, #3 %*, .,>-".,5

$"F, "$ 5.&7/ 3.#1 %*, <#B9

Table of random numbers: "3 %*,., &., 1&/2 -/"%$8 *#7,;,.8 %*,

&<#;, %,'*/">-, $##/ <,'#1,$ +&<#."#-$9 E,+,'%"#/ #3 %*, -/"%$ "$

=.,&%+2 3&'"+"%&%,5 &/5 1&5, 1#., &''-.&%, <2 -$"/= & $,% #3 .&/5#1

/-1<,.$ "/ 7*"'* & +&.=, /-1<,. #3 5"="%$ "$ $,% #-% "/ .&/5#1 #.5,.9

)*, 0.#0,.%2 #3 & %&<+, #3 .&/5#1 /-1<,.$ "$ %*&%8 7*"'*,;,. 7&2 "% "$

.,&58 ;,.%"'&++2 "/ '#+-1/$ #. *#."F#/%&++2 "/ .#7$8 %*, #.5,. #3 %*, 5"="%$

"$ .&/5#19 _#7&5&2$8 &/2 $'",/%"3"' '&+'-+&%#. *&$ %*, $&1, 3&'"+"%",$9

B) Systematic Sampling

:/5";"5-&+$ &., '*#$,/ &% .,=-+&. "/%,.;&+$ L 3#. ,B&10+,8 ,;,.2 C %*N 3.#1

%*, $&10+"/= 3.&1,9 )*, 3".$% -/"% %# <, $,+,'%,5 "$ %&C,/ &% .&/5#1 3.#1

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$,+,'%,5 3.#1 OR[[ $%-5,/%$ #3 & $'*##+9 )*, $&10+, $"F, "$ 5,'"5,5 %#

<, O[[9 )*, $&10+"/= 3.&'%"#/ "$c O[[ @OR[[ q O@OR9 4,/',8 %*, $&10+,

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)*, /-1<,. #3 %*, 3".$% $%-5,/% %# <, "/'+-5,5 "/ %*, $&10+, "$ '*#$,/

.&/5#1+28 3#. ,B&10+, <2 <+"/5+2 0"'C"/= #/, #-% #3 %7,+;, 0",',$ #3

0&0,.8 /-1<,.,5 O %# OR9 :3 /-1<,. Y "$ 0"'C,58 ,;,.2 %7,+3%* $%-5,/%

7"++ <, "/'+-5,5 "/ %*, $&10+,8 $%&.%"/= 7"%* $%-5,/% /-1<,. Y8 -/%"+ O[[

$%-5,/%$ &., $,+,'%,59 )*, /-1<,.$ $,+,'%,5 7#-+5 <, Y8O]8S[8VR8,%'9

Merits

• E2$%,1&%"' $&10+"/= "$ -$-&++2 +,$$ %"1, '#/$-1"/= &/5 ,&$",.

%# 0,.3#.1 %*&/ $"10+, .&/5#1 $&10+"/=9 :% 0.#;"5,$ & =##5

&00.#B"1&%"#/ %# ETE9

• A/+"C, ETE8 $2$%,1&%"' $&10+"/= '&/ <, '#/5-'%,5 7"%*#-% &

$&10+"/= 3.&1, L-$,3-+ "/ $#1, $"%-&%"#/$ 7*,., & $&10+"/=

3.&1, "$ /#% .,&5"+2 &;&"+&<+,N9

?=98 :/ 0&%",/%$ &%%,/5"/= & *,&+%* ',/%,.8 7*,., "% "$ /#% 0#$$"<+, %#

0.,5"'% "/ &5;&/', 7*# 7"++ <, &%%,/5"/=9

Demerits

• :3 %*,., "$ &/2 $#.% #3 '2'+"' 0&%%,./ "/ %*, #.5,."/= #3 %*, $-<J,'%$

7*"'* '#"/'"5,$ 7"%* %*, $&10+"/= "/%,.;&+8 %*, $&10+, 7"++ /#% <,

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Examples

` P"$% #3 1&..",5 '#-0+,$ &..&/=,5 7"%* 1,/e$ /&1,$ &+%,./&%";,+2

7"%* %*, 7#1,/e$ /&1,$ L,;,.2 R/58 V%* 8 ,%'9N 7"++ .,$-+% "/ &

$&10+, #3 &++ 1,/ #. 7#1,/N9

` :3 7, 7&/% %# $,+,'% & .&/5#1 $&10+, #3 & ',.%&"/ 5&2 L$&10+"/=

3.&'%"#/ #/ 7*"'* %# '#-/% '+"/"' &%%,/5&/',8 %*"$ 5&2 1&2 3&++ #/ %*,

$&1, 5&2 #3 %*, 7,,C8 7*"'* 1"=*%8 3#. ,B&10+, <, & 1&.C,% 5&29

C) Stratified Sampling

:% "$ &00.#0."&%, 7*,/ %*, 5"$%."<-%"#/ #3 %*, '*&.&'%,."$%"' %# <,

$%-5",5 "$ $%.#/=+2 &33,'%,5 <2 ',.%&"/ ;&."&<+, L*,%,.#=,/,#-$

0#0-+&%"#/N9 )*, 0#0-+&%"#/ "$ 3".$% 5";"5,5 "/%# =.#-0$ L$%.&%&N

&''#.5"/= %# & '*&.&'%,."$%"' #3 "/%,.,$% L,=98 $,B8 =,#=.&0*"' &.,&8

0.,;&+,/', #3 5"$,&$,8 ,%'9N9 H $,0&.&%, $&10+, "$ %*,/ %&C,/

"/5,0,/5,/%+2 3.#1 ,&'* $%.&%-18 <2 $"10+, .&/5#1 #. $2$%,1&%"'

$&10+"/=9

• proportional allocation ` "3 %*, $&1, $&10+"/= 3.&'%"#/ "$ -$,5 3#.

,&'* $%.&%-19

• non- proportional allocation ` "3 & 5"33,.,/% $&10+"/= 3.&'%"#/ "$ -$,5

3#. ,&'* $%.&%-1 #. "3 %*, $%.&%& &., -/,>-&+ "/ $"F, &/5 & 3"B,5

/-1<,. #3 -/"%$ "$ $,+,'%,5 3.#1 ,&'* $%.&%-19



!"#$%&%"$%"'$

OU]

Merit

` )*, .,0.,$,/%&%";,/,$$ #3 %*, $&10+, "$ "10.#;,59 )*&% "$8 &5,>-&%,

.,0.,$,/%&%"#/ #3 1"/#."%2 $-<=.#-0$ #3 "/%,.,$% '&/ <, ,/$-.,5 <2

$%.&%"3"'&%"#/ &/5 <2 ;&.2"/= %*, $&10+"/= 3.&'%"#/ <,%7,,/ $%.&%& &$

.,>-".,59

DEMERIT

` E&10+"/= 3.&1, 3#. %*, ,/%"., 0#0-+&%"#/ *&$ %# <, 0.,0&.,5

$,0&.&%,+2 3#. ,&'* $%.&%-19

D) Cluster sampling

:/ %*"$ $&10+"/= $'*,1,8 $,+,'%"#/ #3 %*, .,>-".,5 $&10+, "$ 5#/, #/

=.#-0$ #3 $%-52 -/"%$ L'+-$%,.$N "/$%,&5 #3 ,&'* $%-52 -/"% "/5";"5-&++29

)*, $&10+"/= -/"% "$ & '+-$%,.8 &/5 %*, $&10+"/= 3.&1, "$ & +"$% #3

%*,$, '+-$%,.$9

procedure

` )*, .,3,.,/', 0#0-+&%"#/ L*#1#=,/,#-$N "$ 5";"5,5 "/%# '+-$%,.$9

)*,$, '+-$%,.$ &., #3%,/ =,#=.&0*"' -/"%$ L,= 5"$%."'%$8 ;"++&=,$8

,%'9N

` H $&10+, #3 $-'* '+-$%,.$ "$ $,+,'%,5

` H++ %*, -/"%$ "/ %*, $,+,'%,5 '+-$%,.$ &., $%-5",5



!"#$%&%"$%"'$

OUX

:% "$ 0.,3,.&<+, %# $,+,'% & +&.=, /-1<,. #3 $1&++ '+-$%,.$ .&%*,. %*&/ &

$1&++ /-1<,. #3 +&.=, '+-$%,.$9

Merit

H +"$% #3 &++ %*, "/5";"5-&+ $%-52 -/"%$ "/ %*, .,3,.,/', 0#0-+&%"#/ "$ /#%

.,>-".,59 :% "$ $-33"'",/% %# *&;, & +"$% #3 '+-$%,.$9

Demerit

:% "$ <&$,5 #/ %*, &$$-10%"#/ %*&% %*, '*&.&'%,."$%"' %# <, $%-5",5 "$

-/"3#.1+2 5"$%."<-%,5 %*.#-=*#-% %*, .,3,.,/', 0#0-+&%"#/8 7*"'* 1&2

/#% &+7&2$ <, %*, '&$,9 4,/',8 $&10+"/= ,..#. "$ -$-&++2 *"=*,. %*&/

3#. & $"10+, .&/5#1 $&10+, #3 %*, $&1, $"F,9

E) Multi-stage sampling

)*"$ 1,%*#5 "$ &00.#0."&%, 7*,/ %*, .,3,.,/', 0#0-+&%"#/ "$ +&.=, &/5

7"5,+2 $'&%%,.,5 9 E,+,'%"#/ "$ 5#/, "/ $%&=,$ -/%"+ %*, 3"/&+ $&10+"/= -/"%

L,=98 *#-$,*#+5$ #. 0,.$#/$N &., &..";,5 &%9 )*, 0."1&.2 $&10+"/= -/"%

L^EAN "$ %*, $&10+"/= -/"% L-$-&++2 +&.=, $"F,N "/ %*, 3".$% $&10+"/=$%&=,9 )*, $,'#/5&.2 $&10+"/= -/"% LEEAN "$ %*, $&10+"/= -/"% "/ %*,

$,'#/5 $&10+"/= $%&=,8 ,%'9



!"#$%&%"$%"'$

OY[

Example - )*, ^EA$ '#-+5 <, kebeles &/5 %*, EEA$ '#-+5 <,

*#-$,*#+5$9

Merit - G-%$ %*, '#$% #3 0.,0&."/= $&10+"/= 3.&1,

Demerit - E&10+"/= ,..#. "$ "/'.,&$,5 '#10&.,5 7"%* & $"10+, .&/5#1

$&10+,9

6-+%"$%&=, $&10+"/= =";,$ +,$$ 0.,'"$, ,$%"1&%,$ %*&/ $&10+, .&/5#1

$&10+"/= 3#. %*, $&1, $&10+, $"F,8 <-% %*, .,5-'%"#/ "/ '#$% -$-&++2

3&. #-%7,"=*$ %*"$8 &/5 &++#7$ 3#. & +&.=,. $&10+, $"F,9

6.5 Errors in sampling

D*,/ 7, %&C, & $&10+,8 #-. .,$-+%$ 7"++ /#% ,B&'%+2 ,>-&+ %*, '#..,'%

.,$-+%$ 3#. %*, 7*#+, 0#0-+&%"#/9 )*&% "$8 #-. .,$-+%$ 7"++ <, $-<J,'% %#

,..#.$9

6.5.1 Sampling error (random error)

H $&10+, "$ & $-<$,% #3 & 0#0-+&%"#/9 !,'&-$, #3 %*"$ 0.#0,.%2 #3

$&10+,$8 .,$-+%$ #<%&"/,5 3.#1 %*,1 '&//#% .,3+,'% %*, 3-++ .&/=, #3

;&."&%"#/ 3#-/5 "/ %*, +&.=,. =.#-0 L0#0-+&%"#/N9 )*"$ %20, #3 ,..#.8

&."$"/= 3.#1 %*, $&10+"/= 0.#',$$ "%$,+38 "$ '&++,5 $&10+"/= ,..#.8



!"#$%&%"$%"'$

OYO

7*"'* "$ & 3#.1 #3 .&/5#1 ,..#.9 E&10+"/= ,..#. '&/ <, 1"/"1"F,5 <2

"/'.,&$"/= %*, $"F, #3 %*, $&10+,9 D*,/ / q _ ⇒ $&10+"/= ,..#. q [

6.5.2 Non-sampling error (bias)

:% "$ & %20, #3 $2$%,1&%"' ,..#. "/ %*, 5,$"=/ #. '#/5-'% #3 & $&10+"/=

0.#',5-., 7*"'* .,$-+%$ "/ 5"$%#.%"#/ #3 %*, $&10+,8 $# %*&% "% "$ /#

+#/=,. .,0.,$,/%&%";, #3 %*, .,3,.,/', 0#0-+&%"#/9 D, '&/ ,+"1"/&%, #.

.,5-', %*, /#/`$&10+"/= ,..#. L<"&$N <2 '&.,3-+ 5,$"=/ #3 %*, $&10+"/=

0.#',5-., &/5 /#% <2 "/'.,&$"/= %*, $&10+, $"F,9

Example: :3 2#- %&C, 1&+, $%-5,/%$ #/+2 3.#1 & $%-5,/% 5#.1"%#.2 "/

?%*"#0"& "/ #.5,. %# 5,%,.1"/, %*, 0.#0#.%"#/ #3 $1#C,.$8 2#- 7#-+5

.,$-+% "/ &/ #;,.,$%"1&%,8 $"/', 3,1&+,$ &., +,$$ +"C,+2 %# $1#C,9

:/'.,&$"/= %*, /-1<,. #3 1&+, $%-5,/%$ 7#-+5 /#% .,1#;, %*, <"&$9

• )*,., &., $,;,.&+ 0#$$"<+, $#-.',$ #3 <"&$ "/ $&10+"/= L,=98

&'',$$"<"+"%2 <"&$8 ;#+-/%,,. <"&$8 ,%'9N

• )*, <,$% C/#7/ $#-.', #3 <"&$ "$ /#/ .,$0#/$,9 :% "$ %*, 3&"+-., %#

#<%&"/ "/3#.1&%"#/ #/ $#1, #3 %*, $-<J,'%$ "/'+-5,5 "/ %*, $&10+, %#

<, $%-5",59

• _#/ .,$0#/$, .,$-+%$ "/ $"=/"3"'&/% <"&$ 7*,/ %*, 3#++#7"/= %7#

'#/5"%"#/$ &., <#%* 3-+3"++,59



!"#$%&%"$%"'$

OYR

` D*,/ /#/`.,$0#/5,/%$ '#/$%"%-%, & $"=/"3"'&/% 0.#0#.%"#/ #3 %*,

$&10+, L&<#-% OUi #. 1#.,N

` D*,/ /#/`.,$0#/5,/%$ 5"33,. $"=/"3"'&/%+2 3.#1 .,$0#/5,/%$9

• )*,., &., $,;,.&+ 7&2$ %# 5,&+ 7"%* %*"$ 0.#<+,1 &/5 .,5-', %*,

0#$$"<"+"%2 #3 <"&$c

&N K&%& '#++,'%"#/ %##+$ L>-,$%"#//&".,N *&;, %# <, 0.,`%,$%,59

<N :3 /#/ .,$0#/$, "$ 5-, %# &<$,/', #3 %*, $-<J,'%$8 .,0,&%,5

&%%,10%$ $*#-+5 <, '#/$"5,.,5 %# '#/%&'% $%-52 $-<J,'%$ 7*#

7,., &<$,/% &% %*, %"1, #3 %*, "/"%"&+ ;"$"%9

'N )# "/'+-5, &55"%"#/&+ 0,#0+, "/ %*, $&10+,8 $# %*&% /#/`

.,$0#/5,/%$ 7*# 7,., &<$,/% 5-."/= 5&%& '#++,'%"#/ '&/ <,

.,0+&',5 L1&C, $-., %*&% %*,". &<$,/', "$ /#% .,+&%,5 %# %*, %#0"'

<,"/= $%-5",5N9

NB: The number of non-responses should be documented

according to type, so as to facilitate an assessment of the extent of

bias introduced by non-response.



!"#$%&%"$%"'$

OYS

CHAPTER SEVEN

ESTIMATION

7.1 Learning objectives

H% %*, ,/5 #3 %*"$ '*&0%,. %*, $%-5,/% 7"++ <, &<+, %#c

O9 A/5,.$%&/5 %*, '#/',0%$ #3 $&10+, $%&%"$%"'$ &/5 0#0-+&%"#/

0&.&1,%,.$

R9 A/5,.$%&/5 %*, 0."/'"0+,$ #3 $&10+"/= 5"$%."<-%"#/$ #3 1,&/$ &/5

0.#0#.%"#/$ &/5 '&+'-+&%, %*,". $%&/5&.5 ,..#.$

S9 A/5,.$%&/5 %*, 0."/'"0+,$ #3 ,$%"1&%"#/ &/5 5"33,.,/%"&%, <,%7,,/

0#"/% &/5 "/%,.;&+ ,$%"1&%"#/$

V9 G#10-%, &00.#0."&%, '#/3"5,/', "/%,.;&+$ 3#. 0#0-+&%"#/ 1,&/$

&/5 0.#0#.%"#/$ &/5 "/%,.0.,% %*, 3"/5"/=$

U9 K,$'."<, 1,%*#5$ #3 $&10+, $"F, '&+'-+&%"#/ 3#. '.#$$ f $,'%"#/&+

$%-5",$

7.2 Introduction

:/ %*"$ '*&0%,. %*, '#/',0%$ #3 $&10+, $%&%"$%"'$ &/5 0#0-+&%"#/

0&.&1,%,.$ &., 5,$'."<,59 )*, $&10+, 3.#1 & 0#0-+&%"#/ "$ -$,5 %#

0.#;"5, %*, ,$%"1&%,$ #3 %*, 0#0-+&%"#/ 0&.&1,%,.$9 )*, $%&/5&.5

,..#.8 #/, #3 %*, 1#$% "10#.%&/% '#/',0%$ "/ $%&%"$%"'&+ "/3,.,/',8 "$



!"#$%&%"$%"'$

OYV

"/%.#5-',59 6,%*#5$ 3#. '&+'-+&%"/= '#/3"5,/', "/%,.;&+$ 3#.

0#0-+&%"#/ 1,&/$ &/5 0.#0#.%"#/$ &., =";,/9 )*, "10#.%&/', #3 %*,

/#.1&+ 5"$%."<-%"#/ Ld 5"$%."<-%"#/N "$ $%.,$$,5 %*.#-=*#-% %*, '*&0%,.9

7.3 Point Estimation

Definitionc H 0&.&1,%,. "$ & /-1,."'&+ 5,$'."0%";, 1,&$-., #3 &

0#0-+&%"#/ L μ "$ &/ ,B&10+, #3 & 0&.&1,%,.N9 H $%&%"$%"' "$ & /-1,."'&+

5,$'."0%";, 1,&$-., #3 & $&10+, L X "$ &/ ,B&10+, #3 & $%&%"$%"'N9

)# ,&'* $&10+, $%&%"$%"' %*,., '#..,$0#/5$ & 0#0-+&%"#/ 0&.&1,%,.9 D,

-$, X 8 ER8 E 8 08 ,%'9 %# ,$%"1&%, μ8 σR8 σ8 ^ L#. πN8 ,%'9

Sample statistic Corresponding population parameter

X L$&10+, 1,&/N μ L0#0-+&%"#/ 1,&/N

ER L $&10+, ;&."&/',N σR L 0#0-+&%"#/ ;&."&/',N

E L$&10+, E%&/5&.5 5,;"&%"#/N σL0#0-+&%"#/ $%&/5&.5 5,;"&%"#/N

0 L $&10+, 0.#0#.%"#/N ^ #. π L^#0-+&%"#/ 0.#0#.%"#/N

D, *&;, &+.,&52 $,,/ %*&% %*, 1,&/ X #3 & $&10+, '&/ <, -$,5 %#

,$%"1&%, μ9)*"$ 5#,$ /#%8 #3 '#-.$,8 "/5"'&%, %*&% %*, 1,&/ #3 ,;,.2

$&10+, 7"++ ,>-&+ %*, 0#0-+&%"#/ 1,&/9

K,3"/"%"#/c H 0#"/% ,$%"1&%, #3 $#1, 0#0-+&%"#/ 0&.&1,%,. Q "$ & $"/=+,



!"#$%&%"$%"'$

OYU

;&+-, ‚ #3 & $&10+, $%&%"$%"'9

?=9 )*, 1,&/ $-.;";&+ %"1, #3 XO +&<#.&%#.2 .&%$ &3%,. .,1#;&+ #3 %*,

%*2.#"5 =+&/5 7&$ ]R 5&2$ 7"%* & $%&/5&.5 5,;"&%"#/ #3 O[ 5&2$ L&$$-1,

%*, .&%$ 7,., .&/5#1+2 $,+,'%,5N9

:/ %*, &<#;, ,B&10+,8 %*, 0#"/% ,$%"1&%,$ 3#. %*, 0#0-+&%"#/ 0&.&1,%,.$

μ &/5 σ L 7"%* .,=&.5 %# %*, $-.;";&+ %"1, #3 &++ +&<#.&%#.2 .&%$ &3%,.

.,1#;&+ #3 %*, %*2.#"5 =+&/5N &., ]R 5&2$ &/5 O[ 5&2$ .,$0,'%";,+29

7.4 Sampling Distribution of Means

)*, $&10+"/= 5"$%."<-%"#/ #3 1,&/$ "$ #/, #3 %*, 1#$% 3-/5&1,/%&+

'#/',0%$ #3 $%&%"$%"'&+ "/3,.,/',8 &/5 "% *&$ .,1&.C&<+, 0.#0,.%",$9

E"/', "% "$ & 3.,>-,/'2 5"$%."<-%"#/ "% *&$ "%$ #7/ 1,&/ &/5 $%&/5&.5

5,;"&%"#/ 9

One may generate the sampling distribution of means as follows:

ON Q<%&"/ & $&10+, #3 / #<$,.;&%"#/$ $,+,'%,5 '#10+,%,+2 &% .&/5#1

3.#1 & +&.=, 0#0-+&%"#/ 9 K,%,.1"/, %*,". 1,&/ &/5 %*,/ .,0+&', %*,

#<$,.;&%"#/$ "/ %*, 0#0-+&%"#/9

RN Q<%&"/ &/#%*,. .&/5#1 $&10+, #3 / #<$,.;&%"#/$ 3.#1 %*,0#0-+&%"#/8 5,%,.1"/, %*,". 1,&/ &/5 &=&"/ .,0+&', %*,

#<$,.;&%"#/$9

SN T,0,&% %*, $&10+"/= 0.#',5-., "/5,3"/"%,+28 '&+'-+&%"/= %*, 1,&/ #3

%*, .&/5#1 $&10+, #3 / ,&'* %"1, &/5 $-<$,>-,/%+2 .,0+&'"/= %*,



!"#$%&%"$%"'$

OYY

#<$,.;&%"#/$ "/ %*, 0#0-+&%"#/9

VN )*, .,$-+% "$ & $,.",$ #3 1,&/$ #3 $&10+,$ #3 $"F, /9 :3 ,&'* 1,&/ "/

%*, $,.",$ "$ /#7 %.,&%,5 &$ &/ "/5";"5-&+ #<$,.;&%"#/ &/5 &..&2,5

"/ & 3.,>-,/'2 5"$%."<-%"#/8 #/, 5,%,.1"/,$ %*, $&10+"/= 5"$%."<-%"#/

#3 1,&/$ #3 $&10+,$ #3 $"F, /9

!,'&-$, %*, $'#.,$ L X $N "/ %*, $&10+"/= 5"$%."<-%"#/ #3 1,&/$ &.,

%*,1$,+;,$ 1,&/$ L#3 "/5";"5-&+ $&10+,$N8 7, $*&++ -$, %*, /#%&%"#/

σ X 3#. %*, $%&/5&.5 5,;"&%"#/ #3 %*, 5"$%."<-%"#/9 )*, $%&/5&.5

5,;"&%"#/ #3 %*, $&10+"/= 5"$%."<-%"#/ #3 1,&/$ "$ '&++,5 %*, $%&/5&.5

,..#. #3 %*, 1,&/9

?=9 • Q<%&"/ .,0,&% $&10+,$ #3 RU 3.#1 & +&.=, 0#0-+&%"#/ #3 1&+,$9

• K,%,.1"/, %*, 1,&/ $,.-1 -."' &'"5 +,;,+ "/ ,&'* $&10+, <2

.,0+&'"/= %*, RU #<$,.;&%"#/$ ,&'* %"1,9

• H..&2 %*, 1,&/$ "/%# & 5"$%."<-%"#/9

• )*,/ 2#- 7"++ =,/,.&%, %*, $&10+"/= 5"$%."<-%"#/ #3 1,&/

$,.-1 -."' &'"5 +,;,+$ #3 $&10+,$ #3 $"F, RU9

Properties

O9 )*, 1,&/ #3 %*, $&10+"/= 5"$%."<-%"#/ #3 1,&/$ "$ %*, $&1, &$

%*, 0#0-+&%"#/ 1,&/8 μ 9



!"#$%&%"$%"'$

OYW

R9 )*, EK #3 %*, $&10+"/= 5"$%."<-%"#/ #3 1,&/$ "$ σ @ √/ 9

S9 )*, $*&0, #3 %*, $&10+"/= 5"$%."<-%"#/ #3 1,&/$ "$ &00.#B"1&%,+2 &

/#.1&+ '-.;,8 .,=&.5+,$$ #3 %*, $*&0, #3 %*, 0#0-+&%"#/ 5"$%."<-%"#/

&/5 0.#;"5,5 / "$ +&.=, ,/#-=* LG,/%.&+ +"1"% %*,#.,1N9

:/ 0.&'%"',8 %*, &00.#B"1&%"#/ "$ & 7#.C&<+, #/, "3 / "$ S[ #. 1#.,9

?= O9 E-00#$, 2#- *&;, & 0#0-+&%"#/ *&;"/= 3#-. 1,1<,.$ 7"%* ;&+-,$

O[8R[8S[ &/5 V[ 9 :3 2#- %&C, &++ '#/',";&<+, $&10+,$ #3 $"F, R

7"%* .,0+&',1,/%c

&N D*&% "$ %*, 3.,>-,/'2 5"$%."<-%"#/ #3 %*, $&10+, 1,&/$ a

<N \"/5 %*, 1,&/ &/5 $%&/5&.5 5,;"&%"#/ #3 %*, 5"$%."<-%"#/ L$%&/5&.5

,..#. #3 %*, 1,&/N9

^#$$"<+, $&10+,$ x " L $&10+, 1,&/ N

LO[8 R[N #. LR[8 O[N OU

LO[8 S[ N #. LS[8 O[N R[

LO[8 V[N #. LV[8 O[N RU

LR[8 S[N #. LS[8 R[N RU

LR[8 V[N #. LV[8 R[N S[

LS[8 V[N #. LV[8 S[N SU

LO[8 O[N O[

LR[8 R[N R[

LS[8 S[N S[

LV[8 V[N V[



!"#$%&%"$%"'$

OY]

&N 3.,>-,/'2 5"$%."<-%"#/ #3 $&10+, 1,&/$

sample mean L i x N frequency (fi)

O[ OOU R

R[ S

RU V

S[ S

SU R

V[ O

<N "N )*, 1,&/ #3 %*, $&10+"/= 5"$%."<-%"#/ q ∑ x "3" @ ∑3"

q V[[ @ OY q RU""N )*, $%&/5&.5 5,;"&%"#/ #3 %*, 1,&/ q σ x q ∑ L x " ` μNR @ ∑3"

q {∑ LO[ ` RUNR r LOU ` RUNR r u9 r L V[ RUNR } @ OY

q O[[[ @ OY q √YR9U q W9X

?=R9 \#. %*, 0#0-+&%"#/ =";,/ &<#;, LO[8R[8S[ &/5 V[N

3N \"/5 %*, 0#0-+&%"#/ 1,&/9 E*#7 %*&% %*, 0#0-+&%"#/ 1,&/ L μ N q %*,

1,&/ #3 %*, $&10+"/= 5"$%."<-%"#/

=N \"/5 %*, 0#0-+&%"#/ $%&/5&.5 5,;"&%"#/ &/5 $*#7 %*&% %*, $%&/5&.5

,..#. #3 %*, 1,&/ X σ U q σB @√/ N



!"#$%&%"$%"'$

OYX

L%*&% "$8 %*, $%&/5&.5 ,..#. #3 %*, 1,&/ "$ ,>-&+ %# %*, 0#0-+&%"#/

$%&/5&.5 5,;"&%"#/ 5";"5,5 <2 %*, $>-&., .##% #3 %*, $&10+, $"F,N

Answers to example 2

&N μ q ∑ B" @ _ q LO[ r R[ r S[ r V[N @ V q RU

<N σR q ∑LB" ` μNR @ _ q L RRUr RUr RU r RRUN @ V q ORU

4,/',8 σ q √ ORU q OO9O] &/5 σB L$%&/5&.5 ,..#.N qσB @ √/ q OO9O][ @

O9VOV q W9X

7.5 Interval Estimation (large samples)

H 0#"/% ,$%"1&%, 5#,$ /#% =";, &/2 "/5"'&%"#/ #/ *#7 3&. &7&2 %*,

0&.&1,%,. +",$9 H 1#., -$,3-+ 1,%*#5 #3 ,$%"1&%"#/ "$ %# '#10-%, &/

"/%,.;&+ 7*"'* *&$ & *"=* 0.#<&<"+"%2 #3 '#/%&"/"/= %*, 0&.&1,%,.9

Definition: H/ "/%,.;&+ ,$%"1&%, "$ & $%&%,1,/% %*&% & 0#0-+&%"#/

0&.&1,%,. *&$ & ;&+-, +2"/= <,%7,,/ %7# $0,'"3",5 +"1"%$9

7.5.1 Confidence interval for a single mean

G#/$"5,. %*, $%&/5&.5 /#.1&+ 5"$%."<-%"#/ &/5 %*, $%&%,1,/% ^. L`O9XY≤

d ≤O9XYN q 9 XU



!"#$%&%"$%"'$

OW[

)*"$ "$ 1,.,+2 & $*#.%*&/5 &+=,<.&"' $%&%,1,/% %*&% XUi #3 %*, $%&/5&.5

/#.1&+ '-.;, +",$ <,%7,,/ r O9XY &/5 fO9XY9 :3 #/, '*##$,$ %*,

$&10+"/= 5"$%."<-%"#/ #3 1,&/$ L& /#.1&+ '-.;, 7"%* 1,&/ μ &/5

$%&/5&.5 5,;"&%"#/ σ @√ /N8 %*,/ 8

Pr(-1.96 ≤ ( X - μ)/(σ /√n) ≤ 1.96) = .95

H +"%%+, 1&/"0-+&%"#/ 7"%*#-% &+%,."/= %*, 0.#<&<"+"%2 ;&+-, #3 XU 0,.',/%

=";,$

Pr( X - 1.96(σ /√n) ≤ μ ≤ X + 1.96(σ /√n) ) = .95

)*, .&/=, X -1.96(σ

/√

nN %# X + 1.96(σ

/√

n) ) "$ '&++,5 %*, XUi

'#/3"5,/', "/%,.;&+j

X -1.96(σ

/√

nN "$ %*, +#7,. '#/3"5,/', +"1"% 7*"+, X + 1.96(σ

/√

n) "$

%*, -00,. '#/3"5,/', +"1"%9

)*, '#/3"5,/',8 ,B0.,$$,5 &$ & 0.#0#.%"#/8 %*&% %*, "/%,.;&+ X -1.96(σ

/√

n) to X + 1.96(σ

/√

n) '#/%&"/$ %*, -/C/#7/ 0#0-+&%"#/ 1,&/ "$

'&++,5 %*, '#/3"5,/', '#,33"'",/%9 D*,/ %*"$ '#,33"'",/% "$ 9XU &$ =";,/

&<#;,8 %*, 3#++#7"/= 3#.1&+ 5,3"/"%"#/ #3 '#/3"5,/', "/%,.;&+ "$ =";,/9 :3

1&/2 5"33,.,/% .&/5#1 $&10+,$ &., %&C,/8 &/5 "3 %*, '#/3"5,/', "/%,.;&+



!"#$%&%"$%"'$

OWO

3#. ,&'* "$ 5,%,.1"/,58 %*,/ "% "$ ,B0,'%,5 %*&% XUi #3 %*,$, '#10-%,5

"/%,.;&+$ 7"++ '#/%&"/ %*, 0#0-+&%"#/ 1,&/ L μ N 9

G+,&.+28 %*,., &00,&.$ %# <, /# .&%"#/&+, L+#="'&+ <&$"$ N 3#. %&C"/=

.,0,&%,5 $&10+,$ #3 $"F, / &/5 5,%,.1"/, %*, '#..,$0#/5"/=

'#/3"5,/', "/%,.;&+$9 4#7,;,.8 %*, C/#7+,5=, #3 %*, 0.#0,.%",$ #3 %*,$,

$&10+"/= 5"$%."<-%"#/$ #3 1,&/$ L"3 #/, *20#%*,%"'&++2 #<%&"/,5 %*,$,

.,0,&%,5 $&10+,$N 0,.1"%$ #/, %# 5.&7 & '#/'+-$"#/ <&$,5 -0#/ #/,

$&10+, &/5 %*"$ 7&$ $*#7/ .,0,&%,5+2 "/ %*, 0.,;"#-$ $,'%"#/$9

\.#1 %*, &<#;, 5,3"/"%"#/ #3 G#/3"5,/', "/%,.;&+ LG9:9N8 %*, 7"5,+2 -$,5

5,3"/"%"#/ "$ 5,.";,59 )*&% "$8 7*,/ #/, '+&"1$ X ± 1.96 (σ

/√

n) &$ %*,

+"1"%$ #/ μ8 %*,., "$ & XUi '*&/', %*&% %*, $%&%,1,/% "$ '#..,'% L %*&% μ "$ '#/%&"/,5 7"%*"/ %*, "/%,.;&+N9

:3 1#., %*&/ XUi ',.%&"/%2 .,=&.5"/= %*, 0#0-+&%"#/ 1,&/ ` $&28 & XXi

G9:9 7,., 5,$".,58 %*, #/+2 '*&/=, /,,5,5 "$ %# -$, ±R9U] L%*, 0#"/%

,/'+#$"/= XXi #3 %*, $%&/5&.5 /#.1&+ '-.;,N8 7*"'* =";,$ X ± 2.58

(σ /√n) .

?= O9 )*, 1,&/ .,&5"/= $0,,5 #3 & .&/5#1 $&10+, #3 ]O &5-+%$ "$ SRU7#.5$ 0,. 1"/-%,9 \"/5 & X[i G9:9 \#. %*, 1,&/ .,&5"/= $0,,5 #3 &++

&5-+%$ LμN "3 "% "$ C/#7/ %*&% %*, $%&/5&.5 5,;"&%"#/ 3#. &++ &5-+%$ "$ VU

7#.5$ 0,. 1"/-%,9



!"#$%&%"$%"'$

OWR

Given

/ q ]O

σ q VU

B q SRU

d q ± O9YV L %*, 0#"/% ,/'+#$"/= X[i #3 %*, $%&/5&.5 /#.1&+ '-.;,N

H X[i G9:9 3#. μ "$ x ± 1.64 (σ /√n) = 325 ± (1.64 x 5 ) = 325 ± 8.2

= (316.8, 333.2)

Therefore, A 90% CI. For μ is 316.8 to 333.2words per minute.

?= R9 H .&/5#1 $&10+, #3 O[[ 5.-=`%.,&%,5 0&%",/%$ *&$ & 1,&/

$-.;";&+ %"1, #3 VY9X 1#/%*$9 :3 %*, EK #3 %*, 0#0-+&%"#/ "$ VS9S 1#/%*$83"/5 & XUi '#/3"5,/', "/%,.;&+ 3#. %*, 0#0-+&%"#/ 1,&/9

L)*, 0#0-+&%"#/ '#/$"$%$ #3 $-.;";&+ %"1,$ #3 '&/',. 0&%",/%$ 7*# *&;,

<,,/ %.,&%,5 7"%* & /,7 5.-=N

46.9±

(1.96) (43.3 /√

100) = 46.9±

8.5 = (38.4 to 55.4 months)

4,/',8 %*,., "$ XUi ',.%&"/%2 %*&% %*, +"1"%$ L S]9V 8 UU9VN ,1<.&', %*,1,&/ $-.;";&+ %"1,$ "/ %*, 0#0-+&%"#/ 3.#1 7*"'* %*, $&10+, &.#$,9



!"#$%&%"$%"'$

OWS

7.5.2 Confidence interval for the difference of means

G#/$"5,. %7# 5"33,.,/% 0#0-+&%"#/$9 )*, 3".$% 0#0-+&%"#/ L k N *&$ 1,&/

μB &/5 $%&/5&.5 5,;"&%"#/ σB8 %*, $,'#/5 L o N *&$ 1,&/ μ2 &/5

$%&/5&.5 5,;"&%"#/ σ29 \.#1 %*, 3".$% 0#0-+&%"#/ %&C, & $&10+, #3 $"F, /B

&/5 '#10-%, "%$ 1,&/ x j 3.#1 %*, $,'#/5 0#0-+&%"#/ %&C,

"/5,0,/5,/%+2 & $&10+, #3 $"F, /2 &/5 '#10-%, y j %*,/ 5,%,.1"/, x `

y 9 K# %*"$ 3#. &++ 0&".$ #3 $&10+,$ %*&% '&/ <, '*#$,/ "/5,0,/5,/%+2

3.#1 %*, %7# 0#0-+&%"#/$9 )*, 5"33,.,/',$8 x ` y 8 &., & /,7 $,% #3

$'#.,$ 7*"'* 3#.1 %*, $&10+"/= 5"$%."<-%"#/ #3 5"33,.,/',$ #3 1,&/$9

The characteristics of the sampling distribution of differences ofmeans are:

ON )*, 1,&/ #3 %*, $&10+"/= 5"$%."<-%"#/ #3 5"33,.,/',$ #3 1,&/$

,>-&+$ %*, 5"33,.,/', #3 %*, 0#0-+&%"#/ 1,&/$ L 6,&/ q μB ` μ2 N9

RN )*, $%&/5&.5 5,;"&%"#/ #3 %*, $&10+"/= 5"$%."<-%"#/ #3 5"33,.,/',$ #3

1,&/$8 &+$# '&++,5 %*, $%&/5&.5 ,..#. #3 5"33,.,/',$ #3 1,&/$ "$

5,/#%,5 <2 σ L x ` y N 9

σ L x ` y N q √ L σR x r σR y N 7*,., σ x "$ %*, $%&/5&.5 ,..#. #3 %*,

1,&/ #3 %*, 3".$% 0#0-+&%"#/ &/5 σ y "$ %*, $%&/5&.5 ,..#. #3 %*, 1,&/

#3 %*, $,'#/5 0#0-+&%"#/9 LσR x q σR

B @/B j σR y q j σR

2 @ /2 N



!"#$%&%"$%"'$

OWV

SN )*, $&10+"/= 5"$%."<-%"#/ "$ /#.1&+ "3 <#%* 0#0-+&%"#/$ &., /#.1&+8

&/5 "$ &00.#B"1&%,+2 /#.1&+ "3 %*, $&10+,$ &., +&.=, ,/#-=*

L,;,/ "3 %*, 0#0-+&%"#/$ &.,/b% /#.1&+N9 :/ 0.&'%"',8 "% "$ &$$-1,5

%*&% %*, $&10+"/= 5"$%."<-%"#/ #3 5"33,.,/',$ #3 1,&/$ "$ /#.1&+ "3

<#%* /B &/5 /2 &., ≥S[9

H 3#.1-+& 3#. G9: "$ 3#-/5 <2 $#+;"/= d q {L x ` y N ` L μB ` μ2 N} @ σ L x `

y N 3#. μB ` μ2 j *,/', G9:9 3#. %*, 5"33,.,/', #3 1,&/$ "$ L x ` y N ±

d9σ L x ` y N

?=O9 :3 & .&/5#1 $&10+, #3 U[ /#/`$1#C,.$ *&;, & 1,&/ +"3, #3 WY

2,&.$ 7"%* & $%&/5&.5 5,;"&%"#/ #3 ] 2,&.$8 &/5 & .&/5#1 $&10+, #3 YU

$1#C,.$ +";, Y] 2,&.$ 7"%* & $%&/5&.5 5,;"&%"#/ #3 X 2,&.$8

HN D*&% "$ %*, 0#"/% ,$%"1&%, 3#. %*, 5"33,.,/', #3 %*, 0#0-+&%"#/

1,&/$a

!N \"/5 & XUi G9:9 3#. %*, 5"33,.,/', #3 1,&/ +"3,%"1, #3 /#/`$1#C,.$

&/5 $1#C,.$9

Given

^#0-+&%"#/ BL/#/`$1#C,.$N /BqU[ 8 x q WY8 EB q ]8 σR x q ERB @ /B8 q

]R@U[ qO9R] 2,&.$



!"#$%&%"$%"'$

OWU

^#0-+&%"#/ 2 L$1#C,.$N /2qYU 8 y q Y]8 E2 q X8 σR y q ER

2 @ /28 q

XR@YU qO9RU 2,&.$

HN H 0#"/% ,$%"1&%, 3#. %*, 5"33,.,/', #3 0#0-+&%"#/ 1,&/$ (μx- μy) = x -

y =76-68 = 8 years

!N H% & XUi '#/3"5,/', +,;,+8 d q ± O9XY8 σL x - y N q $GS"$:S" +

q G%S$ q O9UX 2,&.$

4,/',8 XUi G9:9 3#. μB` μ2 q L x ` y N ± O9XY σL x ` y N q ] ± O9XY LO9UXN

q ] ± S9OR q LV9]] %# OO9OR 2,&.$N

Exercise

H/ &/%*.#0#+#="$% 7*# 7&/%,5 %# $%-52 %*, *,"=*%$ #3 &5-+% 1,/ &/57#1,/ %##C & .&/5#1 $&10+, #3 OR] &5-+% 1,/ &/5 O[[ &5-+% 7#1,/

&/5 3#-/5 %*, 3#++#7"/= $-11&.2 .,$-+%$9

Mean height Standard deviation

H5-+% 1,/ OW[ '1$ ] '1&

H5-+% 7#1,/ OYV '1$ Y '1$

\"/5 & XUi G9: 3#. %*, 5"33,.,/', #3 1,&/ *,"=*% #3 &5-+% 1,/ &/5

7#1,/9



!"#$%&%"$%"'$

OWY

7.5.3 Confidence interval for a single proportion

_#%&%"#/c ^ L#. πN q 0.#0#.%"#/ #3 l$-'',$$,$m "/ & 0#0-+&%"#/

L0&.&1,%,.N

p q O`^ q 0.#0#.%"#/ #3 l3&"+-.,$m "/ & 0#0-+&%"#/

0 q 0.#0#.%"#/ #3 $-'',$$,$ "/ & $&10+,

> q O`0 0.#0#.%"#/ #3 l3&"+-.,$m "/ & $&10+,

σ0q E%&/5&.5 5,;"&%"#/ #3 %*, $&10+"/= 5"$%."<-%"#/ #3

0.#0#.%"#/$

q E%&/5&.5 ,..#. #3 0.#0#.%"#/$

/ q $"F, #3 %*, $&10+,

)*, 0#0-+&%"#/ .,0.,$,/%$ '&%,=#."'&+ 5&%& 7*"+, %*, $'#.,$ "/ %*,

$&10+"/= 5"$%."<-%"#/ &., 0.#0#.%"#/$ <,%7,,/ [ &/5 O9)*"$ $,% #3

0.#0#.%"#/$ *&$ & 1,&/ &/5 $%&/5&.5 5,;"&%"#/9 )*, $&10+"/=

5"$%."<-%"#/ #3 0.#0#.%"#/$ *&$ %*, 3#++#7"/= '*&.&'%,."$%"'$c

O9 :%$ 1,&/ q ^8 %*, 0.#0#.%"#/ "/ %*, 0#0-+&%"#/9

R9 σ0 q n PQ &

S )*, $*&0, "$ &00.#B"1&%,+2 /#.1&+ 0.#;"5,5 / "$ $-33"'",/%+2 +&.=, `

"/ %*"$ '&$,8 /^ ≥U &/5 /p ≥ U &., %*, .,>-".,1,/%$ 3#. $-33"'",/%+2

+&.=, / L ',/%.&+ +"1"% %*,#.,1 3#. 0.#0#.%"#/$N 9



!"#$%&%"$%"'$

OWW

)*, '#/3"5,/', "/%,.;&+ 3#. %*, 0#0-+&%"#/ 0.#0#.%"#/ L^N "$ =";,/ <2 %*,

3#.1-+&c

0 ± d σ0 L %*&% "$8 0 ` d σ0 &/5 0 r d σ0 N

0 q $&10+, 0.#0#.%"#/8 σ0 q $%&/5&.5 ,..#. #3 %*, 0.#0#.%"#/ L q

n PQ & N9

Examplec H/ ,0"5,1"#+#="$% "$ 7#..",5 &<#-% %*, ,;,. "/'.,&$"/=

%.,/5 #3 1&+&."& "/ & ',.%&"/ +#'&+"%2 &/5 7&/%$ %# ,$%"1&%, %*,

0.#0#.%"#/ #3 0,.$#/$ "/3,'%,5 "/ %*, 0,&C 1&+&."& %.&/$1"$$"#/ 0,."#59

:3 *, %&C,$ & .&/5#1 $&10+, #3 OU[ 0,.$#/$ "/ %*&% +#'&+"%2 5-."/= %*,

0,&C %.&/$1"$$"#/ 0,."#5 &/5 3"/5$ %*&% Y[ #3 %*,1 &., 0#$"%";, 3#.

1&+&."&8 3"/5

&N XUi <NX[i 'NXXi '#/3"5,/', "/%,.;&+$ 3#. %*, 0.#0#.%"#/ #3

%*, 7*#+, "/3,'%,5 0,#0+, "/ %*&% +#'&+"%2 5-."/= %*, 0,&C 1&+&."&

%.&/$1"$$"#/ 0,."#5 9

E&10+, 0.#0#.%"#/ q Y[ @ OU[ q 9V

)*, $%&/5&.5 ,..#. #3 0.#0#.%"#/ 5,0,/5$ #/ %*, 0#0-+&%"#/ ^9

4#7,;,.8 %*, 0#0-+&%"#/ 0.#0#.%"#/ L^N "$ -/C/#7/9 :/ $-'*

$"%-&%"#/$8 n pq & '&/ <, -$,5 &$ &/ &00.#B"1&%"#/ %# σ0 q

"G7&#9S8US x q 9[V

&N H XUi G9: 3#. %*, 0#0-+&%"#/ 0.#0#.%"#/ L %*, 0.#0#.%"#/ #3 %*, 7*#+,



!"#$%&%"$%"'$

OW]

"/3,'%,5 0,#0+, "/ %*&% +#'&+"%2N q 9V ± O9XY L9[VN q L9V ± 9[W]N q L9SRR8

9VW]N9

<N H X[i G9: 3#. %*, 0#0-+&%"#/ 0.#0#.%"#/ L %*, 0.#0#.%"#/ #3 %*, 7*#+,

"/3,'%,5 0,#0+, "/ %*&% +#'&+"%2N q 9V ± O9YV L9[VN q L9V ± 9[YYN q

L9SSV8 9VYYN9

H XXi G9: 3#. %*, 0#0-+&%"#/ 0.#0#.%"#/ L %*, 0.#0#.%"#/ #3 %*, 7*#+,

"/3,'%,5 0,#0+, "/ %*&% +#'&+"%2N q 9V ± R9U] L9[VN q L9V ± 9O[SN q

L9RXW8 9U[SN9

7.5.4 Confidence interval for the difference of two

proportions

!2 %*, $&1, &/&+#=28 %*, G9:9 3#. %*, 5"33,.,/', #3 0.#0#.%"#/$ L^B ` ^2N

"$ =";,/ <2 %*, 3#++#7"/= 3#.1-+&9

G9:9 3#. Px - Py = (px - py) ± Zσ

(Px - Py) 9 D*,., d "$ 5,%,.1"/,5 <2 %*,

'#/3"5,/', '#,33"'",/% &/5 σ L^B 2N =√

{ (px qx)/nx + (px qx)/nx }

Example: ?&'* #3 %7# =.#-0$ '#/$"$%$ #3 O[[ 0&%",/%$ 7*# *&;,+,-C&,1"&9 H /,7 5.-= "$ =";,/ %# %*, 3".$% =.#-0 <-% /#% %# %*, $,'#/5

L%*, '#/%.#+ =.#-0N9 :% "$ 3#-/5 %*&% "/ %*, 3".$% =.#-0 WU 0,#0+, *&;,

.,1"$$"#/ 3#. R 2,&.$j <-% #/+2 Y[ "/ %*, $,'#/5 =.#-09 \"/5 XUi



!"#$%&%"$%"'$

OWX

'#/3"5,/', +"1"%$ 3#. %*, 5"33,.,/', "/ %*, 0.#0#.%"#/ #3 &++ 0&%",/%$ 7"%*

+,-C&,1"& 7*# *&;, .,1"$$"#/ 3#. R 2,&.$9

_#%, %*&% /B0B q O[[ B 9WU q WU •U

/B>2 q O[[ B 9RU q RU •U

/202 q O[[ B 9Y[ q Y[ •U

/2>2 q O[[ B 9V[ q V[ •U

0B q 9WU8 >B q 9RU8 /B q O[[8 σR^B q 0B>B @ /B q 9WU B 9RU @ O[[ q

9[[O]WU

02 q 9Y[8 >B q 9V[8 /2 q O[[8 σR^2 q 02>2 @ /2 q 9Y[ B 9V[ @ O[[ q

9[[RV

4,/',8 σRL^B`^2N q √ LσR

^B r σR^BN q √ L 0B>B @ /BN r L02>2 @ /2N q √

9[[O]WUr9[[RV q 9[YU

H% & XUi G#/3"5,/', +,;,+8 d q ± O9XY &/5 %*, 5"33,.,/', #3 %*, %7#

"/5,0,/5,/% .&/5#1 $&10+,$ "$ L9WU ` 9Y[N q 9OU 9 )*,.,3#.,8 & XU i G9

:9 3#. %*, 5"33,.,/', "/ %*, 0.#0#.%"#/ 7"%* R`2,&. .,1"$$"#/ "$ L9OU ± O9XY

L9[YUN N q L9OU ± 9OSN q L 9[R %# 9R]N9

7.6 Sample Size Estimation in cross – sectional studies

:/ 0+&//"/= &/2 "/;,$%"=&%"#/ 7, 1-$% 5,'"5, *#7 1&/2 0,#0+, /,,5

%# <, $%-5",5 "/ #.5,. %# &/$7,. %*, $%-52 #<J,'%";,$9 :3 %*, $%-52 "$



!"#$%&%"$%"'$

O][

%## $1&++ 7, 1&2 3&"+ %# 5,%,'% "10#.%&/% ,33,'%$8 #. 1&2 ,$%"1&%,

,33,'%$ %## "10.,'"$,+29 :3 %*, $%-52 "$ %## +&.=, %*,/ 7, 7"++ 7&$%,

.,$#-.',$9

:/ =,/,.&+8 "% "$ 1-'* <,%%,. %# "/'.,&$, %*, &''-.&'2 #3 5&%&'#++,'%"#/ L<2 "10.#;"/= %*, %.&"/"/= #3 5&%& '#++,'%#.$ &/5 5&%&

'#++,'%"#/ %##+$N %*&/ %# "/'.,&$, %*, $&10+, $"F, after a certain

point.

)*, ,;,/%-&+ $&10+, $"F, "$ -$-&++2 & '#10.#1"$, <,%7,,/ 7*&% "$

5,$".&<+, &/5 7*&% "$ 3,&$"<+,9 )*, 3,&$"<+, $&10+, $"F, "$

5,%,.1"/,5 <2 %*, &;&"+&<"+"%2 #3 .,$#-.',$9 :% "$ &+$# "10#.%&/% %#.,1,1<,. %*&% .,$#-.',$ &., /#% #/+2 /,,5,5 %# '#++,'% %*,

"/3#.1&%"#/8 <-% &+$# %# &/&+2$, "%9

7.6.1 Estimating a proportion

• ,$%"1&%, *#7 <"= %*, 0.#0#.%"#/ 1"=*% <, L^N

• '*##$, %*, 1&.="/ #3 ,..#. 2#- 7"++ &++#7 "/ %*, ,$%"1&%, #3 %*,

0.#0#.%"#/ L$&2 ± 7N

• '*##$, %*, +,;,+ #3 '#/3"5,/', %*&% %*, 0.#0#.%"#/ "/ %*, 7*#+,

0#0-+&%"#/ "$ "/5,,5 <,%7,,/ L0`7N &/5 L0r7N9 D, '&/ /,;,. <,

O[[i $-.,9 K# 2#- 7&/% %# <, XUi $-.,a



!"#$%&%"$%"'$

O]O

• %*, 1"/"1-1 $&10+, $"F, .,>-".,58 3#. & ;,.2 +&.=, 0#0-+&%"#/

L_≥O[8[[[N

"$c

n = Z2 p(1-p) / w2

E*#7 *#7 %*, &<#;, 3#.1-+& "$ #<%&"/,59

H XUi G9:9 3#. ^ q 0 ± O9XY $, 8 "3 7, 7&/% #-. '#/3"5,/', "/%,.;&+ %#

*&;, & 1&B"1-1 7"5%* #3 ± 78

O9XY $, q 7

O9XY √0LO`0N@/ q 7O9XYR 0LO`0N@/ q 7R 8 4,/',8 / q O9XYR 0LO`0N@7R

Example 1

&N 0 q [9RY 8 7 q [9[S 8 d q O9XY L "9,98 3#. & XUi G9:9N

/ q LO9XYNR L9RY × 9WVN @ L9[SNR q ]RO9RU ≈ ]RR

)*-$ 8 %*, $%-52 $*#-+5 "/'+-5, &% +,&$% ]RR $-<J,'%$9

<N :3 %*, &<#;, $&10+, "$ %# <, %&C,/ 3.#1 & .,+&%";,+2 $1&++ 0#0-+&%"#/

L$&2 _ q S[[[N 8 %*, .,>-".,5 1"/"1-1 $&10+, 7"++ <, #<%&"/,5 3.#1 %*,



!"#$%&%"$%"'$

O]R

&<#;, ,$%"1&%, <2 1&C"/= $#1, &5J-$%1,/% 9

]RO9RU @ LOr L]RO9RU@S[[[NN q YVV9W ≈ YVU $-<J,'%$

7.6.2 Estimating a mean

)*, $&1, &00.#&'* "$ -$,5 <-% 7"%* E? q σ @ √/

)*, .,>-".,5 L1"/"1-1N $&10+, $"F, 3#. & ;,.2 +&.=, 0#0-+&%"#/ "$ =";,/

<2c

n = Z2 σ

2 / w2

?=9 H *,&+%* #33"',. 7"$*,$ %# ,$%"1&%, 1,&/ *&,1#=+#<"/ +,;,+ "/ &

5,3"/,5 '#11-/"%29 \.#1 0.,+"1"/&.2 '#/%&'% *, %*"/C$ %*"$ 1,&/ "$

&<#-% OU[ 1=@+ 7"%* & $%&/5&.5 5,;"&%"#/ #3 SR 1@+9 :3 *, "$ 7"++"/= %#

%#+,.&%, & $&10+"/= ,..#. #3 -0 %# U 1=@+ "/ *"$ ,$%"1&%,8 *#7 1&/2

$-<J,'%$ $*#-+5 <, "/'+-5,5 "/ *"$ $%-52a Lα qUi8 %7# $"5,5N

` :3 %*, 0#0-+&%"#/ $"F, "$ &$$-1,5 %# <, ;,.2 +&.=,8 %*, .,>-".,5

$&10+, $"F, 7#-+5 <,c

/ q LO9XYNR LSRNR @ LUNR q OUW9V ≈ OU] 0,.$#/$

` :3 %*, 0#0-+&%"#/ $"F, "$ 8 $&28 R[[[ 8

)*, .,>-".,5 $&10+, $"F, 7#-+5 <, OVY 0,.$#/$9



!"#$%&%"$%"'$

O]S

NB: σR '&/ <, ,$%"1&%,5 3.#1 0.,;"#-$ $"1"+&. $%-5",$ #.

'#-+5 <, #<%&"/,5 <2 '#/5-'%"/= & $1&++ 0"+#% $%-529

7.6.3 Comparison of two Proportions (sample size in each

region)

/ q L0O>O r 0R>RN L3Lα8βNN @ LL0O ` 0RN

α q %20, : ,..#. L+,;,+ #3 $"=/"3"'&/',N

β q %20, :: ,..#. L O`β q 0#7,. #3 %*, $%-52N

0#7,. q %*, 0.#<&<"+"%2 #3 =,%%"/= & $"=/"3"'&/% .,$-+%

3 Lα8βN qO[9U8 7*,/ %*, 0#7,. q X[i &/5 %*, +,;,+ #3

$"=/"3"'&/', q Ui

?=9 )*, 0.#0#.%"#/ #3 /-.$,$ +,&;"/= %*, *,&+%* $,.;"', "$ '#10&.,5

<,%7,,/ %7# .,="#/$9 :/ #/, .,="#/ S[i #3 /-.$,$ "$ ,$%"1&%,5 %# +,&;,

%*, $,.;"', 7"%*"/ S 2,&.$ #3 =.&5-&%"#/9 :/ #%*,. .,="#/ "% "$ 0.#<&<+2

OUi9

Solution

)*, .,>-".,5 $&10+, %# $*#78 7"%* & X[i +"C,+"*##5 L0#7,.N 8 %*&% %*,

0,.',/%&=, #3 /-.$,$ "$ 5"33,.,/% "/ %*,$, %7# .,="#/$ 7#-+5 <,c

L&$$-1, & '#/3"5,/', +,;,+ #3 XUiN



!"#$%&%"$%"'$

O]V

/ q LO9R]rO9XYNR LL9S×9WN rL9OU ×9]UNN @ L9S[ ` 9OUNR q OU]

158 /-.$,$ &., .,>-".,5 "/ ,&'* .,="#/

7.6.4 Comparison of two means (sample size in each group)

/ q L$OR r $R

RN 3Lα8βN @ L1O ` 1RNR

1O &/5 $OR &., 1,&/ &/5 ;&."&/', #3 =.#-0 O .,$0,'%";,+29

1R &/5 $RR &., 1,&/ &/5 ;&."&/', #3 =.#-0 R .,$0,'%";,+29

?=9 )*, <".%* 7,"=*%$ "/ 5"$%."'%$ H &/5 ! 7"++ <, '#10&.,59 :/ 5"$%."'% H

%*, 1,&/ <".%* 7,"=*% "$ ,B0,'%,5 %# <, S[[[ =.&1$ 7"%* & $%&/5&.5

5,;"&%"#/ #3 U[[ =.&1$9 :/ 5"$%."'% ! %*, 1,&/ "$ ,B0,'%,5 %# <, SR[[

=.&1$ 7"%* & $%&/5&.5 5,;"&%"#/ #3 U[[ =.&1$9 )*, .,>-".,5 $&10+,

$"F, %# 5,1#/$%.&%, L7"%* & +"C,+"*##5 #3 X[i 8 %*&% "$ 7"%* & 0#7,. #3

X[iN & $"=/"3"'&/% 5"33,.,/', <,%7,,/ %*, 1,&/ <".%* 7,"=*%$ "/ 5"$%."'%$

H &/5 ! 7#-+5 <,c

N = (1.96 + 1.28 )! (500 + 500)! / (3200 – 3000)!

= 131 newborn babies in each district

Note that f(α

,β

) = 10.5

That is , α = . 05 (two sided ) ⇒ Z = 1.96

β = ( 1- .9 ) = .1 (one sided ) ⇒ Z = 1.28



!"#$%&%"$%"'$

O]U

7.7 Exercises

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!"#$%&%"$%"'$

O]Y

CHAPTER EIGHT

HYPOTHESIS TESTING


H% %*, ,/5 #3 %*"$ '*&0%,. %*, $%-5,/% 7"++ <, &<+, %#c

O9 A/5,.$%&/5 %*, '#/',0%$ #3 /-++ &/5 &+%,./&%";, *20#%*,$"$

R9 ?B0+&"/ %*, 1,&/"/= &/5 &00+"'&%"#/ #3 $%&%"$%"'&+ $"=/"3"'&/',

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$&10+,$ &., +&.=, &/5 $1&++

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Y9 A/5,.$%&/5 %*, '#/',0%$ #3 5,=.,,$ #3 3.,,5#1

8.2 Introduction

:/ '*&0%,. W 7, 5,&+% 7"%* ,$%"1&%"#/ 7*"'* "$ #/, 3#.1 #3 $%&%"$%"'&+

"/3,.,/',9 :/ %*"$ '*&0%,. 7, $*&++ "/%.#5-', & 5"33,.,/% 3#.1 #3

"/3,.,/',8 %*, $"=/"3"'&/', %,$% #. *20#%*,$"$ %,$%9 H $"=/"3"'&/', %,$%

,/&<+,$ -$ %# 1,&$-., %*, $%.,/=%* #3 %*, ,;"5,/', 7*"'* %*, 5&%&

$-00+2 '#/',./"/= $#1, 0.#0#$"%"#/ #3 "/%,.,$%9



!"#$%&%"$%"'$

O]W

Definition cH $%&%"$%"'&+ *20#%*,$"$ "$ &/ &$$-10%"#/ #. & $%&%,1,/%

7*"'* 1&2 #. 1&2 /#% <, %.-, '#/',./"/= #/, #. 1#., 0#0-+&%"#/$9

?=9 ON )*, 1,&/ *,"=*% #3 %*, I#/5&. G#++,=, #3 6,5"'&+ E'",/',$

LIG6EN $%-5,/%$ "$ O9YS19

RN )*,., "$ /# 5"33,.,/', <,%7,,/ %*, 5"$%."<-%"#/ #3 ^3 &/5 ^;

1&+&."& "/ ?%*"#0"& L&., 5"$%."<-%,5 "/ ,>-&+ 0.#0#.%"#/$9N

In general, hypothesis testing in statistics involves the following

steps:

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*20#%*,$"$ "$ .,J,'%,59

S9 G*##$, & .-+, 3#. 1&C"/= & 5,'"$"#/ &<#-% 7*,/ %# .,J,'% %*,

#."="/&+ *20#%*,$"$ &/5 7*,/ %# 3&"+ %# .,J,'% "%9

V9 G*##$, & .&/5#1 $&10+, 3.#1 %*, &00.#0."&%, 0#0-+&%"#/ &/5

'#10-%, &00.#0."&%, $%&%"$%"'$c %*&% "$8 1,&/8 ;&."&/', &/5 $# #/9

U9 6&C, %*, 5,'"$"#/9



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O]]

8.3 The null and alternative hypotheses

)*, 1&"/ *20#%*,$"$ 7*"'* 7, 7"$* %# %,$% "$ '&++,5 %*, /-++ *20#%*,$"$8

$"/', &'',0%&/', #3 "% '#11#/+2 "10+",$ l/# ,33,'%m #. l /# 5"33,.,/',9m :%

"$ 5,/#%,5 <2 %*, $21<#+ 4Q 9

4Q "$ &+7&2$ & $%&%,1,/% &<#-% & 0&.&1,%,. L 1,&/8 0.#0#.%"#/8 ,%'9 #3

& 0#0-+&%"#/N9 :% "$ /#% &<#-% & $&10+,8 /#. &., $&10+, $%&%"$%"'$ -$,5 "/

3#.1-+&%"/= %*, /-++ *20#%*,$"$9 4Q "$ &/ ,>-&+"%2 L μ q OVN .&%*,. %*&/ &/

"/,>-&+"%2 L μ ≥ OV #. μ t OVN9

Examples ON 4Q c μ q O9YS 1 L3.#1 %*, 0.,;"#-$ ,B&10+,N9

RN H% 0.,$,/% #/+2 Y[i #3 0&%",/%$ 7"%* +,-C&,1"& $-.;";,

1#., %*&/ Y 2,&.$9 H 5#'%#. 5,;,+#0$ & /,7 5.-=9 Q3

V[ 0&%",/%$8 '*#$,/ &% .&/5#18 #/ 7*#1 %*, /,7

5.-= "$ %,$%,58 RY &., &+";, &3%,. Y 2,&.$9 :$ %*, /,7

5.-= <,%%,. %*&/ %*, 3#.1,. %.,&%1,/%a

4,.,8 7, &., >-,$%"#/"/= 7*,%*,. %*, 0.#0#.%"#/ #3 0&%",/%$ 7*#

.,'#;,. -/5,. %*, /,7 %.,&%1,/% "$ $%"++ 9Y[ L &/5 *#0, %*&% "% 7"++ <,

"10.#;,5j %*"$ 7"++ <, $*#7/ "/ #-. '*#"', #3 4 H "/ %*, /,B% $,'%"#/N9

)*, /-++ *20#%*,$"$ #3 %*, &<#;, $%&%,1,/% "$ 7."%%,/ &$ c HO : P = .60



!"#$%&%"$%"'$

O]X

Choosing the Alternative Hypothesis (HA)

)*, /#%&%"#/ 4 H L#. 4O N "$ -$,5 3#. %*, *20#%*,$"$ %*&% 7"++ <, &'',0%,5

"3 4Q "$ .,J,'%,59 4 H 1-$% &+$# <, 3#.1-+&%,5 <,3#., & $&10+, "$ %,$%,58

$# "%8 +"C, %*, /-++ *20#%*,$"$ L4QN8 5#,$ /#% 5,0,/5 #/ $&10+, ;&+-,$9

:3 %*, 1,&/ *,"=*% #3 %*, IG6E $%-5,/%$ L 4 Q c μ q O9YS 1 N "$

>-,$%"#/,58 %*,/ %*, &+%,./&%";, *20#%*,$"$ L4 HN "$ $,% μ ≠ OYS 19 Q%*,.

&+%,./&%";,$ &., &+$#c

4 H c μ • O9YS 19

4 H c μ t O9YS 19

Possible choices of HA

If HO is then HA is

μ q H L$"/=+, 1,&/N μ ≠ H #. μ t H #. μ • H

^ q ! L$"/=+, 0.#0#.%"#/N ^ ≠ ! #. ^ t ! #. ^ • !

μB ` μ2 q G L5"33,.,/', #3 1,&/$N μB ` μ2 ≠ G #. μB ` μ2 t G #. μB ` μ2

• G

^B ` 2 q KL5"33,.,/', #3 0.#0#.%"#/$N ^B ` ^2 ≠ K #. ^B` ^2 t K #. ^B` 2

• K

D*,.,8 H8 !8 G &/5 K &., '#/$%&/%$9



!"#$%&%"$%"'$

OX[

Consider the previous example (patients with leukaemia)

HO : P = .60

HA : P > .60

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#/ %*, /,7 5.-=9 :3 %*, 0.#0#.%"#/ #3 0&%",/%$ 7*# +";, &% +,&$% Y 2,&.$ "$

/#% "/'.,&$,5 -/5,. %*, /,7 %.,&%1,/% #. "$ "/'.,&$,5 #/+2 <2 &/

&1#-/% 5-, %# $&10+"/= 3+-'%-&%"#/8 *, 7"++ +##C 3#. &/#%*,. 5.-=9 !-% "3

%*, 0.#0#.%"#/ 7*# &., &"5,5 "$ $"=/"3"'&/%+2 +&.=,. L%*&% "$8 "3 *, "$ &<+, %#

'#/'+-5, %*&% %*, 0#0-+&%"#/ 0.#0#.%"#/ "$ =.,&%,. %*&/ 9Y[N ` %*,/ *,

7"++ '#/%"/-, *"$ %,$%$9

Exercises

State HO and HA for each of the following

ON :$ %*, &;,.&=, *,"=*% #3 %*, IG6E $%-5,/%$ O9YS 1 #. "$ "% 1#.,a

RN :$ %*, &;,.&=, *,"=*% #3 %*, IG6E $%-5,/%$ O9YS 1 #. "$ "% +,$$a

SN :$ %*, &;,.&=, *,"=*% #3 %*, IG6E $%-5,/%$ O9YS 1 #. "$ "%

$#1,%*"/= 5"33,.,/%a

VN )*,., "$ & <,+",3 %*&% O[i #3 %*, $1#C,.$ 5,;,+#0 +-/= '&/',. "/

'#-/%.2 B9

UN H., 1,/ &/5 7#1,/ "/3,'%,5 7"%* 1&+&."& "/ ,>-&+ 0.#0#.%"#/$8 #. "$

& *"=*,. 0.#0#.%"#/ #3 1,/ =,% 1&+&."& "/ ?%*"#0"&a



!"#$%&%"$%"'$

OXO

8.4 Level of significance

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.,J,'%,59 H/ ,..#. "$ 1&5,8 *#7,;,.8 "3 "% "$ %.-, <-% .,J,'%,58 #. "3 "% "$

3&+$, &/5 7, 3&"+ %# .,J,'% "%9

H .&/5#1 $&10+, #3 $"F, / "$ %&C,/ &/5 %*, "/3#.1&%"#/ 3.#1 %*, $&10+,

"$ -$,5 %# .,J,'% #. &'',0% L3&"+ %# .,J,'%N %*, /-++ *20#%*,$"$9 :% "$ /#%

&+7&2$ 0#$$"<+, %# 1&C, & '#..,'% 5,'"$"#/ $"/', 7, &., 5,&+"/= 7"%*

.&/5#1 $&10+,$9 )*,.,3#.,8 7, 1-$% +,&./ %# +";, 7"%* 0.#<&<"+"%",$ #3

%20, : LαN &/5 %20, :: LβN ,..#.$9

Definitions:

H )20, : ,..#. "$ 1&5, 7*,/ 4Q "$ %.-, <-% .,J,'%,59

H )20, :: ,..#. "$ 1&5, 7*,/ 4Q "$ 3&+$, <-% 7, 3&"+ %# .,J,'% "% 9

Notation:α "$ %*, 0.#<&<"+"%2 #3 & %20, : ,..#.9 :% "$ '&++,5 %*, level of

significance.

β

"$ %*, 0.#<&<"+"%2 #3 & %20, :: ,..#.9



!"#$%&%"$%"'$

OXR

)*, 3#++#7"/= %&<+, $-11&."$,$ %*, .,+&%"#/$*"0$ <,%7,,/ %*, /-++

*20#%*,$"$ &/5 %*, 5,'"$"#/ %&C,/ 9

K,'"$"#/

_-++ *20#%*,$"$ H'',0% 4Q

L\&"+ %# .,J,'% 4QN

T,J,'% 4Q

4Q %.-, G#..,'% )20, : ,..#.

4Q 3&+$, )20, :: ,..#. G#..,'%

:/ 0.&'%"',8 %*, +,;,+ #3 $"=/"3"'&/', L α N "$ '*#$,/ &.<"%.&."+2 &/5 %*,

+"1"%$ 3#. &'',0%"/= 4Q &., 5,%,.1"/,59 :3 & $&10+, $%&%"$%"' "$ #-%$"5,

%*#$, +"1"%$8 4Q "$ .,J,'%,5 L&/5 4 H "$ &'',0%,5N9 )*, 3#.1 #3 4 H 7"++

5,%,.1"/, %*, C"/5 #3 +"1"%$ %# <, $,% -0 L,"%*,. #/, %&"+,5 #. %7# %&"+,5

%,$%$ N 9

Consider the situation when HA includes the symbol " ≠ " . That is,

HA: μ ≠ , P ≠ , μx - μy ≠ ., Px - Py ≠ . etc (two tailed test)

O9 α L+,;,+ #3 $"=/"3"'&/',N "$ &.<"%.&."+2 '*#$,/8 ,>-&+ %# & $1&++

/-1<,. L-$-&++2 9[O8 9[U8 ,%'99N

R9 d ;&+-,$ &., 5,%,.1"/,5 $# %*&% %*, &.,& "/ ,&'* #3 %*, %&"+$ #3 %*,/#.1&+ 5"$%."<-%"#/ "$ α @ R9



!"#$%&%"$%"'$

OXS

The most common values of z are:

α

.10 .05 .01

Z± 1.64 ± 1.96 ±

2.58

S9 )*, ,B0,."1,/% "$ '&..",5 #-% &/5 %*, d ;&+-, #3 %*, &00.#0."&%,

$&10+, $%&%"$%"' L B8 08 B`28 0B ` 02 N "$ 5,%,.1"/,59 :3 %*, '#10-%,5 d

;&+-, 3&++$ 7"%*"/ %*, +"1"%$ 5,%,.1"/,5 "/ $%,0 R &<#;,8 7, 3&"+ %# .,J,'%

4Qj "3 %*, '#10-%,5 d ;&+-, "$ #-%$"5, %*#$, +"1"%$8 4Q "$ .,J,'%,5 L&/5

4 H "$ &'',0%,5N9 E"/', %*,2 $,0&.&%, %*, l3&"+ %# .,J,'%m &/5 l .,J,'%m

.,="#/$8 %*, +"1"%$ 5,%,.1"/,5 "/ $%,0 R 7"++ <, .,3,..,5 %# &$ %*, '."%"'&+

;&+-,$ #3 d9

8.5 Tests of Significance on means and Proportions (large

samples)

:% "$ "10#.%&/% %# .,1,1<,. %*&% & %,$% #3 $"=/"3"'&/', &+7&2$ .,3,.$ %# &

/-++ *20#%*,$"$9 )*, '#/',./ *,., "$ 7"%* &/ -/C/#7/ 0#0-+&%"#/

0&.&1,%,.8 &/5 %*, /-++ *20#%*,$"$ $%&%,$ %*&% "% "$ $#1, 0&.%"'-+&.

;&+-,9

)*, %,$% #3 $"=/"3"'&/', &/$7,.$ %*, >-,$%"#/c :$ '*&/', L$&10+"/=N

;&."&%"#/ & +"C,+2 ,B0+&/&%"#/ #3 %*, 5"$'.,0&/'2 <,%7,,/ & $&10+,

.,$-+% &/5 %*, '#..,$0#/5"/= /-++ *20#%*,$"$ 0#0-+&%"#/ ;&+-,a H l2,$m



!"#$%&%"$%"'$

OXV

&/$7,. f & 5"$'.,0&/'2 %*&% "$ +"C,+2 %# #''-. <2 '*&/', ;&."&%"#/f

"/5"'&%,$ %*, $&10+, .,$-+% "$ '#10&%"<+, 7"%* %*, '+&"1 %*&% %*,

$&10+"/= "$ 3.#1 & 0#0-+&%"#/ "/ 7*"'* %*, /-++ *20#%*,$"$ 0.,;&"+$9 )*"$

"$ %*, 1,&/"/= #3 l/#% $%&%"$%"'&++2 $"=/"3"'&/%9m H l/#m &/$7,. f &

5"$'.,0&/'2 %*&% "$ -/+"C,+2 %# #''-. <2 '*&/', ;&."&%"#/ f "/5"'&%,$ %*&%

%*, $&10+, .,$-+% "$ /#% '#10&%"<+, 7"%* %*, '+&"1 %*&% $&10+"/= "$ 3.#1

& 0#0-+&%"#/ "/ 7*"'* %*, /-++ *20#%*,$"$ 0.,;&"+$9 )*"$ "$ %*, 1,&/"/= #3

l$%&%"$%"'&++2 $"=/"3"'&/%9m

H$ $*#7/ ,&.+",.8 %*, +,;,+ $"=/"3"'&/', $,+,'%,58 <, "% U 0,.',/%8 O

0,.',/%8 #. #%*,.7"$,8 1-$% <, '+,&.+2 "/5"'&%,59 H $%&%,1,/% %*&% %*,

.,$-+%$ 7,., l$%&%"$%"'&++2 $"=/"3"'&/%m 7"%*#-% =";"/= 3-.%*,. 5,%&"+$ "$

7#.%*+,$$9

P – Values

^ f ;&+-,$ &<#-/5 "/ 1,5"'&+ &/5 0-<+"' *,&+%* .,$,&.'* 0&0,.$8 $# "% "$

,$$,/%"&+ %# -/5,.$%&/5 0.,'"$,+2 7*&% %*,2 1,&/9

4&;"/= $,% -0 %*, /-++ *20#%*,$"$8 7, %*,/ ,;&+-&%, %*, 0.#<&<"+"%2 %*&%

7, '#-+5 *&;, #<%&"/,5 %*, #<$,.;,5 5&%& L#. 5&%& %*&% 7,., 1#.,

,B%.,1,N "3 %*, /-++ *20#%*,$"$ 7,., %.-,9 )*"$ 0.#<&<"+"%2 "$ -$-&++2

'&++,5 %*, ^ f ;&+-,9 :3 "% "$ $1&++8 '#/;,/%"#/&++2 +,$$ %*&/ [9[U8 %*, /-++

*20#%*,$"$ "$ .,J,'%,5 &$ "10+&-$"<+,9 :/ #%*,. 7#.5$8 &/ #-%'#1, %*&%

'#-+5 #''-. +,$$ %*&/ #/, %"1, "/ R[ 7*,/ %*, /-++ *20#%*,$"$ "$ %.-,



!"#$%&%"$%"'$

OXU

7#-+5 +,&5 %# %*, .,J,'%"#/ #3 %*, /-++ *20#%*,$"$9 :/ %*"$ 3#.1-+&%"#/8

7*,/ 7, .,J,'% %*, /-++ *20#%*,$"$ 7, &'',0% & '#10+,1,/%&.2

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"/$-33"'",/% "/3#.1&%"#/ "$ &;&"+&<+, %# 5"$'#-/% %*, /-++ *20#%*,$"$9

D*,/ ^ "$ <,+#7 %*, '-% #33 +,;,+LαN8 $&2 [9[U8 %*, .,$-+% "$ '&++,5

statistically significantL &/5 <,+#7 $#1, +#7,. +,;,+8 $-'* &$ [9[O8 "%

1&2 <, '&++,5 highly significantNj 7*,/ &<#;, [9[U "% "$ '&++,5 not

significant9 :% "$ "10#.%&/% %# 5"$%"/=-"$* <,%7,,/ %*, $"=/"3"'&/',

+,;,+ &/5 %*, 0 f ;&+-,9 )*, $"=/"3"'&/', +,;,+ α "$ %*, 0.#<&<"+"%2 #3

1&C"/= & %20, : ,..#.9 )*"$ "$ $,% <,3#., %*, %,$% "$ '&..",5 #-%9 )*, ^ f

;&+-, "$ %*, .,$-+% #<$,.;,5 &3%,. %*, $%-52 "$ '#10+,%,5 &/5 "$ <&$,5

#/ %*, #<$,.;,5 5&%&9

:% 7#-+5 <, <,%%,. L"/3#.1&%";,N %# =";, %*, ,B&'% ;&+-,$ #3 ^j $-'* &$8

^ q [9[R #. ^ q [9OU .&%*,. %*&/ ^ t [9[U #. ^ • [9[U 9 :% "$ /#7

"/'.,&$"/=+2 '#11#/ %# $,, %*, ,B0.,$$"#/ #3 ,B&'% ;&+-,$ +&.=,+2

5-, %# %*, &;&"+&<"+"%2 #3 '#10-%,. 0.#=.&1$ 7*"'* =";, %*, ,B&'% ^ f

;&+-,$9

8.5.1 Tests of significance on a single mean and comparison of

two means

)*, 0.,',5"/= 5"$'-$$"#/$ 0.#;"5, %*, /,',$$&.2 ,>-"01,/% 3#.

'#/5-'%"/= %,$%$ #3 $"=/"3"'&/', #/ & $"/=+, 1,&/ &/5 '#10&."$#/ #3

%7# 1,&/$9



!"#$%&%"$%"'$

OXY

A statistical test of significance on a single mean

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-/C/#7/ 0#0-+&%"#/ 1,&/9 )*, $%&%"$%"'&+ "/3,.,/', '#/$"$%$ #3 5.&7"/=

#/, #3 %*, 3#++#7"/= %7# '#/'+-$"#/$ .,=&.5"/= %*"$ $%&%,1,/%c

:N T,J,'% %*, '+&"1 &<#-% %*, 0#0-+&%"#/ 1,&/ <,'&-$, %*,., "$

$-33"'",/% ,;"5,/', %# 5#-<% "%$ ;&+"5"%29

::N K# /#% .,J,'% %*, '+&"1 &<#-% %*, 0#0-+&%"#/ 1,&/8 <,'&-$, %*,., "$

/#% $-33"'",/% ,;"5,/', %# 5#-<% "%$ ;&+"5"%29

)*, &/&+2$"$ '#/$"$%$ #3 5,%,.1"/"/= %*, '*&/', #3 #<$,.;"/= & 1,&/

&$ 5,;"&/% &$ #. 1#., 5,;"&/% %*&/ %*, $&10+, 1,&/8 -/5,. %*,

&$$-10%"#/ %*&% %*, $&10+, '&1, 3.#1 & 0#0-+&%"#/ 7*#$, 1,&/ "$ μQ9

Q/, %*,/ '#10&.,$ %*"$ '*&/', 7"%* %*, 0.,5,%,.1"/,5 l$-33"'",/%+2

$1&++m '*&/', <2 .,3,.."/= %# %*, %&<+, #3 %*, d 5"$%."<-%"#/ L %*,

$%&/5&.5 /#.1&+ 5"$%."<-%"#/N 9 )*, '."%"'&+ .&%"# Ld $%&%"$%"'N "$ '&+'-+&%,5

&$c d q L x ` μQ N @ L σ / n N 9

Examplec H$$-1, %*&% "/ & ',.%&"/ 5"$%."'% %*, 1,&/ $2$%#+"' <+##50.,$$-., #3 0,.$#/$ &=,5 R[ %# V[ "$ OS[ 11 4= 7"%* & $%&/5&.5

5,;"&%"#/ #3 O[ 11 4= 9 H .&/5#1 $&10+, #3 YV 0,.$#/$ &=,5 R[ %# V[

3.#1 ;"++&=, B #3 %*, $&1, 5"$%."'% *&$ & 1,&/ $2$%#+"' <+##5 0.,$$-.,

#3 OSR 11 4=9 K#,$ %*, 1,&/ $2$%#+"' <+##5 0.,$$-., #3 %*, 57,++,.$



!"#$%&%"$%"'$

OXW

#3 %*, ;"++&=, L&=,5 R[ %# V[N 5"33,. 3.#1 %*&% #3 %*, "/*&<"%&/%$ #3 %*,

5"$%."'% L&=,5 R[ %# V[N "/ =,/,.&+8 &% & Ui P,;,+ #3 $"=/"3"'&/',a

HO :μ

= 130 L %*, 1,&/ $2$%#+"' <+##5 0.,$$-., #3 %*, ;"++&=,

"$ %*, $&1, &$ %*, 1,&/ E!^ #3 %*, 5"$%."'% N

HA: μ ≠ 130

α q 9[U L %*&% "$8 %*, 0.#<&<"+"%2 #3 .,J,'%"/= 4Q 7*,/ "% "$ %.-, "$ %# <,

9[UN9

)*, &.,& #3 ,&'* $*&5,5 h%&"+ h #3 %*, $%&/5&.5 /#.1&+ '-.;, "$ 9[RU

&/5 %*, '#..,$0#/5"/= d $'#.,$ L d %&<-+&%,5N &% %*, <#-/5&.",$ &., ±

O9XY9

E&10+,c / q YV8 X q OSR

)*, d $'#., 3#. %*, .&/5#1 $&10+, #3 YV 0,.$#/$ #3 %*, ;"++&=, &=,5 R[

%# V[ 2,&.$c

Z calc = (132-130) / (10/√

64) = 2 / 1.25 = 1.6

)*"$ $'#., 3&++$ "/$"5, %*, l3&"+ %# .,J,'% .,="#/m 3.#1 fO9XY %# rO9XY 9

:3 %*, '&+'-+&%,5 d ;&+-, "$ 0#$"%";,8 %*, .-+, $&2$c

reject HO if Z calculated ( Z calc) > Z tabulated (Z tab)

or accept HO if Z calc < Z tab.



!"#$%&%"$%"'$

OX]

Q/ %*, #%*,. *&/58 "3 %*, '&+'-+&%,5 d ;&+-, "$ /,=&%";,c

reject HO if Z calculated ( Z calc) < Z tabulated (Z tab)

L4,.,8 <#%* d'&+'-+&%,5 &/5 d%&<-+&%,5 &., /,=&%";, ;&+-,$N

4,/',8 %*, /-++ *20#%*,$"$ #3 %*, &<#;, ,B&10+, "$ &'',0%,59 )*&% "$8

%*, 1,&/ $2$%#+"' <+##5 0.,$$-., #3 0,.$#/$ L &=,5 R[ %# V[ N +";"/= "/

;"++&=, B "$ %*, $&1, &$ %*, 1,&/ $2$%#+"' <+##5 0.,$$-., #3 %*,

"/*&<"%&/%$ L&=,5 R[ %# V[N #3 %*, 5"$%."'%9

)*, $&1, '#/'+-$"#/ 7"++ <, .,&'*,5 <2 .,3,.."/= %# %*, '#..,$0#/5"/=

^`;&+-,9 D, -$, %&<+, V %# 3"/5 %*, ^`;&+-, &$$#'"&%,5 7"%* &/ #<$,.;,5

L'&+'-+&%,5N ;&+-, #3 d 7*"'* "$ O9Y9 \.#1 %*, %&<+, 1,/%"#/,5 &<#;,

7, =,% ^q9OO L/#%, %*&% %*"$ 0`;&+-, "$ =.,&%,. %*&/ %*, =";,/ +,;,+ #3

$"=/"3"'&/',N9

Comparison of two Means

)*, 0-.0#$, #3 %*"$ $,'%"#/ "$ %# ,B%,/5 %*, &.=-1,/%$ #3 %*, $"/=+,

1,&/ %# %*, '#10&."$#/ #3 %7# $&10+, 1,&/$9 E"/', 1,5"'"/, L0-<+"'

*,&+%*N "$8 <2 /&%-.,8 '#10&.&%";,8 %*"$ "$ & .&%*,. 7"5,$0.,&5 $"%-&%"#/8

1#., '#11#/ %*&/ %*&% #3 %*, $"/=+, 1,&/ #3 %*, 0.,',5"/= $,'%"#/9

:/ %*, '#10&."$#/ #3 %7# 1,&/$8 %*,., &., %7# $&10+,$ #3 #<$,.;&%"#/$

3.#1 %7# -/5,.+2"/= 0#0-+&%"#/$ L#3%,/ %.,&%1,/% &/5 '#/%.#+ =.#-0$N

7*#$, 1,&/$ &., 5,/#%,5 <2 μ% &/5 μ' &/5 7*#$, $%&/5&.5 5,;"&%"#/$



!"#$%&%"$%"'$

OXX

&., 5,/#%,5 <2 σ% &/5 σ' 9 T,'&++"/= %*&% & %,$% #3 $"=/"3"'&/', "/;#+;,$ &

/-++ *20#%*,$"$ %*&% $0,'"3",$ ;&+-,$ 3#. 0#0-+&%"#/ >-&/%"%",$8 %*,

.,+,;&/% /-++ *20#%*,$"$ "$ %*&% %*, 1,&/$ &., "5,/%"'&+8 "9,98

HO :μt = μc or HO :

μt - μc = 0

)*, .&%"#/&+, 3#. %*, %,$% #3 $"=/"3"'&/', "$ &$ <,3#.,9 H$$-1"/= %*, /-++

*20#%*,$"$ "$ %.-, L"9,98 %*&% %*,., "$ /# 5"33,.,/', "/ %*, 0#0-+&%"#/

1,&/$N8 #/, 5,%,.1"/,$ %*, '*&/', #3 #<%&"/"/= 5"33,.,/',$ "/ $&10+,

1,&/$ &$ 5"$'.,0&/% &$ #. 1#., 5"$'.,0&/% %*&/ %*&% #<$,.;,59 :3 %*"$

'*&/', "$ $-33"'",/%+2 $1&++8 %*,., "$ .,&$#/&<+, ,;"5,/', %# 5#-<% %*,

;&+"5"%2 #3 %*, /-++ *20#%*,$"$j *,/',8 #/, '#/'+-5,$ %*,., "$ &

$%&%"$%"'&++2 $"=/"3"'&/% 5"33,.,/', <,%7,,/ %*, 1,&/$ #3 %*, %7#

0#0-+&%"#/$ L"9,98 #/, .,J,'%$ %*, /-++ *20#%*,$"$N9

Consider the Example Given in Section 7.5.2

),$% %*, *20#%*,$"$ %*&% %*,., "$ /# 5"33,.,/', <,%7,,/ %*, 1,&/

+"3,%"1,$ #3 #/ $1#C,.$ &/5 $1#C,.$ &% & 9[O +,;,+ #3 $"=/"3"'&/',9

^#0-+&%"#/ B L/#/`$1#C,.$N /BqU[ 8 x q WY8 EB q ]8 σ

R

x q E

R

B @ /B8 q]R@U[ qO9R] 2,&.$

^#0-+&%"#/ 2 L$1#C,.$N /2qYU 8 y q Y]8 E2 q X8 σR y q ER

2 @ /28 q XR@YU

qO9RU 2,&.$



!"#$%&%"$%"'$

R[[

Hypotheses: HO : μt = μc or HO : μt - μc = 0

HA : μt ≠ μc or HA : μt - μc≠

0

α

= .01 ( two tailed )⇒

Z (tabulated) =±

2.58

Z calc = { ( x - y ) ` (μx- μy) } ⁄ Standard error of the difference of

means

Standard error of the difference of means = σL x - y N q $GS"$:S" +

q O9UX 2,&.$

Hence, Zcalc = (76 – 68) / 1.59 = 8 / 1.59 = 5.03

The corresponding P-value is less than .003.

!,'&-$, d'&+' • d%&< L"9,98 P-value < the given α value), %*, /-++

*20#%*,$"$ L 4Q N "$ .,J,'%,59 )*&% "$8 %*,., "$ & $%&%"$%"'&++2 $"=/"3"'&/%

5"33,.,/', "/ %*, 1,&/ +"3,%"1,$ #3 /#/$1C,.$ &/5 $1#C,.$

8.5.2 Test of significance on a single proportion and comparison

of proportions

Test of Significance on a Single Proportion

_#%, %*&% %*, ,B0.,$$"#/ =";,/ "/ $,'%"#/ W9U9S &00+",$ "/ & $"1"+&. 7&2

%# %*"$ $,'%"#/9



!"#$%&%"$%"'$

R[O

Example: H1#/= $-$',0%"<+, "/5";"5-&+$ ,B0#$,5 %# & 0&.%"'-+&.

"/3,'%"#-$ &=,/%8 SY 0,.',/% =,/,.&++2 5,;,+#0 '+"/"'&+ 5"$,&$,9 H1#/=

& $'*##+ =.#-0 #3 OVV 0,.$#/$ $-$0,'%,5 #3 ,B0#$-., %# %*, &=,/%8 #/+2

SU 5,;,+#0,5 '+"/"'&+ 5"$,&$,9 :$ %*"$ .,$-+% 7"%*"/ '*&/', ;&."&%"#/ Lα q

9[UN9

)# -$, %*, %&<+, #3 %*, $%&/5&.5 /#.1&+ 5"$%."<-%"#/8 #/, /,,5$ %#

'&+'-+&%, '."%"'&+ .&%"#$9 )*,., "$8 *#7,;,.8 &/ &55"%"#/&+ '#/$"5,.&%"#/

5,.";,5 3.#1 %*, 3&'% %*&% %*, '."%"'&+ .&%"# "$ <&$,5 #/ $1##%* /#.1&+

'-.;,$ -$,5 &$ $-<$%"%-%,$ 3#. 5"$'.,%, <"/#1"&+ 5"$%."<-%"#/$9 H

'#..,'%"#/ '#/%"/-"%2 "$ %*,.,3#., &00.#0."&%,9

d qw P p − ` L n$

"

Nx @ $,L^N

q w⏐"88

%G f 9SY⏐`

$::

" x @ "88&98S%9S × q R9]V

)*, #<$,.;,5 ;&+-, #3 d L7*"'* "$ R9]VN '#..,$0#/5$ %# & ^`;&+-, #3

9[[U9

4,/',8 &% & 9[U +,;,+ #3 $"=/"3"'&/',8 %*, /-++ *20#%*,$"$ "$ .,J,'%,59

)*,.,3#.,8 %*, $&10+, 3"/5"/= "$ /#% '#10&%"<+, 7"%* %*, 0#0-+&%"#/

0.#0#.%"#/9

Comparison of two Proportions

H $"1"+&. &00.#&'* "$ &5#0%,5 7*,/ 0,.3#.1"/= & *20#%*,$"$ %,$% %#

'#10&., %7# 0.#0#.%"#/$9 )*, $%&/5&.5 ,..#. #3 %*, 5"33,.,/', "/



!"#$%&%"$%"'$

R[R

0.#0#.%"#/$ "$ &=&"/ '&+'-+&%,58 <-% <,'&-$, 7, &., ,;&+-&%"/= %*,

0.#<&<"+"%2 #3 %*, 5&%& #/ %*, &$$-10%"#/ %*&% %*, /-++ *20#%*,$"$ "$ %.-,

7, '&+'-+&%, & $+"=*%+2 5"33,.,/% $%&/5&.5 ,..#.9 :3 %*, /-++ *20#%*,$"$ "$

%.-,8 %*, %7# $&10+,$ '#1, 3.#1 0#0-+&%"#/$ *&;"/= %*, $&1, %.-,

0.#0#.%"#/ #3 "/5";"5-&+$ 7"%* %*, '*&.&'%,."$%"' #3 "/%,.,$%8 $&28 ^ 9 D,

5# /#% C/#7 ^8 <-% <#%* 0O &/5 0R &., ,$%"1&%,$ #3 ^9 Q-. <,$% ,$%"1&%,

#3 ^ "$ =";,/ <2 '&+'-+&%"/= c

0 q$"

$"

nn

r r

++

)*, $%&/5&.5 ,..#. #3 Ô ` ^R -/5,. %*, /-++ *20#%*,$"$ "$ %*-$

'&+'-+&%,5 #/ %*, &$$-10%"#/ %*&% %*, 0.#0#.%"#/ "/ ,&'* =.#-0 "$ 0 8 $#

%*&% 7, *&;,

$,LÔ ` ^RN q #$&""&"#U"U nn p p +−

)*, $&10+"/= 5"$%."<-%"#/ #3 Ô ` ^R "$ /#.1&+8 $# 7, '&+'-+&%, &

$%&/5&.5 /#.1&+ 5,;"&%, Ld $%&%"$%"'N 8 &$

dq#$"U

$" p p se p p

−−

?B&10+,c H *,&+%* #33"',. "$ %.2"/= %# $%-52 %*, 1&+&."& $"%-&%"#/ #3

?%*"#0"&9 \.#1 %*, .,'#.5$ #3 $,&$#/&+ <+##5 $-.;,2 LE!EN .,$-+%$ *,



!"#$%&%"$%"'$

R[S

'&1, %# -/5,.$%&/5 %*&% %*, 0.#0#.%"#/ #3 0,#0+, *&;"/= 1&+&."& "/

?%*"#0"& 7&$ S9]i "/ OXW] L?%*9 G&+N9 )*, $"F, #3 %*, $&10+,

'#/$"5,.,5 7&$ OU[[[9 4, &+$# .,&+"$,5 %*&% 5-."/= %*, 2,&. %*&%

3#++#7,5 LOXWXN8 <+##5 $&10+,$ 7,., %&C,/ 3.#1 O[8[[[ .&/5#1+2

$,+,'%,5 0,.$#/$9 )*, .,$-+% #3 %*, OXWX $,&$#/&+ <+##5 $-.;,2 $*#7,5

%*&% R[[ 0,.$#/$ 7,., 0#$"%";, 3#. 1&+&."&9 4,+0 %*, *,&+%* #33"',. "/

%,$%"/= %*, *20#%*,$"$ %*&% %*, 1&+&."& $"%-&%"#/ #3 OXWX 5"5 /#% $*#7

&/2 $"=/"3"'&/% 5"33,.,/', 3.#1 %*&% #3 OXW] L%&C, %*, +,;,+ #3

$"=/"3"'&/',8 α q9[ON9

4Q c OXW] q ÔXWX L #. ÔXW] ` ÔXWX q [ N

4 H c OXW] ≠ ÔXWX L #. ÔXW] ` ÔXWX ≠ [ N

0OXW] q 9[S] 8 /OXW] q OU8[[[

0OXWX q 9[R 8 /OXWX q O[8[[[

d %&< L α q 9[O N q R9U]

G#11#/ L0##+,5N 0.#0#.%"#/8 0 q"7777"G777

$77GO7

++

q [9[S[]

&/5 %*, $%&/5&.5 ,..#. q #"7777&""G777&"UR9R$S7%:S +×

q 9[[RR

4,/',8 d '&+' q L9[S] ` 9[R[N @ 9[[RR q 9[O] @ 9[[RR q ]9 R

D*"'* '#..,$0#/5$ %# & ^`;&+-, #3 +,$$ %*&/ 9[[S9



!"#$%&%"$%"'$

R[V

Decision: reject Ho (because Z calc > Z tab); in other words, the p-

value is less than the level of significance (i.e., α = .01)

)*,.,3#.,8 "% "$ '#/'+-5,5 %*&% %*,., 7&$ & $%&%"$%"'&++2 $"=/"3"'&/%

5"33,.,/', "/ %*, 0.#0#.%"#/ #3 1&+&."& 0&%",/%$ <,%7,,/ OXW] &/5 OXWX

&% & 9[O +,;,+ #3 $"=/"3"'&/',9

_#%, %*&% %*, ,33,'% #3 %*, '#/%"/-"%2 '#..,'%"#/ "$ &+7&2$ %# .,5-', %*,

1&=/"%-5, #3 %*, /-1,.&%#. #3 %*, '."%"'&+ .&%"#9 :/ %*"$ $,/$,8 %*,

'#/%"/-"%2 '#..,'%"#/ "$ '#/$,.;&%";, "/ %*&% "%$ -$,8 '#10&.,5 7"%* "%$ /#%

<,"/= -$,58 7"++ +&<,+ $#1,7*&% 3,7,. #<$,.;,5 $&10+, 5"33,.,/',$ &$

<,"/= $%&%"$%"'&++2 $"=/"3"'&/%9 Q/ %*, #%*,. *&/58 7*,/ 5,&+"/= 7"%*

;&+-,$ #3 /π &/5 /LO` πN %*&% &., 7,++ &<#;, U8 %*, '#/%"/-"%2 '#..,'%"#/

*&$ & /,=+"="<+, ,33,'%9 )*"$ /,=+"="<+, ,33,'% '&/ <, ,&$"+2 $,,/ 3.#1 %*,

&<#;, ,B&10+,9

8.6 One tailed tests

:/ %*, 0.,',5"/= $,'%"#/$ 7, *&;, <,,/ 5,&+"/= 7"%* %7# %&"+,5 L$"5,5N

%,$%$9

'#/$"5,. %*, $"%-&%"#/ 7*,/ 4 H "/'+-5,$ %*, $21<#+ l • #. t m9 )*&% "$8

HA:μ

> __ , HA:μ

< __, HA : P > __, HA : P < __, HA :μx -μy > ___,

HA :μx -μy < ___, etc. (One tailed test).

:/ %*, 0.,;"#-$ $,'%"#/$ 7, +,&./,5 *#7 %# $,% -0 & /-++ *20#%*,$"$ 4Q

&/5 &/ &+%,./&%";, *20#%*,$"$ 4 H 8 &/5 *#7 %# .,&'* & 5,'"$"#/ #/



!"#$%&%"$%"'$

R[U

7*,%*,. #. /#% %# .,J,'% %*, /-++ *20#%*,$"$ &% & =";,/ +,;,+ #3

$"=/"3"'&/', 7*,/ 4 H "/'+-5,$ %*, $21<#+ l ≠ m9 )*, 5,'"$"#/ 7&$

.,&'*,5 <2 3#++#7"/= & $0,'"3"' '#/;,/%"#/ L"9,98 %*, &.,& "/ ,&'* %&"+ #3

%*, $&10+"/= 5"$%."<-%"#/ "$ &$$-1,5 %# <, α@R N 9 )*"$ '#/;,/%"#/

5,%,.1"/,$ %7# ;&+-,$ #3 d 7*"'* $,0&.&%, & l 3&"+ %# .,J,'% .,="#/m 3.#1

%7# .,J,'%"#/ .,="#/$9 )*,/ %*, d`;&+-, #3 & $%&%"$%"' "$ '#10-%,5 3#. &

.&/5#1 $&10+,8 &/5 4Q "$ $%"++ &'',0%,5 "3 %*, d`;&+-, 3&++$ "/ %*, h 3&"+ %#

.,J,'% .,="#/ h 0.,;"#-$+2 5,%,.1"/,5j #%*,.7"$,8 4Q "$ .,J,'%,59

H #/, %&"+,5 %,$% L 4 Hc μ • 8 ,%'9N "$ &+$# J-$%"3",5 7*,/ %*, "/;,$%"=&%#.

'&/ $%&%, &% %*, #-%$,% %*&% "% "$ ,/%".,+2 "/'#/',";&<+, %*&% %*, %.-,

0#0-+&%"#/ 1,&/ "$ <,+#7 L #. &<#;,N %*&% #3 %*, /-++ *20#%*,$"$9 )#

5,3,/5 %*"$ 0#$"%"#/8 %*,., 1-$% <, $#+"5 &/5 '#/;"/'"/= $-00#.%"/=

,;"5,/',9 :/ 1,5"'&+ L0-<+"' *,&+%*N &00+"'&%"#/$8 *#7,;,.8 & #/, %&"+,5

%,$% "$ /#% '#11#/+2 ,/'#-/%,.,59

D*,/ & #/,`%&"+,5 %,$% "$ '&..",5 #-%8 "% "$ "10#.%&/% %# /#%"', %*&% %*,

.,J,'%"#/ .,="#/ "$ '#/',/%.&%,5 "/ #/, %&"+ #3 %*, $&10+"/= 5"$%."<-%"#/ #3

1,&/$9 )*, &.,& #3 %*"$ #/, %&"+ "$ α8 .&%*,. %*&/ %*, α@R -$,5

0.,;"#-$+28 &/5 %*, '."%"'&+ ;&+-, #3 d '*&/=,$ &''#.5"/=+29



!"#$%&%"$%"'$

R[Y

Example:

Two-tailed test one-tailed test

4Q c μ q O[[ 4Q c μ q O[[

4Q c μ ≠O[[ 4 H c μ • O[[

α q 9[U α q 9[U

The most frequently used values ofα

and the corresponding

critical values of Z are:

α (level of significance) two-tailed one -tailed, < one-tailed, >

.10 ± 1.64 - 1.28 1.28

.05 ± 1.96 - 1.64 1.64

.01±

2.58 - 2.33 2.33

P#="'&++28 &/ "/;,$%"=&%#. '*##$,$ <,%7,,/ & #/, &/5 %7#`%&"+,5 %,$%

<,3#., *, #<%&"/$ *"$ $&10+, .,$-+%8 "9,98 %*, '*#"', "$ /#% "/3+-,/',5 <2

%*, $&10+, #-%'#1,9 )*, "/;,$%"=&%#. &$C$c :$ "% "10#.%&/% %# <, &<+, %#

5,%,'% &+%,./&%";,$ #/+2 &<#;, %*, /-++ *20#%*,$"$ 1,&/8 #. "$ "%

"10#.%&/% %# <, &<+, %# 5,%,'% &+%,./&%";,$ %*&% 1&2 <, ,"%*,. &<#;, #.

<,+#7 %*"$ 1,&/a 4"$ &/$7,.8 5,0,/5$ #/ %*, 0&.%"'-+&. '".'-1$%&/',$

#3 %*, "/;,$%"=&%"#/9

?B&10+,Oc :% "$ C/#7/ %*&% O`2,&. #+5 5#=$ *&;, & 1,&/ =&"/ "/ 7,"=*%

#3 O9[ 0#-/5 0,. 1#/%* 7"%* & $%&/5&.5 5,;"&%"#/ #3 9V[ 0#-/59 H

$0,'"&+ 5",% $-00+,1,/% "$ =";,/ %# & .&/5#1 $&10+, #3 U[ O`2,&. #+5



!"#$%&%"$%"'$

R[W

5#=$ 3#. & 1#/%*j %*,". 1,&/ =&"/ "/ 7,"=*% 0,. 1#/%* "$ O9OU 0#-/5$8

7"%* & $%&/5&.5 5,;"&%"#/ #3 9S[ 0#-/59 K#,$ 7,"=*% =&"/ "/ O`2,&. #+5

5#=$ "/'.,&$, "3 & $0,'"&+ 5",% $-00+,1,/% "$ "/'+-5,5 "/ %*,". -$-&+ 5",%a

L9[O +,;,+ #3 $"=/"3"'&/',N

4Q c μ q O9[

4 Hc μ • O9[ L#/, %&"+,5 %,$%N

d %&< Lα q9[ON q R9SS &/5 .,J,'% 4Q "3 d '&+' • R9SS9

d '&+' q LO9OU ` O9[N @ L9V@√U[N q [9OU@[9[UYY q R9YU L)*"$ '#..,$0#/5$

%# & ^`;&+-, #3 9[[VN

4,/',8 &% & 9[O +,;,+ #3 $"=/"3"'&/', 7,"=*% =&"/ "$ "/'.,&$,5 "3 & $0,'"&+

5",% $-00+,1,/% "$ "/'+-5,5 "/ %*, -$-&+ 5",% #3 O`2,&. #+5 5#=$9

Example 2: H 0*&.1&',-%"'&+ '#10&/2 '+&"1$ %*&% & 5.-= 7*"'* "%

1&/-3&'%-.,$ .,+",;,$ '#+5 $210%#1$ 3#. & 0,."#5 #3 O[ *#-.$ "/ X[i #3

%*#$, 7*# %&C, "%9 :/ & .&/5#1 $&10+, #3 V[[ 0,#0+, 7"%* '#+5$ 7*#

%&C, %*, 5.-=8 SU[ 3"/5 .,+",3 3#. O[ *#-.$9 H% & 9[U +,;,+ #3 $"=/"3"'&/',8

"$ %*, 1&/-3&'%-.,.b$ '+&"1 '#..,'%a

4Q c ^ q 9X[

4 H c^ t 9X[

d %&< Lα q 9[UN q `O9YV &/5 .,J,'% 4Q "3 d '&+' t -1.649

d '&+' q L9]WU ` 9X[N @ √L9X[ B 9O[ @V[[N q L9]WU`9X[N@9[OU q -1.67



!"#$%&%"$%"'$

R[]

)*, '#..,$0#/5"/= ^`;&+-, "$ 9[VWU

4,/',8 4Q "$ .,J,'%,5c %*, 1&/-3&'%-.,.e$ '+&"1 "$ /#% -0*,+59

8.7 comparing the means of small samples

D, *&;, $,,/ "/ %*, 0.,',5"/= $,'%"#/$ *#7 %*, E%&/5&.5 /#.1&+

5"$%."<-%"#/ '&/ <, -$,5 %# '&+'-+&%, '#/3"5,/', "/%,.;&+$ &/5 %# '&..2

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The t-distribution

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But, what happens when n<30 and σ is unknown?

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Degrees of Freedom

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Table of t-distributions

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?=S9 :3 53 qU8 7*&% "$ %*, 0.#<&<"+"%2 %*&% & % $'#., "$ &<#;, R9[R #.

<,+#7 `R9[Ra

Solutions

1) P##C &% %*, %&<+, L%`5"$%."<-%"#/ N9 H+#/= %*, .#7 +&<,++,5 l#/, %&"+m %#

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%*, 53 '#+-1/ =";,$ %*, ;&+-, #3 t = 1.86.



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%*, 53 '#+-1/ =";,$ to = 2.1839

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R9[R9

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Computation of Confidence Intervals and Tests of Hypothesis

using the t - distribution

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4 H c μ ≠ WR

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% %&< L 7"%* α q 9[U &/5 ] 53N q ± R9SO

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%*&/ fR9SO9

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\#.1-+& 3#. *20#%*,$"$ %,$%$: t = (d - μd ) / (Sd / √n ) and C.I. for μd = d

± tα

(Sd / √n)

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!N \"/5 %*, XUi G9:9 3#. %*, 1,&/ #3 %*, 0#0-+&%"#/ 5"33,.,/',$9

solutions

HN 4Q c μ5 q [

4 Hc μ5 ≠ [

d q S8 μ5 q [8 E5 q √{∑L5"`5NR @ L/`ON} q √ { LY`SNR r LV`SNR r

u rLV`SNR }@ LO[`ON

q √ LOS]@XN q S9XR



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t calc = (d - μd ) / (Sd / √n) = ( 3- 0) / (3.92 /√

10) = 3 / 1.24 = 2.4

L)*"$ '#..,$0#/5$ %# & ^`;&+-, #3 +,$$ %*&/ 9[UN

t tab (α

= .05, df = 9) = 2.26

% '&+' "$ • % %&< ⇒ .,J,'% 4Q9

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√/N

q S ± R9RYLO9RVN q S ± R9]

q L[9R 8 U9]N

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%*, ;&+-,$ 9O[8 9[R &/5 9[O9 D*&% 5# 2#- -/5,.$%&/5 3.#1 %*"$a

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Placebo Caffeine

WR WY

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Mean 68 75

Variance 31.11 28.67

HN ),$% %*, *20#%*,$"$ %*&% '&33,"/, *&$ /# ,33,'% #/ %*, 0-+$, .&%,$ #3

2#-/= 1,/ Lα q 9[UN9

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L0+&',<#N =.#-0

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\#. %*, 5&%& =";,/ "/ %*, &<#;, ,B&10+,8

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A 95% confidence interval =";,$ & 0+&-$"<+, .&/=, #3 ;&+-,$ %*&%

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1&%*,1&%"'$ 7"++ +,&5 %# %*, $&1, '#/'+-$"#/ &<#-% 7*,%*,. &

0#0-+&%"#/ 5"33,.,/', ,B"$%$9 G#/3"5,/', "/%,.;&+$ &., 0.,3,.&<+,

<,'&-$, %*,2 =";, "/3#.1&%"#/ &<#-% %*, $"F, #3 &/2 5"33,.,/', "/ %*,

0#0-+&%"#/8 &/5 %*,2 &+$# L'.-'"&++2N "/5"'&%, %*, &1#-/% #3 -/',.%&"/%2

.,1&"/"/= &<#-% %*, $"F, #3 %*, 5"33,.,/',9



!"#$%&%"$%"'$

RRO

8.9 Test of significance using the chi-square and fisher’s exact

tests

8.9.1 The Chi – square test

H '*" $>-&., LχR N 5"$%."<-%"#/ "$ & 0.#<&<"+"%2 5"$%."<-%"#/9 )*, '*"`$>-&.,

"$ -$,3-+ "/ 1&C"/= $%&%"$%"'&+ "/3,.,/',$ &<#-% '&%,=#."'&+ 5&%& "/ 7*"'*

%*, '&%,=#.",$ &., %7# &/5 &<#;, 9

Definition H $%&%"$%"' 7*"'* 1,&$-.,$ %*, 5"$'.,0&/'2 <,%7,,/ n

#<$,.;,5 3.,>-,/'",$ QO8 QR8 9 QC &/5 %*, '#..,$0#/5"/= ,B0,'%,5

3.,>-,/'",$ ,O8 ,R 9 ,C9

Chi square = χ

2 = ∑{ (Oi - ei)2 } / ei

)*, $&10+"/= 5"$%."<-%"#/ #3 %*, '*"`$>-&., $%&%"$%"' "$ C/#7/ &$ %*, '*"

$>-&., 5"$%."<-%"#/9 H$ "/ % 5"$%."<-%"#/$8 %*,., "$ & 5"33,.,/% χR

5"$%."<-%"#/ 3#. ,&'* 5"33,.,/% ;&+-, #3 5,=.,,$ #3 3.,,5#18 <-% &++ #3

%*,1 $*&., %*, 3#++#7"/= '*&.&'%,."$%"'$9

Characteristics

O9 ?;,.2 χR 5"$%."<-%"#/ ,B%,/5$ "/5,3"/"%,+2 %# %*, ."=*% 3.#1 [9

R9 ?;,.2 χR 5"$%."<-%"#/ *&$ #/+2 #/, L."=*% N %&"+9



!"#$%&%"$%"'$

RRR

S9 H$ 53 "/'.,&$,$8 %*, χR '-.;,$ =,% 1#., <,++ $*&0,5 &/5 &00.#&'*

%*, /#.1&+ '-.;, "/ &00,&.&/', L<-% .,1,1<,. %*&% & '*" $>-&., '-.;,

$%&.%$ &% [8 /#% &% ` ∞ N

:3 %*, ;&+-, #3 χR "$ F,.#8 %*,/ %*,., "$ & 0,.3,'% &=.,,1,/% <,%7,,/ %*,

#<$,.;,5 &/5 %*, ,B0,'%,5 3.,>-,/'",$9 )*, =.,&%,. %*, 5"$'.,0&/'2

<,%7,,/ %*, #<$,.;,5 &/5 ,B0,'%,5 3.,>-,/'",$8 %*, +&.=,. 7"++ <, %*,

;&+-, #3 χR9

:/ #.5,. %# %,$% %*, $"=/"3"'&/', #3 %*, χR8 %*, '&+'-+&%,5 ;&+-, #3 χR "$

'#10&.,5 7"%* %*, %&<-+&%,5 ;&+-, 3#. %*, =";,/ 53 &% & ',.%&"/ +,;,+ #3

$"=/"3"'&/',9

Example1c :/ &/ ,B0,."1,/% 7"%* 0,&$ #/, #<$,.;,5 SY[ .#-/5 &/5

2,++#78 OS[ .#-/5 &/5 =.,,/8 OO] 7."/C+,5 &/5 2,++#7 &/5 SR 7."/C+,5

&/5 =.,,/9 H''#.5"/= %# %*, 6,/5,+"&/ %*,#.2 #3 *,.,5"%2 %*, /-1<,.$

$*#-+5 <, "/ %*, .&%"# XcScScO9 :$ %*,., &/2 ,;"5,/', #3 5"33,.,/', 3.#1

%*, 0+&/%$ &% Ui +,;,+ #3 $"=/"3"'&/',a

Solution

420#%*,$"$c HO : Ratio is 9:3:3:1HA : Ratio is not 9:3:3:1



!"#$%&%"$%"'$

RRS

G&%,=#.2 Q" 0.#0#.%"#/ ,"

To SY[ X@OY SY[

TI OS[ S@OY OR[

Do OO] S@OY OR[

DI SR O@OY V[

χR '&+' q LSY[ ` SY[NR @ SY[ r LOS[`OR[NR @ OS[ r LOO]`OR[NR @ OR[ r

LSR`V[NR @V[

q [ r 9]SS r 9[SS r O9Y[

q R9VYY ≈ R9VW

χR %&< L α q 9[U8 53 qSN q W9]

χR '&+' t χR %&< ⇒ &'',0% 4Q

)*,.,3#.,8 T&%"# "$ XcScScO9

Example2c )*, 3#++#7"/= %&<+, $*#7$ %*, .,+&%"#/ <,%7,,/ %*, /-1<,.

#3 &''"5,/%$ "/ O 2,&. &/5 %*, &=, #3 %*, 5.";,. "/ & .&/5#1 $&10+,

#3 U[[ 5.";,.$ <,%7,,/ O] &/5 U[9 ),$%8 &% & [O +,;,+ #3

$"=/"3"'&/',8 %*, *20#%*,$"$ %*&% %*, /-1<,. #3 &''"5,/%$ "$

"/5,0,/5,/% #3 %*, 5.";,.e$ &=,9

)*,., &., WU 5.";,.$ <,%7,,/ O] &/5 RU 7*# *&;, /# &''"5,/%$8 OOU

<,%7,,/ RY &/5 V[ 7"%* /# &''"5,/%$8 &/5 $# #/8 $-'* & %&<+, "$ '&++,5

& '#/%"/=,/'2 %&<+,9 ?&'* l<#Bm '#/%&"/"/= & 3.,>-,/'2 "$ '&++,5 & ',++9

)*"$ "$ & S B S '#/%"/=,/'2 %&<+,9



!"#$%&%"$%"'$

RRV

Observed frequencies

H=, #3 5.";,.

_-1<,. #3

&''"5,/%$

O] ` RU RY ` V[ • V[ %#%&+

[ WU OOU OO[ S[[

O U[ YU SU OU[

≥ R RU R[ U U[

)#%&+ OU[ R[[ OU[ U[[

Expected frequencies

H=, #3 5.";,.

_-1<,. #3

&''"5,/%$

O] ` RU RY ` V[ • V[ %#%&+

[ X[ OR[ X[ S[[

O VU Y[ VU OU[

≥ R OU R[ OU U[

)#%&+ OU[ R[[ OU[ U[[

G&+'-+&%"#/ #3 ,B0,'%,5 3.,>-,/'",$c H %#%&+ #3 OU[ 5.";,.$ &=,5 O]`RU8

&/5 S[[@U[[ q S@U #3 &++ 5.";,.$ *&;, *&5 /# &''"5,/%$9 :3 %*,., "$ /#



!"#$%&%"$%"'$

RRU

.,+&%"#/ <,%7,,/ 5.";,. &=, &/5 /-1<,. #3 &''"5,/%$8 7, ,B0,'% %*&%

S@ULOU[N q X[ 5.";,.$ &=,5 O]`RU 7#-+5 *&;, /# &''"5,/%$9 :9,98

,OO qG77

%77"G7 × q X[

E"1"+&.+28,ORL.#7O &/5 '#+-1/ RN q R[[BS[[ @U[[ q OR[

,OS L .#7O &/5 '#+-1/ SN q OU[BS[[ @U[[ q X[

,RR q LR[[BOU[N@U[[ q Y[

,RS q LOU[BOU[N@U[[ q VU

,SO L OU[BU[@U[[N q OU

,SR q LR[[BU[N@U[[ q R[

,SS q LOU[BOU[N@U[[ q OU

Hypothesisc HO : There is no relation between age of driver and

number of accidents

HA : The variables are dependent (related)

The degrees of freedom (df) in a contingency table with R rows

and C columns is:

df = ( R – 1) ( C – 1)

Hence, χ2 tab with df = 4, at .01 level of significance = 13.3



!"#$%&%"$%"'$

RRY

χ2 calc = (75 –90)! /90 + (115 – 120 )! /120 + (110 – 90)! /90 + +

(5 – 15 )! /15

= 1 + 0.208 + 4.444 + 0.556 + 0.417 + 2.222 + 6.667 + 0 + 6.667

= 22. 2 (This corresponds to a P-value of less than .001)

Therefore, there is a relationship between number of accidents

and age of the driver.

]9X9R Fisher’s exact test

)*, '*"`$>-&., %,$% 5,$'."<,5 ,&.+",. "$ & +&.=, $&10+, %,$%9 )*,'#/;,/%"#/&+ '."%,."#/ 3#. %*, χR %,$% %# <, ;&+"5 L0.#0#$,5 <2 D9I9

G#'*.&/ &/5 /#7 7"5,+2 &'',0%,5N $&2$ %*&% &% +,&$% ][ 0,.',/% #3

%*, ,B0,'%,5 3.,>-,/'",$ $*#-+5 ,B',,5 U &/5 &++ %*, ,B0,'%,5

3.,>-,/'",$ $*#-+5 ,B',,5 O9 _#%, %*&% %*"$ '#/5"%"#/ &00+",$ %# %*,

,B0,'%,5 3.,>-,/'",$8 /#% %*, #<$,.;,5 3.,>-,/'",$9 :% "$ >-"%,

&'',0%&<+, 3#. &/ #<$,.;,5 3.,>-,/'2 %# <, [8 0.#;"5,5 %*, ,B0,'%,5

3.,>-,/'",$ 1,,% %*, '."%,."#/9

:3 %*, '."%,."#/ "$ /#% $&%"$3",5 7, '&/ -$-&++2 '#1<"/, #. 5,+,%, .#7$

&/5 '#+-1/$ %# =";, <"==,. ,B0,'%,5 ;&+-,$9 4#7,;,.8 %*"$ 0.#',5-.,

'&//#% <, &00+",5 3#. R <2 R %&<+,$9



!"#$%&%"$%"'$

RRW

:/ & '#10&."$#/ #3 %*, 3.,>-,/'2 #3 #<$,.;&%"#/$ "/ & 3#-.3#+5 %&<+,8 "3

#/, #. 1#., #3 %*, ,B0,'%,5 ;&+-,$ &., +,$$ %*&/ U8 %*, #.5"/&.2 χR f

%,$% '&//#% <, &00+",59

)*, 1,%*#5 -$,5 "/ $-'* $"%-&%"#/$ "$ '&++,5 \"$*,.b$ ,B&'% %,$%9 )*,

,B&'% 0.#<&<"+"%2 5"$%."<-%"#/ 3#. %*, %&<+, '&/ #/+2 <, 3#-/5 7*,/ %*,

.#7 &/5 '#+-1/ %#%&+$ L1&.="/&+ %#%&+$N &., =";,/9

Eg1: E-00#$, 7, '&..2 #-% & '+"/"'&+ %."&+ &/5 .&/5#1+2 &++#'&%, Y

0&%",/%$ %# %.,&%1,/% H &/5 Y %# %.,&%1,/% ! 9)*, #-%'#1, "$ &$

3#++#7$c

).,&%1,/% %20, E-.;";,5 K",5 )#%&+

H S S Y

! U O Y

)#%&+ ] V OR

),$% %*, *20#%*,$"$ %*&% %*,., "$ /# &$$#'"&%"#/ <,%7,,/ %.,&%1,/% &/5

$-.;";&+ &% Ui +,;,+ #3 $"=/"3"'&/',9

H$ '&/ <, #<$,.;,5 3.#1 %*, =";,/ 5&%&8 &++ ,B0,'%,5 3.,>-,/'",$ &.,

+,$$ %*&/ U9 )*,.,3#.,8 7, -$, \"$*,.b$ ,B&'% 0.#<&<"+"%2 %,$%9

\#. %*, =,/,.&+ '&$, 7, '&/ -$, %*, 3#++#7"/= /#%&%"#/c

& < .O

' 5 .R

'O 'R _



!"#$%&%"$%"'$

RR]

)*, ,B&'% 0.#<&<"+"%2 3#. &/2 =";,/ %&<+, "$ /#7 5,%,.1"/,5 3.#1 %*,

3#++#7"/= 3#.1-+&c

.Oz .Rz 'Oz 'Rz @ _z &z <z 'z 5z

)*, ,B'+&1&%"#/ 1&.C 5,/#%,$ l3&'%#."&+m &/5 1,&/$ $-'',$$";,

1-+%"0+"'&%"#/ <2 '&.5"/&+ /-1<,.$ "/ 5,$',/5"/= $,.",$8 %*&% "$ Uz

1,&/$ UBVBSBRBOq OR[8 !2 '#/;,/%"#/ [z q O9

)*,., "$ /# /,,5 %# ,/-1,.&%, &++ %*, 0#$$"<+, %&<+,$9 )*, 0.#<&<"+"%2 #3

%*, #<$,.;,5 #. 1#., ,B%.,1, %&<+,$ &."$"/= <2 '*&/', '&/ <, 3#-/5

3.#1 %*, $"10+, 3#.1-+& =";,/ &<#;,9

^. L#<$,.;,5 %&<+,N q ]z Vz Yz Yz @ ORz Sz Uz Sz Oz q 9RV

^. L1#., ,B%.,1, %&<+,N q ]z Vz Yz Yz @ ORz Rz Yz Vz [z q 9[S

G#/$,>-,/%+28 %*, 0.#<&<"+"%2 %*&% %*, 5"33,.,/', "/ 1#.%&+"%2 <,%7,,/

%*, %7# %.,&%1,/%$ "$ 5-, %# '*&/', "$ 2 x (.24 +.03) = .54

4,/',8 %*, *20#%*,$"$ %*&% %*,., "$ /# &$$#'"&%"#/ <,%7,,/ %.,&%1,/%

&/5 $-.;";&+ '&//#% <, .,J,'%,59



!"#$%&%"$%"'$

RRX

_!c :3 %*, %#%&+ 0.#<&<"+"%2 "$ $1&++ L $&2 +,$$ %*&/ 9[U N %*, 5&%& &.,

"/'#/$"$%,/% 7"%* %*, /-++ *20#%*,$"$ &/5 7, '&/ '#/'+-5, %*&% %*,., "$

,;"5,/', %*&% &/ &$$#'"&%"#/ ,B"$%$9

8.10 Exercises

O9 G#/$"5,. %*, 3#++#7"/= 5&%& #/ +";"/= &.,& #3 1#%*,.$ &/5

<".%*7,"=*%$ LC=N #3 %*,". '*"+5.,/ 7*"'* 7,., .&/5#1+2 %&C,/ 3.#1 %*,

.,'#.5$ #3 & =";,/ *,&+%* ',/%,.9

6#%*,.b$ +";"/=

&.,&

!".%*7,"=*% #3 '*"+5

T-.&+ R9X[ 8 S9R]8 R9S]8 S9[Y8 R9Y[8 R9][

A.<&/ S9OX8 S9R[8 S9RV8 S9OY8 R9XR8 S9Y]8 S9V[8 S9SO8

R9UO8 R9][

),$% %*, *20#%*,$"$ %*&% <&<",$ <#./ %# 1#%*,.$ '#1"/= 3.#1 .-.&+ &/5

-.<&/ &.,&$ *&;, ,>-&+ <".%*7,"=*%$9

LH$$-1, %*&% %*, 5"$%."<-%"#/ "$ /#% $C,7,5 &/5 %&C, %*, +,;,+ #3

$"=/"3"'&/', &$ UiN

R9 Q3 S[ 1,/ ,10+#2,5 "/ & $1&++ 0.";&%, '#10&/2 O] 7#.C,5 "/ #/,

5,0&.%1,/% &/5 OR "/ &/#%*,. 5,0&.%1,/%9 :/ #/, 2,&. U #3 %*, O] 1,/

.,0#.%,5 $"'C 7"%* $,0%"' *&/5$ &/5 #3 %*, OR 1,/ O 5"5 $#9 D*&% "$ %*,



!"#$%&%"$%"'$

RS[

0.#<&<"+"%2 %*&% $-'* & 5"33,.,/', <,%7,,/ $"'C/,$$ .&%,$ "/ %*, %7#

5,0&.%1,/%$ 7#-+5 *&;, &."$,/ <2 '*&/',a



!"#$%&%"$%"'$

RSO

CHAPTER NINE

CORRELATION AND REGRESSION


H% %*, ,/5 #3 %*"$ '*&0%,. %*, $%-5,/% 7"++ <, &<+, %#c

O9 ?B0+&"/ %*, 1,&/"/= &/5 &00+"'&%"#/ #3 +"/,&. '#..,+&%"#/

R9 K"33,.,/%"&%, <,%7,,/ %*, 0.#5-'% 1#1,/% '#..,+&%"#/ &/5

.&/C '#..,+&%"#/

S9 A/5,.$%&/5 %*, '#/',0% #3 $0-."#-$ '#..,+&%"#/

V9 ?B0+&"/ %*, 1,&/"/= &/5 &00+"'&%"#/ #3 +"/,&. .,=.,$$"#/

U9 A/5,.$%&/5 %*, -$, #3 $'&%%,. 5"&=.&1$

Y9 A/5,.$%&/5 %*, 1,%*#5$ #3 +,&$% $>-&.,$

9.2 Introduction

:/ %*"$ '*&0%,. 7, $*&++ $,, %*, .,+&%"#/$*"0$ <,%7,,/ 5"33,.,/%

;&."&<+,$ &/5 '+#$,+2 .,+&%,5 %,'*/">-,$ #3 '#..,+&%"#/ &/5 +"/,&.

.,=.,$$"#/ 3#. "/;,$%"=&%"/= %*, +"/,&. &$$#'"&%"#/ <,%7,,/ %7#

'#/%"/-#-$ ;&."&<+,$9 G#..,+&%"#/ 1,&$-.,$ %*, '+#$,/,$$ #3 %*,

&$$#'"&%"#/8 7*"+, +"/,&. .,=.,$$"#/ =";,$ %*, ,>-&%"#/ #3 %*, $%.&"=*%

+"/, %*&% <,$% 5,$'."<,$ "% &/5 ,/&<+,$ %*, 0.,5"'%"#/ #3 #/, ;&."&<+,

3.#1 %*, #%*,.9 \#. ,B&10+,8 "/ %*, +&<#.&%#.28 *#7 5#,$ &/ &/"1&+b$



!"#$%&%"$%"'$

RSR

.,$0#/$, %# & 5.-= '*&/=, &$ %*, 5#$&=, #3 %*, 5.-= '*&/=,$a :/ %*,

'+"/"'8 "$ %*,., & .,+&%"#/ <,%7,,/ %7# 0*2$"#+#="'&+ #. <"#'*,1"'&+

5,%,.1"/&%"#/$ 1,&$-.,5 "/ %*, $&1, 0&%",/%$a :/ %*, '#11-/"%28

7*&% "$ %*, .,+&%"#/ <,%7,,/ ;&."#-$ "/5"',$ #3 *,&+%* &/5 %*, ,B%,/%

%# 7*"'* *,&+%* '&., "$ &;&"+&<+,a H++ %*,$, >-,$%"#/$ '#/',./ %*,

.,+&%"#/$*"0 <,%7,,/ %7# ;&."&<+,$8 ,&'* 1,&$-.,5 #/ %*, $&1, -/"%$

#3 #<$,.;&%"#/8 <, %*,2 &/"1&+$8 0&%",/%$8 #. '#11-/"%",$9 G#..,+&%"#/

&/5 .,=.,$$"#/ '#/$%"%-%, %*, $%&%"$%"'&+ %,'*/">-,$ 3#. "/;,$%"=&%"/=

$-'* .,+&%"#/$*"0$9

9.3 Correlation Analysis

G#..,+&%"#/ "$ %*, 1,%*#5 #3 &/&+2$"$ %# -$, 7*,/ $%-52"/= %*,

0#$$"<+, &$$#'"&%"#/ <,%7,,/ %7# '#/%"/-#-$ ;&."&<+,$9 :3 7, 7&/% %#

1,&$-., %*, 5,=.,, #3 &$$#'"&%"#/8 %*"$ '&/ <, 5#/, <2 '&+'-+&%"/=

%*, '#..,+&%"#/ '#,33"'",/%9 )*, $%&/5&.5 1,%*#5 L^,&.$#/ '#..,+&%"#/N

+,&5$ %# & >-&/%"%2 '&++,5 . 7*"'* '&/ %&C, &/2 ;&+-, 3.#1 `O %# rO9

)*"$ '#..,+&%"#/ '#,33"'",/% . 1,&$-.,$ %*, 5,=.,, #3 e$%.&"=*%`+"/,e

&$$#'"&%"#/ <,%7,,/ %*, ;&+-,$ #3 %7# ;&."&<+,$9 )*-$ & ;&+-, #3 rO9[

#. `O9[ "$ #<%&"/,5 "3 &++ %*, 0#"/%$ "/ & $'&%%,. 0+#% +", #/ & 0,.3,'%

$%.&"=*% +"/, L$,, 3"=-.,$N9

)*, '#..,+&%"#/ <,%7,,/ %7# ;&."&<+,$ "$ 0#$"%";, "3 *"=*,. ;&+-,$ #3

#/, ;&."&<+, &., &$$#'"&%,5 7"%* *"=*,. ;&+-,$ #3 %*, #%*,. &/5

/,=&%";, "3 #/, ;&."&<+, %,/5$ %# <, +#7,. &$ %*, #%*,. =,%$ *"=*,.9 H



!"#$%&%"$%"'$

RSS

'#..,+&%"#/ #3 &.#-/5 F,.# "/5"'&%,$ %*&% %*,., "$ /# +"/,&. .,+&%"#/

<,%7,,/ %*, ;&+-,$ #3 %*, %7# ;&."&<+,$ L"9,9 %*,2 &., -/'#..,+&%,5N9

D*&% &., 7, 1,&$-."/= 7"%* .a :/ ,$$,/', . "$ & 1,&$-., #3 %*,

$'&%%,. #3 %*, 0#"/%$ &.#-/5 &/ -/5,.+2"/= +"/,&. %.,/5c %*, =.,&%,. %*,$0.,&5 #3 %*, 0#"/%$ %*, +#7,. %*, '#..,+&%"#/9

)*, '#..,+&%"#/ '#,33"'",/% -$-&++2 '&+'-+&%,5 "$ '&++,5 ^,&.$#/e$ . #.

%*, e0.#5-'%`1#1,/%e '#..,+&%"#/ '#,33"'",/% L#%*,. '#,33"'",/%$ &.,

-$,5 3#. .&/C,5 5&%&8 ,%'9N9

:3 7, *&;, %7# ;&."&<+,$ B &/5 28 %*, '#..,+&%"#/ <,%7,,/ %*,1

5,/#%,5 <2 . LB8 2N "$ =";,/ <2

∑ ∑−∑ ∑−

∑ ∑ ∑−=

∑ ∑ −−

∑ −−=

&=eA#UAf&=eQ#UQf

Ae&=QfQ

#A(

UA#Q(

UQ

#A(

#UAQ(

UQ2

$$$$$$

y

7*,., B" &/5 2" &., %*, ;&+-,$ #3 k &/5 o 3#. %*, "%* "/5";"5-&+9

)*, ,>-&%"#/ "$ '+,&.+2 $211,%."' &$ "% 5#,$ /#% 1&%%,. 7*"'* ;&."&<+,

"$ B &/5 7*"'* "$ 2 L %*"$ 5"33,.$ 3.#1 %*, '&$, #3 T,=.,$$"#/ &/&+2$"$N9



!"#$%&%"$%"'$

RSV

Example: T,$%"/= 1,%&<#+"' .&%, LT6TN "$ .,+&%,5 7"%* <#52 7,"=*%9

Body Weight (kg) RMR (kcal/24 hrs)

UW9Y OSRU

YV9X OSYU

UX9R OSVR

Y[9[ OSOY

WR9] OS]R

WW9O OVSX

]R9[ OUSY

]Y9R OVYY

XO9Y OUOX

XX9] OYSX

\".$% 7, $*#-+5 0+#% %*, 5&%& -$"/= $'&%%,. 0+#%$9 :% "$ '#/;,/%"#/&+ %#

0+#% %*, o` .,$0#/$, ;&."&<+, #/ ;,.%"'&+ &B"$ &/5 %*, "/5,0,/5,/%

*#."F#/%&+ &B"$9

)*, 0+#% $*#7$ %*&% <#52 7,"=*% %,/5$ %# <, &$$#'"&%,5 7"%* .,$%"/=

1,%&<#+"' .&%, &/5 ;"', ;,.$&9 )*"$ &$$#'"&%"#/ "$ 1,&$-.,5 <2 %*,

'#..,+&%"#/ '#,33"'",/%8 .9

∑ ∑ −−

∑ −=$$ #AAU#QQU

#AA#UQZUQ 2

7*,., B 5,/#%,$ <#52 7,"=*% &/5 2 5,/#%,$ .,$%"/= 1,%&<#+"' .&%,

LT6TN8 &/5 Q &/5 A &., %*, '#..,$0#/5"/= 1,&/$9 )*, '#..,+&%"#/



!"#$%&%"$%"'$

RSU

'#,33"'",/% "$ &+7&2$ & /-1<,. <,%7,,/ fO &/5 rO8 &/5 ,>-&+$ F,.# "3

%*, ;&."&<+,$ &., /#% L+"/,&.+2N &$$#'"&%,59 :% "$ 0#$"%";, "3 B &/5 2 %,/5

%# <, *"=* #. +#7 %#=,%*,.8 &/5 %*, +&.=,. "%$ ;&+-, %*, '+#$,. %*,

&$$#'"&%"#/9 )*, 1&B"1-1 ;&+-, #3 O "$ #<%&"/,5 "3 %*, 0#"/%$ "/ %*,

$'&%%,. 5"&=.&1 +", ,B&'%+2 #/ & $%.&"=*% +"/,9 G#/;,.$,+28 %*,

'#..,+&%"#/ '#,33"'",/% "$ /,=&%";, "3 *"=* ;&+-,$ #3 2 %,/5 %# =# 7"%*

+#7 ;&+-,$ #3 B8 &/5 ;"', ;,.$&9 It is important to note that acorrelation between two variables shows that they are

associated but does not necessarily imply a ‘cause and effect’

relationship.

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)*, $'&%%,. 0+#% #3 <#52 7,"=*% &/5 T6T $-==,$%$ & +"/,&.

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++ β

XX..

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%&C,$ %*, 3#++#7"/= 3#.1c

y =α

+β

x + e

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Definitionc \#. &/2 +"/,&.`.,=.,$$"#/ ,>-&%"#/ #3 %*, 3#.1 y =α

+β

x

+ e8 2 "$ .,3,..,5 %# &$ %*, dependent ;&."&<+, &/5 B &$ %*,

independent ;&."&<+,8 $"/', 7, &., %.2"/= %# 0.,5"'% 2 3.#1 B9

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7"%* 1,&/α

+β

x &/5 ;&."&/', σ

R

9 :3 σ

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&/5 <".%*5&2N9

L&N β • [ L<N β t [

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In other words, the calculated values for ‘a’ and ‘b’ are sample

estimates of the values of the intercept and slope from the

regression line describing the linear association between x and y

in the whole population. They are, therefore, subject to sampling

variation and their precision is measured by their standard

errors.



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%7# 5,=.,,$ #3 3.,,5#1 <,'&-$, 7, *&;, %# ,$%"1&%, <#%* %*, $+#0,

&/5 %*, "/%,.',0% #3 %*, +"/, 3.#1 %*, 5&%&9 H %`5"$%."<-%"#/ #/ ]

5,=.,,$ #3 3.,,5#1 7"++ *&;, XUi #3 "%$ &.,& <,%7,,/ `R9SO &/5 R9SO9

H %`%,$% "$ -$,5 %# %,$% 7*,%*,. < 5"33,.$ $"=/"3"'&/%+2 3.#1 & $0,'"3",5

;&+-,8 5,/#%,5 <2 β9

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\#. #-. 5&%& $,% %*, '&+'-+&%,5 %`;&+-, "$c

":SROG8S7

7R$S9* =−=

This is very far out in the right-hand tail and is strong evidence

against the hypothesis of no relationship. Notice that the output

table gives the t-value and a p-value. (Remember that p is the

probability of obtaining our result or more extreme given that the

null hypothesis is true (true slope = 0)) Again we would then

follow this with a confidence interval for the slope.



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PREDICTION

DEFINITION: )4? ^T?K:G)?K8 QT ?k^?G)?K8 ZHPA? Q\ o \QT

H I:Z?_ ZHPA? Q\ k8 HE Q!)H:_?K !o )4? T?IT?EE:Q_

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:E HPDHoE Q_ )4? T?IT?EE:Q_ P:_?.

:/ $#1, $"%-&%"#/$ "% 1&2 <, -$,3-+ %# -$, %*, .,=.,$$"#/ ,>-&%"#/ %#

0.,5"'% %*, ;&+-, #3 2 3#. & 0&.%"'-+&. ;&+-, #3 B8 $&2 Bb9 )*, 0.,5"'%,5

;&+-, "$c 2b q & r <Bb &/5 "%$ $%&/5&.5 ,..#. "$

∑ −−++=

$

$

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%*,$, &$$-10%"#/$ 5#,$ /#% *#+58 #. "3 %*, .,+&%"#/$*"0 $,,1$ /#/`

+"/,&.9

PRECAUTIONS IN USE AND INTERPRETATION

O9 )*, .,+&%"#/$*"0 1-$% <, .,0.,$,/%&<+, <2 & $%.&"=*% +"/,9 :/

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1#5,+9

Assumptions of the Model - Validation of the model

)*, "10+"'&%"#/ #3 %*"$ 1#5,+ "$ %*&% &++ %*, $2$%,1&%"' ;&."&%"#/ "$

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APPENDIX : STATISTICAL TABLES

TABLE 4: HT?HE :_ Q_? )H:P Q\ )4? E)H_KHTK _QT6HP

GATZ?

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

9U[[[ 9VXY[ 9VXR[ 9V]][ 9V]V[ 9V][O 9VWYO 9VWRO 9VY]O 9VYVO

9VY[R 9VUYR 9VURR 9VV]S 9VVVS 9VV[V 9VSYV 9VSRU 9VR]Y 9VRVW

9VR[W 9VOY] 9VORX 9V[X[ 9V[UR 9V[OS 9SXWV 9SXSY 9S]XW 9S]UX

9S]RO 9SW]S 9SWVU 9SW[W 9SYYX 9SYSR 9SUXV 9SUUW 9SUR[ 9SV]S

9SVVY 9SV[X 9SSWR 9SSSY 9SS[[ 9SRYV 9SRR] 9SOXR 9SOUY 9SORO

9S[]U 9S[U[ 9S[OU 9RX]O 9RXVY 9RXOR 9R]WW 9R]VS 9R]O[ 9RWWY

9RWVS 9RW[X 9RYWY 9RYVS 9RYOO 9RUW] 9RUVY 9RUOV 9RV]S 9RVUO

9RVR[ 9RS]X 9RSU] 9RSRW 9RRXY 9RRYY 9RRSY 9RR[Y 9ROWW 9ROV]

9ROOX 9R[X[ 9R[YO 9R[SS 9R[[U 9OXWW 9OXVX 9OXRR 9O]XV 9O]YW

9O]VO 9O]OV 9OW]] 9OWYR 9OWSY 9OWOO 9OY]U 9OYY[ 9OYSU 9OYOO

9OU]W 9OUYR 9OUSX 9OUOU 9OVXR 9OVYX 9OVVY 9OVRS 9OV[O 9OSWX

9OSUW 9OSSU 9OSOV 9ORXR 9ORWO 9ORUO 9ORS[ 9ORO[ 9OOX[ 9OOW[

9OOUO 9OOSO 9OOOR 9O[XS 9O[WU 9O[UY 9O[S] 9O[R[ 9O[[S 9[X]U

9[XY] 9[XUO 9[XSV 9[XO] 9[X[O 9[]]U 9[]YX 9[]US 9[]S] 9[]RS

9[][] 9[WXS 9[WW] 9[WYV 9[WVX 9[WSU 9[WRO 9[W[] 9[YXV 9[Y]O



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1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.2

3.4

9[YY] 9[YUU 9[YVS 9[YS[ 9[YO] 9[Y[Y 9[UXV 9[U]R 9[UWO 9[UUX

9[UV] 9[USW 9[URY 9[UOY 9[U[U 9[VXU 9[V]U 9[VWU 9[VYU 9[VUU

9[VVY 9[VSY 9XVRW 9[VO] 9[V[X 9[V[O 9[SXR 9[S]V 9[SWU 9[SYW

9[SUX 9[SUO 9[SVV 9[SSY 9[SRX 9[SRR 9[SOV 9[S[W 9[S[O 9[RXV

9[R]W 9[R]O 9[RWV 9[RY] 9[RYR 9[RUY 9[RU[ 9[RVV 9[RSX 9[RSS

9[RR] 9[RRR 9[ROW 9[ROR 9[R[W 9[R[R 9[OXW 9[OXR 9[O]] 9[O]S

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9[OSX 9[OSY 9[OSR 9[ORX 9[ORU 9[ORR 9[OOX 9[OOY 9[OOS 9[OO[

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9[[VW 9[[VU 9[[VV 9[[VS 9[[VO 9[[V[ 9[[SX 9[[S] 9[[SW 9[[SY

9[[SU 9[[SV 9[[SS 9[[SR 9[[SO 9[[S[ 9[[RX 9[[R] 9[[RW 9[[RY

9[[RY 9[[RU 9[[RV 9[[RS 9[[RS 9[[RR 9[[RO 9[[RO 9[[R[ 9[[OX

9[[OX 9[[O] 9[[O] 9[[OW 9[[OY 9[[OY 9[[OU 9[[OU 9[[OV 9[[OV

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3.6

3.8

4.0

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Table 5: ^,.',/%&=, 0#"/%$ #3 %*, % 5"$%."<-%"#/ L%*"$ %&<+, =";,$ %*,

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Area in two tails

[9R [9O [9[U [9[R [9[O [9[[O

Area in one tail 53

[9O [9[U [9[RU [9[O [9[[U [9[[[U

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O9USS R9OSR R9WWY S9WVW V9Y[V ]9YO[

O9VWY R9[OU R9UWO S9SYU V9[SR Y9]UX

O9VV[ O9XVS R9VVW S9OVS S9W[W U9XUX

O9VOU O9]XU R9SYU R9XX] S9VXX U9V[U

O9SXW O9]Y[ R9S[Y R9]XY S9SUU U9[VO

O9S]S O9]SS R9RYR R9]RO S9RU[ V9W]O

O9SWR O9]OR R9RR] R9WYV S9OYX V9U]W

O9SYS O9WXY R9R[O R9WO] S9O[Y V9VSW

O9SUY O9W]R R9OWX R9Y]O S9[UU V9SO]



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O9SOY O9W[] R9[Y[ R9V]U R9W]W S9WRU

O9SOU O9W[Y R9[UY R9VWX R9WWX S9W[W

O9SOV O9W[S R9[UR R9VWS R9WWO S9YX[

O9SOS O9W[O R9[V] R9VYW R9WYS S9YWV

O9SOO O9YXX R9[VU R9VYR R9WUY S9YUX

O9SO[ O9YXW R9[VR R9VUW R9WU[ S9YVY

O9S[S O9Y]V R9[RO R9VRS R9W[V S9UUO

O9RXY O9YWO R9[[[ R9SX[ R9YY[ S9VY[

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Table 6: ^,.',/%&=, 0#"/%$ #3 %*, '*"`$>-&., 5"$%."<-%"#/ L%*"$ %&<+,

=";,$ %*, ;&+-,$ #3 χƒ 3#. 5"33,."/= 53 %*&% '-% #33 $0,'"3",5 0.#0#.%"#/$

#3 %*, -00,. %&"+ #3 '*"`$>-&., %*, % 5"$%."<-%"#/N

Area in upper tail

K3 [9R [9O [9[U [9[R [9[O [9[[O

O

R

S

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O9YVR R9W[Y S9]VO U9VOR Y9YSU O[9]RW

S9ROX V9Y[U U9XXO W9]RV X9RO[ OS9]OU

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W9R]X X9RSY OO9[W[ OS9S]] OU9[]Y R[9UOW

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OV9YSO OW9RWU OX9YWU RR9YO] RV9WRU SO9RYV

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RS9X[[ RW9R[V S[9OVV SS9Y]W SY9OXO VS9]R[

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References

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,59A/";,.$"%2 ^.,$$8 QB3#.59

S9 H+%1&/8 K9I9 LOXXON9 ^.&'%"'&+ E%&%"$%"'$ 3#. 6,5"'&+ T,$,&.'*8

G*&01&/ &/5 4&++8 P#/5#/9

V9 H.1"%&=,8 ^9 &/5 !,..28 I9 LOX]WN9 E%&%"$%"'&+ 6,%*#5$ "/

6,5"'&+

T,$,&.'*8 R/5 ,59 !+&'C7,++8 QB3#.59

U9 6"'*&,+8 M9 LOXXXN9 6,5"'&+ E%&%"$%"'$c H '#11#/$,/$,

&00.#&'*8

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Y9 \+,%'*,.8 69 LOXXRN9 ^."/'"0+,$ &/5 ^.&'%"', #3 ?0"5,1"#+#=28

H55"$ H<&<&9

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E%&%"$%"'$ 8 D4Q8 I,/,;&

]9 I-0%& G9!9 LOX]ON9 H/ :/%.#5-'%"#/ )# E%&%"$%"'$ 6,%*#5$8 X%*

?59 Z"C$$ ^-<+"$*"/= 4#-$, ^;% P%58 :/5"&9

X9 H<.&1$#/ M949 LOXWXN9 E-.;,2 6,%*#5$ :/ G#11-/"%2

6,5"'"/,8 R

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?59 G*-.'*"++ P";"/=$%#/,8 P#/5#/ &/5 _,7 o#.C9O[9 E7"/$'#7 )9K9Z LOX]YN9 E%&%"$%"'$ H% E>-&., Q/,9 P&%"1,.

).,/5 &/5 G#10&/2 P%58 ^+21#-%*8 I.,&% !."%&"/9

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