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7/23/2019 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
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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.
Important Guidelines for Printing and PhotocopyingLimited permission is granted free of charge to print or photocopy all pages of this
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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List of Tables
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List of Figures
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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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CHAPTER TWO
Methods Of Data Collection, Organization And
Presentation
2.1. Learning Objectives
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2.2. Introduction
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2.3. Data Collection Methods
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2.6 Steps in Designing a Questionnaire
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Step1: CONTENT
Take your objectives and variables as your starting point.
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Step 2: FORMULATING QUESTIONS
Formulate one or more questions that will provide the
information needed for each variable.
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Check whether each question measures one thing at a time.
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Avoid leading questions.
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Step 3: SEQUENCING OF QUESTIONS
Design your interview schedule or questionnaire to be
“consumer friendly.”
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Step 4: FORMATTING THE QUESTIONNAIRE
When you finalize your questionnaire, be sure that:
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Your questionnaire should not only be consumer but also user
friendly!
Step 5: TRANSLATION
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2.7 Methods of data organization and presentation
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2.7.1 Frequency Distributions
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Number of movies Number of persons Relative frequency (%)
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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
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CHAPTER FOUR
DEMOGRAPHY AND HEALTH SERVICES
STATISTICS
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CHAPTER FIVE
ELEMENTARY PROBABILITY AND
PROBABILITY DISTRIBUTIONS
5.1 Learning Objectives
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frequentist definition of Probability.
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The Classical Probability Concept
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5.3 Mutually exclusive events and the additive law
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+ pr(B)+ pr(C) +
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!@H E-.;";, &=, RU %# YU =";,/ $-.;";&+ %# &=, RU a
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5.5 Random variables and probability distributions
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Definitionc H .&/5#1 ;&."&<+, 3#. 7*"'* %*,., ,B"$%$ & 5"$'.,%,
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;&."&<+,9
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Definitionc H .&/5#1 ;&."&<+, 7*#$, ;&+-,$ 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
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?=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 &$
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?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#$$"<+, ;&+-,
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3.#1 %*, ,B0,'%,5 ;&+-, <2 "%$ .,$0,'%";, 0.#<&<"+"%2 &/5 $-11"/=
#;,.&++ %*, ;&+-,$ %*&% *&;, 0#$"%";, 0.#<&<"+"%29
Definitionc )*, ;&."&/', #3 & 5"$'.,%, .&/5#1 ;&."&<+, 5,/#%,5 <2 k
"$ 5,3"/,5 <2
ZLkN q ∑ ∑=
−===−==
]
"
] $(
$((
$(
$
"μ#Q@U5Q#Q@U5#μQUσ
i i
D*,., %*, k"b$ &., %*, ;&+-,$ 3#. 7*"'* %*, .&/5#1 ;&."&<+, %&C,$ #/
0#$"%";, 0.#<&<"+"%29 )*, EK #3 & .&/5#1 ;&."&<+, k8 5,/#%,5 <2
EKLkN #. σ "$ 5,3"/,5 <2 $>-&., .##% #3 "%$ ;&."&/',9
Examplec G#10-%, %*, ;&."&/', &/5 EK 3#. %*, .&/5#1 ;&."&<+,
.,0.,$,/%"/= /-1<,. #3 ,0"$#5,$ #3 5"&..*#,& "/ %*, 3".$% R 2,&.$ #3+"3,9
?LkN q μ q R9[V
#Q@U5=
Q ("(
( =∑=
q [RL9ORXN r ORL9RYVN r RRL9RWON r SRL9O]UN r
VRL9[XUN r URL9[SXN r YRL[9[OWN q Y9OR
)*-$8 ZLkN q Y9OR f LR9[VNR q O9XYW &/5 %*, EK #3 k "$
"S87$"SR9Oσ ==
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""""99 ))44?? !!::__QQ66:: H HPP KK::EE))TT::!!AA))::QQ__
Binomial assumptions:
ON )*, $&1, ,B0,."1,/% "$ '&..",5 #-% / %"1,$ L / %."&+$ &., 1&5,N9
RN ?&'* %."&+ *&$ %7# 0#$$"<+, #-%'#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
. $-'',$$,$ "/ / %."&+$ "$c
2 =2 @#U"@2
= 2#@U5 −−⎟⎟
⎞⎜⎜⎝
⎛ == 8 . q [8 O8 R8 u8 /
)*"$ 0.#<&<"+"%2 5"$%."<-%"#/ "$ '&++,5 %*, <"/#1"&+ 5"$%."<-%"#/9
⎟ ⎠ ⎞
⎜⎝ ⎛
2
= q / G. "$ %*, /-1<,. #3 7&2$ #3 '*##$"/= . "%,1$ 3.#1 /8 &/5 "$
& /-1<,. 7, *&;, %# '&+'-+&%,9 )*, =,/,.&+ 3#.1-+& 3#. %*,
'#,33"'",/%$ ⎟⎟ ⎠
⎞⎜⎜⎝
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= "$ ⎟⎟
⎠
⎞⎜⎜⎝
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=q
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:3 %*, %.-, 0.#0#.%"#/ #3 ,;,/%$ #3 "/%,.,$% "$ ^8 %*,/ "/ & $&10+, #3
$"F, / %*, 1,&/ #3 %*, <"/#1"&+ 5"$%."<-%"#/ "$ n×p &/5 %*, $%&/5&.5
5,;"&%"#/ "$ 4#=4U"−
Example: H$$-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
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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.,$$-.,8 *,"=*% &/5
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7,"=*%N9 )*, /#.1&+ 5"$%."<-%"#/ "$ & %*,#.,%"'&+8 '#/%"/-#-$ 0.#<&<"+"%2
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$
$
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1
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⎟ ⎠ ⎞
⎜⎝ ⎛ −
= σ
μ
σ π
x
3#. `∝ t B t r∝
)*, &.,& %*&% .,0.,$,/%$ %*, 0.#<&<"+"%2 <,%7,,/ %7# 0#"/%$ ' &/5 5
#/ &<$'"$$& "$ 5,3"/,5 <2c
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$
σ
μQ
$
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σπ$
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⎟ ⎠ ⎞
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∫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
5,/$"%28 '&//#% <, %&C,/ &$ %*, 0.#<&<"+"%2 #3 & 0&.%"'-+&. ;&+-,9
)*"$ "$ <,'&-$, 3#. & '#/%"/-#-$ ;&."&<+, %*,., &., "/3"/"%,+2 1&/2
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0#$$"<+, ;&+-,$ $# %*&% %*, 0.#<&<"+"%2 #3 &/2 $0,'"3"' ;&+-, "$
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$%&/5&.5 /#.1&+ 5"$%."<-%"#/ LSNDN 7*"'* *&$ & 0-<+"$*,5 %&<+,9
E"/', %*, ;&+-,$ #3 μ
&/5σ
7"++ 5,0,/5 #/ %*, 0&.%"'-+&. 0.#<+,1 "/*&/5 &/5 %&<+,$ #3 %*, /#.1&+ 5"$%."<-%"#/ '&//#% <, 0-<+"$*,5 3#.
&++ ;&+-,$ #3 μ &/5 σ8 '&+'-+&%"#/$ &., 1&5, <2 .,3,.."/= %# %*,
$%&/5&.5 /#.1&+ 5"$%."<-%"#/ 7*"'* *&$ μ = 0 &/5 σ = 1. )*-$ &/
#<$,.;&%"#/ B 3.#1 & /#.1&+ 5"$%."<-%"#/ 7"%* 1,&/ μ &/5 $%&/5&.5
5,;"&%"#/ σ '&/ <, .,+&%,5 %# & E%&/5&.5 /#.1&+ 5"$%."<-%"#/ <2
'&+'-+&%"/= c
SND = Z = (x - μ ) / σ
Area under any Normal curve
)# 3"/5 %*, &.,& -/5,. & /#.1&+ '-.;, L 7"%* 1,&/ μ
&/5 $%&/5&.5
5,;"&%"#/ σN <,%7,,/ Bq& &/5 Bq<8 3"/5 %*, d $'#.,$ '#..,$0#/5"/= %# &
&/5 < L'&++ %*,1 dO &/5 dRN &/5 %*,/ 3"/5 %*, &.,& -/5,. %*, $%&/5&.5
/#.1&+ '-.;, <,%7,,/ dO &/5 dR 3.#1 %*, 0-<+"$*,5 %&<+,9
Z- Scores
H$$-1, & 5"$%."<-%"#/ *&$ & 1,&/ #3 W[ &/5 & $%&/5&.5 5,;"&%"#/ #3 O[9
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4#7 1&/2 $%&/5&.5 5,;"&%"#/ -/"%$ &<#;, %*, 1,&/ "$ & $'#., #3 ]Sa
d q L]S W[N @ O[ q O9S
The number of standard deviation units is called a Z-score or Z-
value.
:/ =,/,.&+8 d q L.&7 $'#., ` 0#0-+&%"#/ 1,&/N @ 0#0-+&%"#/ EK = (x-μ) /σ
:/ %*, &<#;, 0#0-+&%"#/8 7*&% d`$'#., '#..,$0#/5$ %# & .&7 $'#., Y]a
d q LY]`W[N@O[ q ` [9R
d`$'#.,$ &., "10#.%&/% <,'&-$, =";,/ & d f ;&+-, 7, '&/ 3"/5 #-% %*,
0.#<&<"+"%2 #3 #<%&"/"/= & $'#., %*"$ +&.=, #. +&.=,. L#. %*"$ +#7 #. +#7,.N9
L +##C -0 %*, ;&+-, "/ & F`%&<+,N9 )# +##C -0 %*, 0.#<&<"+"%2 #3 #<%&"/"/= &
d`;&+-, &$ +&.=, #. +&.=,. %*&/ & =";,/ ;&+-,8 +##C -0 %*, 3".$% %7# 5"="%$
#3 %*, d`$'#., "/ %*, +,3% *&/5 '#+-1/ &/5 %*,/ .,&5 %*, *-/5.,5%*$
0+&', &'.#$$ %*, %#09
4,/',8 ^L`O t d t rON q [9Y]RW j ^L`O9XY t d t rO9XYN q [9XU &/5
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\.#1 %*, $211,%.2 0.#0,.%",$ #3 %*, $%&%,5 /#.1&+ 5"$%."<-%"#/8
^Ld ≤ `BN q ^Ld ≥ BN q Of ^LF ≤ BN
Example1c E-00#$, & <#.5,.+"/, *20,.%,/$";, "$ 5,3"/,5 &$ &
0,.$#/ 7*#$, K!^ "$ <,%7,,/ X[ &/5 XU 11 4= "/'+-$";,8 &/5 %*,
$-<J,'%$ &., SU`VV`2,&.`#+5 1&+,$ 7*#$, !^ "$ /#.1&++2 5"$%."<-%,5
7"%* 1,&/ ][ &/5 ;&."&/', OVV9 D*&% "$ %*, 0.#<&<"+"%2 %*&% &
.&/5#1+2 $,+,'%,5 0,.$#/ 3.#1 %*"$ 0#0-+&%"#/ 7"++ <, & <#.5,.+"/,
*20,.%,/$";,a
Solutionc P,% k <, K!^8 k € _L][8 OVVN
^ LX[ t k t XUN q ⎟ ⎠ ⎞
⎜⎝ ⎛ −
<−
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σ
μQ
"$
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q ^ Ld t O9RUN − ^Ld t [9]SN q [9]XVV − [9WXYW q [9[X]
)*-$8 &00.#B"1&%,+2 X9]i #3 %*"$ 0#0-+&%"#/ 7"++ <, <#.5,.+"/,
*20,.%,/$";,9
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&N D*&% 0,.',/% #3 <#2$ "/ %*"$ &=, .&/=, *&;, '&.<#*25.&%, "/%&C,
&<#;, OV[=@O[[[ '&+a
<N D*&% 0,.',/% #3 <#2$ "/ %*"$ &=, .&/=, *&;, '&.<#*25.&%, "/%&C,
<,+#7 X[=@O[[[ '&+a
Solutionc P,% k <, '&.<#*25.&%, "/%&C, "/ OR`OV`2,&.`#+5 1&+,$ &/5
k ∼ _ LORV8 V[[N
&N ^Lk • OV[N q ^Ld • LOV[`ORVN@R[N q ^Ld • [9]N
q O− ^Ld t [9]N q O− [9W]]O q [9ROOX
<N ^Lk t X[N q ^Ld t LX[`ORVN@R[N q ^Ld t `O9WN
q ^Ld • O9WN q O− ^Ld t O9WN q O− [9XUUV q [9[VVY
b. Exercises
O9 H$$-1, %*&% &1#/= 5"&<,%"'$ %*, 3&$%"/= <+##5 +,;,+ #3 =+-'#$, "$
&00.#B"1&%,+2 /#.1&++2 5"$%."<-%, 7"%* & 1,&/ #3 O[U 1= 0,. O[[
1+ &/5 EK #3 X 1= 0,. O[[ 1+9
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3. Answer the following questions by referring to the table of the
standard normal distribution.
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<N :3 d q [9O[8 %*, &.,& %# %*, ."=*% #3 d "$ 9
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CHAPTER SIX
SAMPLING METHODS
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$&10+"/= %,.1"/#+#=",$
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V9 K"33,.,/%"&%, <,%7,,/ .&/5#1 ,..#. &/5 <"&$U9 ?/-1,.&%, &5;&/%&=,$ &/5 +"1"%&%"#/$ #3 %*, 5"33,.,/%
$&10+"/= 1,%*#5$
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5,3"/,5 0#0-+&%"#/9 )*, 0#0-+&%"#/ "$ %## +&.=, 3#. -$ %# '#/$"5,.
'#++,'%"/= "/3#.1&%"#/ 3.#1 &++ "%$ 1,1<,.$9 :3 %*, 7*#+, 0#0-+&%"#/ "$%&C,/ %*,., "$ /# /,,5 #3 $%&%"$%"'&+ "/3,.,/',9 A$-&++28 & .,0.,$,/%&%";,
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.,0.,$,/%&%";, $&10+, *&$ &++ %*, "10#.%&/% '*&.&'%,."$%"'$ #3 %*,
0#0-+&%"#/ 3.#1 7*"'* "% "$ 5.&7/9
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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.#',$$"/= & $&10+, '&/ &'%-&++2 0.#5-',
0.,'"$, .,$-+%$9
If we have to draw a sample, we will be confronted with the
following questionsc
&N D*&% "$ %*, =.#-0 #3 0,#0+, L0#0-+&%"#/N 3.#1 7*"'* 7, 7&/% %#
5.&7 & $&10+,a
<N 4#7 1&/2 0,#0+, 5# 7, /,,5 "/ #-. $&10+,a
'N 4#7 7"++ %*,$, 0,#0+, <, $,+,'%,5a
Apart from persons, a population may consist of mosquitoes,
villages, institutions, etc.
6.3 Common terms used in sampling
Reference population (also called source population or target
population) - %*, 0#0-+&%"#/ #3 "/%,.,$%8 %# 7*"'* %*, "/;,$%"=&%#.$
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7#-+5 +"C, %# =,/,.&+"F, %*, .,$-+%$ #3 %*, $%-528 &/5 3.#1 7*"'* &
.,0.,$,/%&%";, $&10+, "$ %# <, 5.&7/9
Study or sample population ` %*, 0#0-+&%"#/ "/'+-5,5 "/ %*, $&10+,9
Sampling unit ` %*, -/"% #3 $,+,'%"#/ "/ %*, $&10+"/= 0.#',$$
Study unit ` %*, -/"% #/ 7*"'* "/3#.1&%"#/ "$ '#++,'%,59
` %*, $&10+"/= -/"% "$ /#% /,',$$&."+2 %*, $&1, &$ %*, $%-52 -/"%9
` "3 %*, #<J,'%";, "$ %# 5,%,.1"/, %*, &;&"+&<"+"%2 #3 +&%."/,8 %*,/ %*,
$%-52 -/"% 7#-+5 <, %*, *#-$,*#+5j "3 %*, #<J,'%";, "$ %# 5,%,.1"/, %*,
0.,;&+,/', #3 %.&'*#1&8 %*,/ %*, $%-52 -/"% 7#-+5 <, %*, "/5";"5-&+9
Sampling frame ` %*, +"$% #3 &++ %*, -/"%$ "/ %*, .,3,.,/', 0#0-+&%"#/8
3.#1 7*"'* & $&10+, "$ %# <, 0"'C,59
Sampling fraction (Sampling interval) ` %*, .&%"# #3 %*, /-1<,. #3
-/"%$ "/ %*, $&10+, %# %*, /-1<,. #3 -/"%$ "/ %*, .,3,.,/', 0#0-+&%"#/
(n/N)
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6.4 Sampling methods (Two broad divisions)
6.4.1 Non-probability Sampling Methods
` A$,5 7*,/ & $&10+"/= 3.&1, 5#,$ /#% ,B"$%
` _# .&/5#1 $,+,'%"#/ L-/.,0.,$,/%&%";, #3 %*, =";,/
0#0-+&%"#/N
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3"/5"/=$ #<%&"/,5 3.#1 & $&10+, %# %*, 0#0-+&%"#/9
Two such non-probability sampling methods are:
A) Convenience samplingc "$ & 1,%*#5 "/ 7*"'* 3#. '#/;,/",/',
$&C, %*, $%-52 -/"%$ %*&% *&00,/ %# <, &;&"+&<+, &% %*, %"1, #3 5&%&
'#++,'%"#/ &., $,+,'%,59
B) Quota samplingc "$ & 1,%*#5 %*&% ,/$-.,$ %*&% & ',.%&"/ /-1<,. #3
$&10+, -/"%$ 3.#1 5"33,.,/% '&%,=#.",$ 7"%* $0,'"3"' '*&.&'%,."$%"'$
&., .,0.,$,/%,59 :/ %*"$ 1,%*#5 %*, "/;,$%"=&%#. "/%,.;",7$ &$ 1&/2
0,#0+, "/ ,&'* '&%,=#.2 #3 $%-52 -/"% &$ *, '&/ 3"/5 -/%"+ *, *&$
3"++,5 *"$ >-#%&9
Both the above methods do not claim to be representative of the
entire population9
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6.4.2 Probability Sampling methods
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$*#-+5 *&;, &/ ,>-&+ #. &% +,&$% & C/#7/ '*&/', #3 <,"/= "/'+-5,5
"/ %*, $&10+,9
` I,/,.&+"F&%"#/ "$ 0#$$"<+, L3.#1 $&10+, %# 0#0-+&%"#/N
A) Simple random sampling (SRS)
` )*"$ "$ %*, 1#$% <&$"' $'*,1, #3 .&/5#1 $&10+"/=9
` ?&'* -/"% "/ %*, $&10+"/= 3.&1, *&$ &/ ,>-&+ '*&/', #3 <,"/=
$,+,'%,5
` .,0.,$,/%&%";,/,$$ #3 %*, $&10+, "$ ,/$-.,59
4#7,;,.8 "% "$ '#$%+2 %# '#/5-'% ETE9 6#.,#;,.8 1"/#."%2 $-<=.#-0$ #3
"/%,.,$% "/ %*, 0#0-+&%"#/ 12 /#% <, 0.,$,/% "/ %*, $&10+, "/ $-33"'",/%
/-1<,.$ 3#. $%-529
To select a simple random sample you need to:
•
6&C, & /-1<,.,5 +"$% #3 &++ %*, -/"%$ "/ %*, 0#0-+&%"#/ 3.#1 7*"'*
2#- 7&/% %# 5.&7 & $&10+,9
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_ L7*,., _ "$ %*, $"F, #3 %*, 0#0-+&%"#/N
• K,'"5, #/ %*, $"F, #3 %*, $&10+,
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• E,+,'% %*, .,>-".,5 /-1<,. #3 $%-52 -/"%$8 -$"/= & l+#%%,.2m
1,%*#5 #. & %&<+, #3 .&/5#1 /-1<,.$9
hLottery” methodc 3#. & $1&++ 0#0-+&%"#/ "% 1&2 <, 0#$$"<+, %# -$, %*,
l+#%%,.2m 1,%*#5c ,&'* -/"% "/ %*, 0#0-+&%"#/ "$ .,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
&1#/= %*, 3".$% C -/"%$9 \#. ,B&10+,8 & $2$%,1&%"' $&10+, "$ %# <,
$,+,'%,5 3.#1 OR[[ $%-5,/%$ #3 & $'*##+9 )*, $&10+, $"F, "$ 5,'"5,5 %#
<, O[[9 )*, $&10+"/= 3.&'%"#/ "$c O[[ @OR[[ q O@OR9 4,/',8 %*, $&10+,
"/%,.;&+ "$ OR9
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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
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%# 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#$$"<+, %#
0.,5"'% "/ &5;&/', 7*# 7"++ <, &%%,/5"/=9
Demerits
• :3 %*,., "$ &/2 $#.% #3 '2'+"' 0&%%,./ "/ %*, #.5,."/= #3 %*, $-<J,'%$
7*"'* '#"/'"5,$ 7"%* %*, $&10+"/= "/%,.;&+8 %*, $&10+, 7"++ /#% <,
.,0.,$,/%&%";, #3 %*, 0#0-+&%"#/9
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Examples
` P"$% #3 1&..",5 '#-0+,$ &..&/=,5 7"%* 1,/e$ /&1,$ &+%,./&%";,+2
7"%* %*, 7#1,/e$ /&1,$ L,;,.2 R/58 V%* 8 ,%'9N 7"++ .,$-+% "/ &
$&10+, #3 &++ 1,/ #. 7#1,/N9
` :3 7, 7&/% %# $,+,'% & .&/5#1 $&10+, #3 & ',.%&"/ 5&2 L$&10+"/=
3.&'%"#/ #/ 7*"'* %# '#-/% '+"/"' &%%,/5&/',8 %*"$ 5&2 1&2 3&++ #/ %*,
$&1, 5&2 #3 %*, 7,,C8 7*"'* 1"=*%8 3#. ,B&10+, <, & 1&.C,% 5&29
C) Stratified Sampling
:% "$ &00.#0."&%, 7*,/ %*, 5"$%."<-%"#/ #3 %*, '*&.&'%,."$%"' %# <,
$%-5",5 "$ $%.#/=+2 &33,'%,5 <2 ',.%&"/ ;&."&<+, L*,%,.#=,/,#-$
0#0-+&%"#/N9 )*, 0#0-+&%"#/ "$ 3".$% 5";"5,5 "/%# =.#-0$ L$%.&%&N
&''#.5"/= %# & '*&.&'%,."$%"' #3 "/%,.,$% L,=98 $,B8 =,#=.&0*"' &.,&8
0.,;&+,/', #3 5"$,&$,8 ,%'9N9 H $,0&.&%, $&10+, "$ %*,/ %&C,/
"/5,0,/5,/%+2 3.#1 ,&'* $%.&%-18 <2 $"10+, .&/5#1 #. $2$%,1&%"'
$&10+"/=9
• proportional allocation ` "3 %*, $&1, $&10+"/= 3.&'%"#/ "$ -$,5 3#.
,&'* $%.&%-19
• non- proportional allocation ` "3 & 5"33,.,/% $&10+"/= 3.&'%"#/ "$ -$,5
3#. ,&'* $%.&%-1 #. "3 %*, $%.&%& &., -/,>-&+ "/ $"F, &/5 & 3"B,5
/-1<,. #3 -/"%$ "$ $,+,'%,5 3.#1 ,&'* $%.&%-19
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Merit
` )*, .,0.,$,/%&%";,/,$$ #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 $-'* '+-$%,.$ "$ $,+,'%,5
` H++ %*, -/"%$ "/ %*, $,+,'%,5 '+-$%,.$ &., $%-5",5
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:% "$ 0.,3,.&<+, %# $,+,'% & +&.=, /-1<,. #3 $1&++ '+-$%,.$ .&%*,. %*&/ &
$1&++ /-1<,. #3 +&.=, '+-$%,.$9
Merit
H +"$% #3 &++ %*, "/5";"5-&+ $%-52 -/"%$ "/ %*, .,3,.,/', 0#0-+&%"#/ "$ /#%
.,>-".,59 :% "$ $-33"'",/% %# *&;, & +"$% #3 '+-$%,.$9
Demerit
:% "$ <&$,5 #/ %*, &$$-10%"#/ %*&% %*, '*&.&'%,."$%"' %# <, $%-5",5 "$
-/"3#.1+2 5"$%."<-%,5 %*.#-=*#-% %*, .,3,.,/', 0#0-+&%"#/8 7*"'* 1&2
/#% &+7&2$ <, %*, '&$,9 4,/',8 $&10+"/= ,..#. "$ -$-&++2 *"=*,. %*&/
3#. & $"10+, .&/5#1 $&10+, #3 %*, $&1, $"F,9
E) Multi-stage sampling
)*"$ 1,%*#5 "$ &00.#0."&%, 7*,/ %*, .,3,.,/', 0#0-+&%"#/ "$ +&.=, &/5
7"5,+2 $'&%%,.,5 9 E,+,'%"#/ "$ 5#/, "/ $%&=,$ -/%"+ %*, 3"/&+ $&10+"/= -/"%
L,=98 *#-$,*#+5$ #. 0,.$#/$N &., &..";,5 &%9 )*, 0."1&.2 $&10+"/= -/"%
L^EAN "$ %*, $&10+"/= -/"% L-$-&++2 +&.=, $"F,N "/ %*, 3".$% $&10+"/=$%&=,9 )*, $,'#/5&.2 $&10+"/= -/"% LEEAN "$ %*, $&10+"/= -/"% "/ %*,
$,'#/5 $&10+"/= $%&=,8 ,%'9
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Example - )*, ^EA$ '#-+5 <, kebeles &/5 %*, EEA$ '#-+5 <,
*#-$,*#+5$9
Merit - G-%$ %*, '#$% #3 0.,0&."/= $&10+"/= 3.&1,
Demerit - E&10+"/= ,..#. "$ "/'.,&$,5 '#10&.,5 7"%* & $"10+, .&/5#1
$&10+,9
6-+%"$%&=, $&10+"/= =";,$ +,$$ 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.#',$$ "%$,+38 "$ '&++,5 $&10+"/= ,..#.8
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7*"'* "$ & 3#.1 #3 .&/5#1 ,..#.9 E&10+"/= ,..#. '&/ <, 1"/"1"F,5 <2
"/'.,&$"/= %*, $"F, #3 %*, $&10+,9 D*,/ / q _ ⇒ $&10+"/= ,..#. q [
6.5.2 Non-sampling error (bias)
:% "$ & %20, #3 $2$%,1&%"' ,..#. "/ %*, 5,$"=/ #. '#/5-'% #3 & $&10+"/=
0.#',5-., 7*"'* .,$-+%$ "/ 5"$%#.%"#/ #3 %*, $&10+,8 $# %*&% "% "$ /#
+#/=,. .,0.,$,/%&%";, #3 %*, .,3,.,/', 0#0-+&%"#/9 D, '&/ ,+"1"/&%, #.
.,5-', %*, /#/`$&10+"/= ,..#. L<"&$N <2 '&.,3-+ 5,$"=/ #3 %*, $&10+"/=
0.#',5-., &/5 /#% <2 "/'.,&$"/= %*, $&10+, $"F,9
Example: :3 2#- %&C, 1&+, $%-5,/%$ #/+2 3.#1 & $%-5,/% 5#.1"%#.2 "/
?%*"#0"& "/ #.5,. %# 5,%,.1"/, %*, 0.#0#.%"#/ #3 $1#C,.$8 2#- 7#-+5
.,$-+% "/ &/ #;,.,$%"1&%,8 $"/', 3,1&+,$ &., +,$$ +"C,+2 %# $1#C,9
:/'.,&$"/= %*, /-1<,. #3 1&+, $%-5,/%$ 7#-+5 /#% .,1#;, %*, <"&$9
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<, $%-5",59
• _#/ .,$0#/$, .,$-+%$ "/ $"=/"3"'&/% <"&$ 7*,/ %*, 3#++#7"/= %7#
'#/5"%"#/$ &., <#%* 3-+3"++,59
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$&10+, L&<#-% OUi #. 1#.,N
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• )*,., &., $,;,.&+ 7&2$ %# 5,&+ 7"%* %*"$ 0.#<+,1 &/5 .,5-', %*,
0#$$"<"+"%2 #3 <"&$c
&N K&%& '#++,'%"#/ %##+$ L>-,$%"#//&".,N *&;, %# <, 0.,`%,$%,59
<N :3 /#/ .,$0#/$, "$ 5-, %# &<$,/', #3 %*, $-<J,'%$8 .,0,&%,5
&%%,10%$ $*#-+5 <, '#/$"5,.,5 %# '#/%&'% $%-52 $-<J,'%$ 7*#
7,., &<$,/% &% %*, %"1, #3 %*, "/"%"&+ ;"$"%9
'N )# "/'+-5, &55"%"#/&+ 0,#0+, "/ %*, $&10+,8 $# %*&% /#/`
.,$0#/5,/%$ 7*# 7,., &<$,/% 5-."/= 5&%& '#++,'%"#/ '&/ <,
.,0+&',5 L1&C, $-., %*&% %*,". &<$,/', "$ /#% .,+&%,5 %# %*, %#0"'
<,"/= $%-5",5N9
NB: The number of non-responses should be documented
according to type, so as to facilitate an assessment of the extent of
bias introduced by non-response.
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CHAPTER SEVEN
ESTIMATION
7.1 Learning objectives
H% %*, ,/5 #3 %*"$ '*&0%,. %*, $%-5,/% 7"++ <, &<+, %#c
O9 A/5,.$%&/5 %*, '#/',0%$ #3 $&10+, $%&%"$%"'$ &/5 0#0-+&%"#/
0&.&1,%,.$
R9 A/5,.$%&/5 %*, 0."/'"0+,$ #3 $&10+"/= 5"$%."<-%"#/$ #3 1,&/$ &/5
0.#0#.%"#/$ &/5 '&+'-+&%, %*,". $%&/5&.5 ,..#.$
S9 A/5,.$%&/5 %*, 0."/'"0+,$ #3 ,$%"1&%"#/ &/5 5"33,.,/%"&%, <,%7,,/
0#"/% &/5 "/%,.;&+ ,$%"1&%"#/$
V9 G#10-%, &00.#0."&%, '#/3"5,/', "/%,.;&+$ 3#. 0#0-+&%"#/ 1,&/$
&/5 0.#0#.%"#/$ &/5 "/%,.0.,% %*, 3"/5"/=$
U9 K,$'."<, 1,%*#5$ #3 $&10+, $"F, '&+'-+&%"#/ 3#. '.#$$ 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 "$
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"/%.#5-',59 6,%*#5$ 3#. '&+'-+&%"/= '#/3"5,/', "/%,.;&+$ 3#.
0#0-+&%"#/ 1,&/$ &/5 0.#0#.%"#/$ &., =";,/9 )*, "10#.%&/', #3 %*,
/#.1&+ 5"$%."<-%"#/ Ld 5"$%."<-%"#/N "$ $%.,$$,5 %*.#-=*#-% %*, '*&0%,.9
7.3 Point Estimation
Definitionc H 0&.&1,%,. "$ & /-1,."'&+ 5,$'."0%";, 1,&$-., #3 &
0#0-+&%"#/ L μ "$ &/ ,B&10+, #3 & 0&.&1,%,.N9 H $%&%"$%"' "$ & /-1,."'&+
5,$'."0%";, 1,&$-., #3 & $&10+, L X "$ &/ ,B&10+, #3 & $%&%"$%"'N9
)# ,&'* $&10+, $%&%"$%"' %*,., '#..,$0#/5$ & 0#0-+&%"#/ 0&.&1,%,.9 D,
-$, X 8 ER8 E 8 08 ,%'9 %# ,$%"1&%, μ8 σR8 σ8 ^ L#. πN8 ,%'9
Sample statistic Corresponding population parameter
X L$&10+, 1,&/N μ L0#0-+&%"#/ 1,&/N
ER L $&10+, ;&."&/',N σR L 0#0-+&%"#/ ;&."&/',N
E L$&10+, E%&/5&.5 5,;"&%"#/N σL0#0-+&%"#/ $%&/5&.5 5,;"&%"#/N
0 L $&10+, 0.#0#.%"#/N ^ #. π L^#0-+&%"#/ 0.#0#.%"#/N
D, *&;, &+.,&52 $,,/ %*&% %*, 1,&/ X #3 & $&10+, '&/ <, -$,5 %#
,$%"1&%, μ9)*"$ 5#,$ /#%8 #3 '#-.$,8 "/5"'&%, %*&% %*, 1,&/ #3 ,;,.2
$&10+, 7"++ ,>-&+ %*, 0#0-+&%"#/ 1,&/9
K,3"/"%"#/c H 0#"/% ,$%"1&%, #3 $#1, 0#0-+&%"#/ 0&.&1,%,. Q "$ & $"/=+,
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;&+-, ‚ #3 & $&10+, $%&%"$%"'9
?=9 )*, 1,&/ $-.;";&+ %"1, #3 XO +&<#.&%#.2 .&%$ &3%,. .,1#;&+ #3 %*,
%*2.#"5 =+&/5 7&$ ]R 5&2$ 7"%* & $%&/5&.5 5,;"&%"#/ #3 O[ 5&2$ L&$$-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+&'"/= %*,
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#<$,.;&%"#/$ "/ %*, 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
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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 $&10+, 1,&/ N
LO[8 R[N #. LR[8 O[N OU
LO[8 S[ N #. LS[8 O[N R[
LO[8 V[N #. LV[8 O[N RU
LR[8 S[N #. LS[8 R[N RU
LR[8 V[N #. LV[8 R[N S[
LS[8 V[N #. LV[8 S[N SU
LO[8 O[N O[
LR[8 R[N R[
LS[8 S[N S[
LV[8 V[N V[
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&N 3.,>-,/'2 5"$%."<-%"#/ #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
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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 %*, $%&%,1,/% ^. L`O9XY≤
d ≤O9XYN q 9 XU
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)*"$ "$ 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,/', "/%,.;&+
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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
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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
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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
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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 $&10+, #3 YU
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HN D*&% "$ %*, 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,&.$
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^#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$
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7#1,/9
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7.5.3 Confidence interval for a single proportion
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0 q 0.#0#.%"#/ #3 $-'',$$,$ "/ & $&10+,
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σ0q E%&/5&.5 5,;"&%"#/ #3 %*, $&10+"/= 5"$%."<-%"#/ #3
0.#0#.%"#/$
q E%&/5&.5 ,..#. #3 0.#0#.%"#/$
/ q $"F, #3 %*, $&10+,
)*, 0#0-+&%"#/ .,0.,$,/%$ '&%,=#."'&+ 5&%& 7*"+, %*, $'#.,$ "/ %*,
$&10+"/= 5"$%."<-%"#/ &., 0.#0#.%"#/$ <,%7,,/ [ &/5 O9)*"$ $,% #3
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O9 :%$ 1,&/ q ^8 %*, 0.#0#.%"#/ "/ %*, 0#0-+&%"#/9
R9 σ0 q n PQ &
S )*, $*&0, "$ &00.#B"1&%,+2 /#.1&+ 0.#;"5,5 / "$ $-33"'",/%+2 +&.=, `
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+&.=, / L ',/%.&+ +"1"% %*,#.,1 3#. 0.#0#.%"#/$N 9
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3#.1-+&c
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0 q $&10+, 0.#0#.%"#/8 σ0 q $%&/5&.5 ,..#. #3 %*, 0.#0#.%"#/ L q
n PQ & N9
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%.,/5 #3 1&+&."& "/ & ',.%&"/ +#'&+"%2 &/5 7&/%$ %# ,$%"1&%, %*,
0.#0#.%"#/ #3 0,.$#/$ "/3,'%,5 "/ %*, 0,&C 1&+&."& %.&/$1"$$"#/ 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
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$"%-&%"#/$8 n pq & '&/ <, -$,5 &$ &/ &00.#B"1&%"#/ %# σ0 q
"G7	S8US x q 9[V
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"/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"$$"#/ 3#. R 2,&.$j <-% #/+2 Y[ "/ %*, $,'#/5 =.#-09 \"/5 XUi
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'#/3"5,/', +"1"%$ 3#. %*, 5"33,.,/', "/ %*, 0.#0#.%"#/ #3 &++ 0&%",/%$ 7"%*
+,-C&,1"& 7*# *&;, .,1"$$"#/ 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 "$
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%## $1&++ 7, 1&2 3&"+ %# 5,%,'% "10#.%&/% ,33,'%$8 #. 1&2 ,$%"1&%,
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.,$#-.',$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
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• %*, 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 %*,
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&<#;, ,$%"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+, $"F, 7#-+5 <, OVY 0,.$#/$9
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NB: σR '&/ <, ,$%"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&$$-1, & '#/3"5,/', +,;,+ #3 XUiN
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/ q LO9R]rO9XYNR LL9S×9WN rL9OU ×9]UNN @ L9S[ ` 9OUNR q OU]
158 /-.$,$ &., .,>-".,5 "/ ,&'* .,="#/
7.6.4 Comparison of two means (sample size in each group)
/ q L$OR r $R
RN 3Lα8βN @ L1O ` 1RNR
1O &/5 $OR &., 1,&/ &/5 ;&."&/', #3 =.#-0 O .,$0,'%";,+29
1R &/5 $RR &., 1,&/ &/5 ;&."&/', #3 =.#-0 R .,$0,'%";,+29
?=9 )*, <".%* 7,"=*%$ "/ 5"$%."'%$ H &/5 ! 7"++ <, '#10&.,59 :/ 5"$%."'% H
%*, 1,&/ <".%* 7,"=*% "$ ,B0,'%,5 %# <, S[[[ =.&1$ 7"%* & $%&/5&.5
5,;"&%"#/ #3 U[[ =.&1$9 :/ 5"$%."'% ! %*, 1,&/ "$ ,B0,'%,5 %# <, SR[[
=.&1$ 7"%* & $%&/5&.5 5,;"&%"#/ #3 U[[ =.&1$9 )*, .,>-".,5 $&10+,
$"F, %# 5,1#/$%.&%, L7"%* & +"C,+"*##5 #3 X[i 8 %*&% "$ 7"%* & 0#7,. #3
X[iN & $"=/"3"'&/% 5"33,.,/', <,%7,,/ %*, 1,&/ <".%* 7,"=*%$ "/ 5"$%."'%$
H &/5 ! 7#-+5 <,c
N = (1.96 + 1.28 )! (500 + 500)! / (3200 – 3000)!
= 131 newborn babies in each district
Note that f(α
,β
) = 10.5
That is , α = . 05 (two sided ) ⇒ Z = 1.96
β = ( 1- .9 ) = .1 (one sided ) ⇒ Z = 1.28
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7.7 Exercises
O9 Q3 VU 0&%",/%$ %.,&%,5 <2 & O *#-. *20/#$"$ $,$$"#/ %# C"'C %*,
$1#C"/= *&<"%8 SY $%#00,5 $1#C"/=8 &% +,&$% 3#. %*, 1#1,/%9 \"/5 &
XUi G9:9 3#. %*, 0.#0#.%"#/ #3 &++ $1#C,.$ 7*# >-"% &3%,. '*##$"/=%*"$ %20, #3 %.,&%1,/%9 L%*, 0&%",/%$ 7,., $,+,'%,5 .&/5#1+2N9
H XUi '#/3"5,/', "/%,.;&+ 3#. %*, 0#0-+&%"#/ 0.#0#.%"#/ L"9,98
0.#0#.%"#/ #3 &++ $1#C,.$ 7*# >-"% $1#C"/= &3%,. '*##$"/= *20/#$"$N
"$ L[9Y] %# [9XRN9
R9 H *#$0"%&+ &51"/"$%.&%#. 7"$*,$ %# C/#7 7*&% 0.#0#.%"#/ #3
5"$'*&.=,5 0&%",/%$ &., -/*&002 7"%* %*, '&., .,',";,5 5-."/=*#$0"%&+"F&%"#/9 :3 XUi G#/3"5,/', "/%,.;&+ "$ 5,$".,5 %# ,$%"1&%, %*,
0.#0#.%"#/ 7"%*"/ Ui8 *#7 +&.=, & $&10+, $*#-+5 <, 5.&7/a
/ q dR 0LO`0N@7R qLO9XYNRL9U×9UN@L9[UNR qS]V9R ≈ S]U 0&%",/%$
NB :3 2#- 5#/b% *&;, &/2 "/3#.1&%"#/ &<#-% ^8 %&C, "% &$ U[i &/5 =,%
%*, 1&B"1-1 ;&+-, #3 ^p 7*"'* "$ O@V LRUiN9
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CHAPTER EIGHT
HYPOTHESIS TESTING
8.1 Learning objectives
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8.2 Introduction
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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
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In general, hypothesis testing in statistics involves the following
steps:
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8.3 The null and alternative hypotheses
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"/,>-&+"%2 L μ ≥ OV #. μ t OVN9
Examples ON 4Q c μ q O9YS 1 L3.#1 %*, 0.,;"#-$ ,B&10+,N9
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Choosing the Alternative Hypothesis (HA)
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:3 %*, 1,&/ *,"=*% #3 %*, IG6E $%-5,/%$ L 4 Q c μ q O9YS 1 N "$
>-,$%"#/,58 %*,/ %*, &+%,./&%";, *20#%*,$"$ L4 HN "$ $,% μ ≠ OYS 19 Q%*,.
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4 H c μ • O9YS 19
4 H c μ t O9YS 19
Possible choices of HA
If HO is then HA is
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^ q ! L$"/=+, 0.#0#.%"#/N ^ ≠ ! #. ^ t ! #. ^ • !
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• G
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• K
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Consider the previous example (patients with leukaemia)
HO : P = .60
HA : P > .60
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7"++ '#/%"/-, *"$ %,$%$9
Exercises
State HO and HA for each of the following
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8.4 Level of significance
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3&+$, &/5 7, 3&"+ %# .,J,'% "%9
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"$ -$,5 %# .,J,'% #. &'',0% L3&"+ %# .,J,'%N %*, /-++ *20#%*,$"$9 :% "$ /#%
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.&/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
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%*#$, +"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)
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The most common values of z are:
α
.10 .05 .01
Z± 1.64 ± 1.96 ±
2.58
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.,="#/$8 %*, +"1"%$ 5,%,.1"/,5 "/ $%,0 R 7"++ <, .,3,..,5 %# &$ %*, '."%"'&+
;&+-,$ #3 d9
8.5 Tests of Significance on means and Proportions (large
samples)
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.,$-+% &/5 %*, '#..,$0#/5"/= /-++ *20#%*,$"$ 0#0-+&%"#/ ;&+-,a H l2,$m
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P – Values
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8.5.1 Tests of significance on a single mean and comparison of
two means
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A statistical test of significance on a single mean
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#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
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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.
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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 %*, ^`;&+-, &$$#'"&%,5 7"%* &/ #<$,.;,5
L'&+'-+&%,5N ;&+-, #3 d 7*"'* "$ O9Y9 \.#1 %*, %&<+, 1,/%"#/,5 &<#;,
7, =,% ^q9OO L/#%, %*&% %*"$ 0`;&+-, "$ =.,&%,. %*&/ %*, =";,/ +,;,+ #3
$"=/"3"'&/',N9
Comparison of two Means
)*, 0-.0#$, #3 %*"$ $,'%"#/ "$ %# ,B%,/5 %*, &.=-1,/%$ #3 %*, $"/=+,
1,&/ %# %*, '#10&."$#/ #3 %7# $&10+, 1,&/$9 E"/', 1,5"'"/, L0-<+"'
*,&+%*N "$8 <2 /&%-.,8 '#10&.&%";,8 %*"$ "$ & .&%*,. 7"5,$0.,&5 $"%-&%"#/8
1#., '#11#/ %*&/ %*&% #3 %*, $"/=+, 1,&/ #3 %*, 0.,',5"/= $,'%"#/9
:/ %*, '#10&."$#/ #3 %7# 1,&/$8 %*,., &., %7# $&10+,$ #3 #<$,.;&%"#/$
3.#1 %7# -/5,.+2"/= 0#0-+&%"#/$ L#3%,/ %.,&%1,/% &/5 '#/%.#+ =.#-0$N
7*#$, 1,&/$ &., 5,/#%,5 <2 μ% &/5 μ' &/5 7*#$, $%&/5&.5 5,;"&%"#/$
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&., 5,/#%,5 <2 σ% &/5 σ' 9 T,'&++"/= %*&% & %,$% #3 $"=/"3"'&/', "/;#+;,$ &
/-++ *20#%*,$"$ %*&% $0,'"3",$ ;&+-,$ 3#. 0#0-+&%"#/ >-&/%"%",$8 %*,
.,+,;&/% /-++ *20#%*,$"$ "$ %*&% %*, 1,&/$ &., "5,/%"'&+8 "9,98
HO :μt = μc or HO :
μt - μc = 0
)*, .&%"#/&+, 3#. %*, %,$% #3 $"=/"3"'&/', "$ &$ <,3#.,9 H$$-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,&.$
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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.,$$"#/ =";,/ "/ $,'%"#/ W9U9S &00+",$ "/ & $"1"+&. 7&2
%# %*"$ $,'%"#/9
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Example: H1#/= $-$',0%"<+, "/5";"5-&+$ ,B0#$,5 %# & 0&.%"'-+&.
"/3,'%"#-$ &=,/%8 SY 0,.',/% =,/,.&++2 5,;,+#0 '+"/"'&+ 5"$,&$,9 H1#/=
& $'*##+ =.#-0 #3 OVV 0,.$#/$ $-$0,'%,5 #3 ,B0#$-., %# %*, &=,/%8 #/+2
SU 5,;,+#0,5 '+"/"'&+ 5"$,&$,9 :$ %*"$ .,$-+% 7"%*"/ '*&/', ;&."&%"#/ Lα q
9[UN9
)# -$, %*, %&<+, #3 %*, $%&/5&.5 /#.1&+ 5"$%."<-%"#/8 #/, /,,5$ %#
'&+'-+&%, '."%"'&+ .&%"#$9 )*,., "$8 *#7,;,.8 &/ &55"%"#/&+ '#/$"5,.&%"#/
5,.";,5 3.#1 %*, 3&'% %*&% %*, '."%"'&+ .&%"# "$ <&$,5 #/ $1##%* /#.1&+
'-.;,$ -$,5 &$ $-<$%"%-%,$ 3#. 5"$'.,%, <"/#1"&+ 5"$%."<-%"#/$9 H
'#..,'%"#/ '#/%"/-"%2 "$ %*,.,3#., &00.#0."&%,9
d qw P p − ` L n$
"
Nx @ $,L^N
q w⏐"88
%G f 9SY⏐`
$::
" x @ "88&98S%9S × q R9]V
)*, #<$,.;,5 ;&+-, #3 d L7*"'* "$ R9]VN '#..,$0#/5$ %# & ^`;&+-, #3
9[[U9
4,/',8 &% & 9[U +,;,+ #3 $"=/"3"'&/',8 %*, /-++ *20#%*,$"$ "$ .,J,'%,59
)*,.,3#.,8 %*, $&10+, 3"/5"/= "$ /#% '#10&%"<+, 7"%* %*, 0#0-+&%"#/
0.#0#.%"#/9
Comparison of two Proportions
H $"1"+&. &00.#&'* "$ &5#0%,5 7*,/ 0,.3#.1"/= & *20#%*,$"$ %,$% %#
'#10&., %7# 0.#0#.%"#/$9 )*, $%&/5&.5 ,..#. #3 %*, 5"33,.,/', "/
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0.#0#.%"#/$ "$ &=&"/ '&+'-+&%,58 <-% <,'&-$, 7, &., ,;&+-&%"/= %*,
0.#<&<"+"%2 #3 %*, 5&%& #/ %*, &$$-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
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'&+'-+&%,5 #/ %*, &$$-10%"#/ %*&% %*, 0.#0#.%"#/ "/ ,&'* =.#-0 "$ 0 8 $#
%*&% 7, *&;,
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dq#$"U
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'&1, %# -/5,.$%&/5 %*&% %*, 0.#0#.%"#/ #3 0,#0+, *&;"/= 1&+&."& "/
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'#/$"5,.,5 7&$ OU[[[9 4, &+$# .,&+"$,5 %*&% 5-."/= %*, 2,&. %*&%
3#++#7,5 LOXWXN8 <+##5 $&10+,$ 7,., %&C,/ 3.#1 O[8[[[ .&/5#1+2
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%*&% R[[ 0,.$#/$ 7,., 0#$"%";, 3#. 1&+&."&9 4,+0 %*, *,&+%* #33"',. "/
%,$%"/= %*, *20#%*,$"$ %*&% %*, 1&+&."& $"%-&%"#/ #3 OXWX 5"5 /#% $*#7
&/2 $"=/"3"'&/% 5"33,.,/', 3.#1 %*&% #3 OXW] L%&C, %*, +,;,+ #3
$"=/"3"'&/',8 α q9[ON9
4Q c OXW] q ^OXWX L #. ^OXW] ` ^OXWX q [ N
4 H c OXW] ≠ ^OXWX L #. ^OXW] ` ^OXWX ≠ [ N
0OXW] q 9[S] 8 /OXW] q OU8[[[
0OXWX q 9[R 8 /OXWX q O[8[[[
d %&< L α q 9[O N q R9U]
G#11#/ L0##+,5N 0.#0#.%"#/8 0 q"7777"G777
$77GO7
++
q [9[S[]
&/5 %*, $%&/5&.5 ,..#. q #"7777&""G777&"UR9R$S7%:S +×
q 9[[RR
4,/',8 d '&+' q L9[S] ` 9[R[N @ 9[[RR q 9[O] @ 9[[RR q ]9 R
D*"'* '#..,$0#/5$ %# & ^`;&+-, #3 +,$$ %*&/ 9[[S9
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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 %*,
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<,"/= -$,58 7"++ +&<,+ $#1,7*&% 3,7,. #<$,.;,5 $&10+, 5"33,.,/',$ &$
<,"/= $%&%"$%"'&++2 $"=/"3"'&/%9 Q/ %*, #%*,. *&/58 7*,/ 5,&+"/= 7"%*
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*&$ & /,=+"="<+, ,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
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7*,%*,. #. /#% %# .,J,'% %*, /-++ *20#%*,$"$ &% & =";,/ +,;,+ #3
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%*, $&10+"/= 5"$%."<-%"#/ "$ &$$-1,5 %# <, α@R N 9 )*"$ '#/;,/%"#/
5,%,.1"/,$ %7# ;&+-,$ #3 d 7*"'* $,0&.&%, & l 3&"+ %# .,J,'% .,="#/m 3.#1
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.,J,'%"#/ .,="#/ "$ '#/',/%.&%,5 "/ #/, %&"+ #3 %*, $&10+"/= 5"$%."<-%"#/ #3
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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
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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
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)*, '#..,$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
#-% %,$%$ #3 $"=/"3"'&/', 3#. %*, 1,&/$ &/5 0.#0#.%"#/$ #3 +&.=, $&10+,$9
:/ %*"$ $,'%"#/ 7, $*&++ $,, *#7 $"1"+&. 1,%*#5$ 1&2 <, -$,5 7*,/ 7,
*&;, $1&++ $&10+,$8 -$"/= %*, %`5"$%."<-%"#/9
The t-distribution
:/ %*, 0.,;"#-$ $,'%"#/$ %*, $%&/5&.5 /#.1&+ 5"$%."<-%"#/ Ld`
5"$%."<-%"#/N 7&$ -$,5 "/ ,$%"1&%"/= <#%* 0#"/% &/5 "/%,.;&+ ,$%"1&%,$9
:% 7&$ &+$# -$,5 %# 1&C, <#%* #/, &/5 %7#`%&"+,5 %,$%$9 4#7,;,.8 "%
$*#-+5 <, /#%,5 %*&% %*, d`%,$% "$ &00+",5 7*,/ %*, 5"$%."<-%"#/ "$
/#.1&+ &/5 %*, 0#0-+&%"#/ $%&/5&.5 5,;"&%"#/ σ "$ C/#7/ #. 7*,/ %*,
$&10+, $"F, / "$ +&.=, L n≥
30N &/5 7"%* -/C/#7/ σ L<2 %&C"/= E &$
,$%"1&%#. #3 σN 9
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But, what happens when n<30 and σ is unknown?
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Mean 68 75
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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*"'*
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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
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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
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G&%,=#.2 Q" 0.#0#.%"#/ ,"
To SY[ X@OY SY[
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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
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#3 U[[ 5.";,.$ <,%7,,/ O] &/5 U[9 ),$%8 &% & [O +,;,+ #3
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Observed frequencies
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.,+&%"#/ <,%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[
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,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
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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
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CHAPTER NINE
CORRELATION AND REGRESSION
9.1 Learning objectives
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In other words, the calculated values for ‘a’ and ‘b’ are sample
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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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H XUi '#/3"5,/', "/%,.;&+ 3#. %*, +#/=`.-/ $+#0, "$8 %*,.,3#.,c
Estimated slope±
t1-α
(standard error of slope)
D, 5# /#% /,,5 %# C/#7 *#7 %# '&+'-+&%, %*, $%&/5&.5 ,..#. #3 $+#0,
&$ %*, '#10-%,. 0."/%$ %*"$ #-% 3#. -$9
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SIGNIFICANT TEST
4#c P#/=`.-/ $+#0, "$ F,.# Lβ=0N
4Oc P#/=`.-/ $+#0, "$ /#% F,.# Lβ≠0N
:3 %*, /-++ *20#%*,$"$ "$ %.-, %*,/ %*, $%&%"$%"'c
.,/41/D.121L1?/E ISbS
7Z.,/41cD.12L1?* =
7"++ 3#++#7 & %`5"$%."<-%"#/ #/ / ` R q ] 5,=.,,$ #3 3.,,5#19 D, +#$,
%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_
P:_?8 :E K?_Q)?K !o Ag qH r !k9 )4AE )4? ^Q:_) Lk8 H r !kN
: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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IIFF TTHHEE BBOODD Y Y WWEEIIGGHHTT WWEERREE 6655 KKGG,, TTHHEENN TTHHEE BBEESSTT PPRREEDDIICCTTIIOONN OOFF
RRMMRR WWOOUULLDD BBEE
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%:S%R"S"G%%S%""TRG%SGG9
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Assumptions: )*,., &., %7# &$$-10%"#/$ -/5,.+2"/= %*, 1,%*#5$
#3 +"/,&. .,=.,$$"#/c \".$%+28 3#. &/2 ;&+-, #3 B8 2 "$ /#.1&++2 5"$%."<-%,5
&/5 $,'#/5+28 %*, 1&=/"%-5, #3 %*, $'&%%,. #3 %*, 0#"/%$ &<#-% %*, +"/,
"$ %*, $&1, %*.#-=*#-% %*, +,/=%* #3 %*, +"/,9 )*"$ $'&%%,. "$
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1,&$-.,5 <2 %*, $%&/5&.5 5,;"&%"#/8 E8 #3 %*, 0#"/%$ &<#-% %*, +"/, &$
5,3"/,5 &<#;,9 H '*&/=, #3 $'&+, 1&2 <, &00.#0."&%, "3 ,"%*,. #3
%*,$, &$$-10%"#/$ 5#,$ /#% *#+58 #. "3 %*, .,+&%"#/$*"0 $,,1$ /#/`
+"/,&.9
PRECAUTIONS IN USE AND INTERPRETATION
O9 )*, .,+&%"#/$*"0 1-$% <, .,0.,$,/%&<+, <2 & $%.&"=*% +"/,9 :/
'&+'-+&%"/= %*, '#..,+&%"#/ '#,33"'",/% 7, &., 0.,$-1"/= %*&% %*,
.,+&%"#/$*"0 <,%7,,/ %*, %7# 3&'%#.$ 7"%* 7*"'* 7, &., 5,&+"/= "$ #/,
7*"'* & $%.&"=*% +"/, &5,>-&%,+2 5,$'."<,$9
R9 )*, +"/, 1-$% /#% <, -/5-+2 L,B%.,1,+2N ,B%,/5,59 :3 %*, $%.&"=*%
+"/, "$ 5.&7/ &/5 %*, .,=.,$$"#/ ,>-&%"#/ "$ 3#-/58 "% "$ 5&/=,.#-$ %#
,B%,/5 %*&% +"/, <,2#/5 %*, .&/=, #3 %*, &'%-&+ #<$,.;&%"#/$ -0#/
7*"'* "% "$ <&$,59 \#. ,B&10+,8 "/ $'*##+ '*"+5.,/ *,"=*% "/'.,&$,$
7"%* &=, "/ $-'* & 7&2 %*&% & $%.&"=*% +"/, 5,$'."<,$ %*, .,+&%"#/$*"0
.,&$#/&<+2 7,++9 !-% %# -$, %*&% +"/, %# 0.,5"'% %*, *,"=*% #3 &5-+%$
7#-+5 <, ."5"'-+#-$9
S9 H$$#'"&%"#/ "$ /#% /,',$$&."+2 '&-$&%"#/9 )*, '#..,+&%"#/
'#,33"'",/% "$ & 1,&$-., #3 &$$#'"&%"#/8 &/58 "/ "/%,.0.,%"/= "%$
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6,&$-., #3 3"% #3 %*, 1#5,+
)*, TR ;&+-,8 #3 XO9Ri 0.#;"5,$ -$ 7"%* & 1,&$-., #3 3"% #3 %*, 1#5,+9
:% "$ %*, 0.#0#.%"#/ #3 ;&."&/', #3 o %*&% '&/ <, &''#-/%,5 3#. <2 %*,
1#5,+9
Assumptions of the Model - Validation of the model
)*, "10+"'&%"#/ #3 %*"$ 1#5,+ "$ %*&% &++ %*, $2$%,1&%"' ;&."&%"#/ "$
,1<#5",5 "/ %*, $%.&"=*% +"/, &/5 /# 3-.%*,. $2$%,1&%"' ;&."&%"#/
$*#-+5 <, 5"$',./"<+, "/ %*, 5&%&9 :3 7, "/$0,'% #-. $'&%%,. 0+#% 7"%*
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:% "$ -$-&+ %# ,+"1"/&%, %*, $%.&"=*%`+"/, ,33,'% 3.#1 %*, 5&%& 3".$% &/5
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E-<%.&'%"/= 3.#1 ,&'* #<$,.;,5 o ;&+-, %*, '#..,$0#/5"/= ;&+-, #/
%*, +"/, ` %*, 3"%%,5 o ;&+-, 5#,$ %*"$c
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%*, O[ 7#1,/9
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T6T +,;,+ "$
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&=&"/$% %*, 3"%%,5 #. 0.,5"'%,5 ;&+-,$9
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)*, .,$"5-&+$ &., .&/5#1+2 $'&%%,.,5 &<#-% F,.# 7"%* /# 5"$',./"<+,
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$2$%,1&%"'&++2 &$ 0.,5"'%,5 ;&+-,$ "/'.,&$,9 _,"%*,. "$ %*,., &/
"/5"'&%"#/ #3 &/2 /#/`+"/,&. 0&%%,./ L%*,., "$ J-$% & *"/% #3 & $-==,$%"#/
%*&% %*, $'&%%,. "$ =.,&%,. 3#. +&.=,. 0.,5"'%,5 ;&+-,$ %*&/ 3#. $1&++
#/,$ <-% 7"%* $# 3,7 #<$,.;&%"#/$ "% 7#-+5 <, 5"33"'-+% %# 5.&7 $-'* &
'#/'+-$"#/j 7, $*#-+5 *#7,;,. <,&. "% "/ 1"/5 "3 1#., 5&%& &., %# <,
'#++,'%,5N9
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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
9[OWX 9[OWV 9[OW[ 9[OYY 9[OYR 9[OU] 9[OUV 9[OU[ 9[OVY 9[OVS
9[OSX 9[OSY 9[OSR 9[ORX 9[ORU 9[ORR 9[OOX 9[OOY 9[OOS 9[OO[
9[O[W 9[O[V 9[O[R 9[[XX 9[[XY 9[[XV 9[[XO 9[[]X 9[[]W 9[[]V
9[[]R 9[[][ 9[[W] 9[[WU 9[[WS 9[[WO 9[[YX 9[[Y] 9[[YY 9[[YV
9[[YR 9[[Y[ 9[[UX 9[[UW 9[[UU 9[[UV 9[[UR 9[[UO 9[[VX 9[[V]
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
9[[OS[
9[[[YX
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3.6
3.8
4.0
9[[[OY
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Table 5: ^,.',/%&=, 0#"/%$ #3 %*, % 5"$%."<-%"#/ L%*"$ %&<+, =";,$ %*,
;&+-,$ #3 % 3#. 5"33,."/= 53 %*&% '-% #33 $0,'"3",5 0.#0#.%"#/$ #3 %*, &.,&
"/ #/, &/5 "/ %7# %&"+$ #3 %*, % 5"$%."<-%"#/N
Area in two tails
[9R [9O [9[U [9[R [9[O [9[[O
Area in one tail 53
[9O [9[U [9[RU [9[O [9[[U [9[[[U
O
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S
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U
Y
W
]
X
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S9[W] Y9SOV OR9W[Y SO9]RO YS9YUW YSY9YOX
O9]]Y R9XR[ V9S[S Y9XYU X9XRU SO9UX]
O9YS] R9SUS S9O]R V9UVO U9]VO OR9XVO
O9USS R9OSR R9WWY S9WVW V9Y[V ]9YO[
O9VWY R9[OU R9UWO S9SYU V9[SR Y9]UX
O9VV[ O9XVS R9VVW S9OVS S9W[W U9XUX
O9VOU O9]XU R9SYU R9XX] S9VXX U9V[U
O9SXW O9]Y[ R9S[Y R9]XY S9SUU U9[VO
O9S]S O9]SS R9RYR R9]RO S9RU[ V9W]O
O9SWR O9]OR R9RR] R9WYV S9OYX V9U]W
O9SYS O9WXY R9R[O R9WO] S9O[Y V9VSW
O9SUY O9W]R R9OWX R9Y]O S9[UU V9SO]
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RUX
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RO
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∞
O9SU[ O9WWO R9OY[ R9YU[ S9[OR V9RRO
O9SVU O9WYO R9OVU R9YRV R9XWW V9OV[
O9SVO O9WUS R9OSO R9Y[R R9XVW V9[WS
O9SSW O9WVY R9OR[ R9U]S R9XRO V9[OU
O9SSS O9WV[ R9OO[ R9UYW R9]X] S9XYU
O9SS[ O9WSV R9O[O R9UUR R9]W] S9XRR
O9SR] O9WRX R9[XS R9USX R9]YO S9]]S
O9SRU O9WRU R9[]Y R9UR] R9]VU S9]U[
O9SRS O9WRO R9[][ R9UO] R9]SO S9]OX
O9SRO O9WOW R9[WV R9U[] R9]OX S9WXR
O9SOX O9WOV R9[YX R9U[[ R9][W S9WYW
O9SO] O9WOO R9[YV R9VXR R9WXW S9WVU
O9SOY O9W[] R9[Y[ R9V]U R9W]W S9WRU
O9SOU O9W[Y R9[UY R9VWX R9WWX S9W[W
O9SOV O9W[S R9[UR R9VWS R9WWO S9YX[
O9SOS O9W[O R9[V] R9VYW R9WYS S9YWV
O9SOO O9YXX R9[VU R9VYR R9WUY S9YUX
O9SO[ O9YXW R9[VR R9VUW R9WU[ S9YVY
O9S[S O9Y]V R9[RO R9VRS R9W[V S9UUO
O9RXY O9YWO R9[[[ R9SX[ R9YY[ S9VY[
O9R]X O9YU] O9X][ R9SU] R9YOW S9SWS
O9R][ O9YVU O9XY[ R9SRY R9UWY S9RXO
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Table 6: ^,.',/%&=, 0#"/%$ #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
V9YVR Y9RUO W9]OU X9]SW OO9SVU OY9RY]
U9X]X W9WWX X9V]] OO9YY] OS9RWW O]9VYU
W9R]X X9RSY OO9[W[ OS9S]] OU9[]Y R[9UOW
]9UU] O[9YVU OR9UXR OU9[SS OY9]OR RR9VUW
X9][S OR9[OW OV9[YW OY9YRR O]9VWU RV9SRR
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OR9RVR OV9Y]V OY9XOX OX9YWX RO9YYY RW9]WW
OS9VVR OU9X]W O]9S[W RO9OYO RS9R[X RX9U]]
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