Cheatsheet

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• When the levels of one factor (diet) are associated with the levels of another factor (location), we say that these two factors are (-5$#--,-&$'"('$�(-('#$($'#&'$&'(')&')*$+-,/$($.811$"70,'"#&)&9$!"#$#&')/('#5$ confounded. Randomization eliminates confounding, simplifying the interpretation of data. ()*+,'!-(!.#%!+$#+#$%-#.),!%#!%&'!-.?$')('!-.!%&'!()*+,'!(-@':!;&'! 0#!0#A!0!4>0#A01A!A'M20>2%#9! • " we reject H0 if the p-value of the test is less than 0.05. (-5$#--,-$,+$'"#$5)++#-#.*#$6#'G##.$(2#-(3#&$,+$).5#0#.5#.'$&(/01#&$)&$ /,*9%#("$('**"*(&'!*',0'0(<%)8(=-:!A(!)!$'(2,%=!"#$!-.(%).?'=!%#!?2%! • The estimated standard error of the difference between averages of independent samples is '!&'0#!%@!>I'!40&?C2#3!A24>12GF>2%#!24!>I'!?%?FC0>2%#! %&'!*)$/-.!#"!'$$#$!-.!&),"=!6#2!.''0!B!%-*'(!)(!*).6!?)('(:!C#(%(=! $$% $%% 0&'>'1*!'2>I'1!-!%1! # 9!/0$I!?010&'>'1!C2'4!0>!>I'!$'#>'1!%@!2>4! $%"!" $ !" # # ' $ &#>'<'$=!%6+-?),,6!$-('!-.!+$#+#$%-#.!%#!-1(>":(5"43&(06'#&(?()%/'0( $ % & & $ % !"*!>I'!&'0#!%@!>I'!40&?C2#3! &?C2#3!A24>12GF>2%#9!Z%1! !" )(!*2?&!%#!(,-?'!%&'!*)$/-.!#"!'$$#$!-.!&),":! • Conditon for doing t-test for se. #"*!>I'!&'0#!%@! >12GF>2%#!24!-*!>I'!?%?FC0>2%#!?1%?%1>2%#9!!Z%1! o SRS condition (Simple Random Sample) (#:&!**(-#*+!"#32&4;$%#;"!-'#'(-(&!/#";?"/&!5*"#@.A%-<#@0A#*2#12GF>2%#!24!#9! o Similar variances '&2;5"8#B2&#*+!"#(C%45$()#!*,"#42&(#;"(3;$#*2#:&!*(#*+!"#32&4;$%#%"$

o Nearly normal '!4>0#A01A!A'M20>2%#!%@!>I'!40&?C2#3!A24>12GF>2%#!24!>I'! 7"<(-,*9'(,(0,/6-'(&"(<'(#''&@(A(0%/6-'(,#0<'*(%0(B/"*'(%0( " " • Two-sample t-tests and two-sample intervals for the difference between two means allow us to compare results $'%Bothconfidence $(% obtained from two samples. procedures $%"!" $ !" # # ' " $ rely on standard error of the difference between two sample averages and the use #A01A!'11%19!Z%1!0#!0M'103'!%@!2#A'?'#A'#>!%G4'1M0>2%#4*! .'))'*:C(.4)(&,),(!"0)()%/'(,#&(/"#'51(7"<(/4!8(%0('#"498@( ' ( of a t-model for the sampling & distribution. &( Experiments provide the ideal data for such comparisons. In an experiment, subjects from a sample are randomly 'assigned to treatment groups defined by levels of the experimental factor. This randomization '>I'1!0!?1%?%1>2%#!%1!#%>*!>I'!@%1&FC0!@%1!>I'!4>0#A01A!'11%1!24! D'$"*'(5"4(.'9%#()"(!"--'!)(&,),:(%)30(,(9""&(%&',()"(E#"<(<8')8'*( & !"#$%&'()*&+*#,&-*./012# avoids confounding that introduces the possibility of lurking factors. 05*!! %&'!()*+,'!(-@'!6#2!?).!)""#$0!>-,,!1'!)0'D2)%'!"#$!>&)%!6#2! $0183$).$'"#$#&')/('#&$+-,/$!(61#$>P%@=$'"#$&'(.5(-5$#--,-$)&$ 16-confidence interval ' !!"#$'!"#$'%%&!'(!)*+!,-+.!/012!34,53221*6!(*5! >).%!%#!,')$.!)1#2%!%&'!+#+2,)%-#.:! %& $'"$&'() % & %( ! $%'(*% !!16/*!/03!(*57+-8! *5!/03!2/869859!355*5:!)*+!.3/!/03!2/869859!355*5!(*5!,5*,*5/1*62;! $%"!""' $• !"( # # ' $D#E8EF#52;-<"$ E"!6#2!F.#>!%&'!.''0'0!*)$/-.!#"!'$$#$=!6#2!?).!'(%-*)%'!%&'! (( (! I'!#F&'10>%1*!'!24!>I'![Q!%@!0!42#3C'!%G4'1M0>2%#*!0#A!2!24!>I'! #!"& ## #'!'00,*5(0,/6-'(0%F'1(2)30("#-5(,(94'00(.'!,40'(5"4(<"#3)(E#"<( !"$%!% $7#0#!"#*+;"$ ! !" #$ # &?C'!42\'9!]I'#!0M'1032#3!2#A2$0>%14!80!.^5!AF&&D!M0120GC' K 2.%-,!6#2!/'%!%&'!()*+,':!G)*+,'!(-@'!?),?2,)%-#.(!)$'!$)$',6!'H)?%:! % " •• A confidence ;#!/'%!)!+)$%-?2,)$!*)$/-.!#"!'$$#$=!(#,<'!"#$!-!-.!%&'!"#$*2,)3! ! a range !"34# parameter based on the data in a sample. "!" $ !"(interval # $ %provides +'&% $ % of plausible values for a population ## ' # !( $',$$ "#-./'*%0&/"#&1%*2*%'-*"0& " !# # "# $%"!" !" " ' $ !"($ # ('() " ! !" ! # !" !":!)*+!@6*A! *162! ! 869!(&!?6=3!)*+!@6*A! #$ #$ " • • Confidence intervals provide # a range of plausible values for a parameter ;&'!.'?'(()$6!()*+,'!(-@'!0'+'.0(!#.! :!I'!?)..#%!'(%-*)%'! #! of a population. The coverage (or confidence level) of a !%&'("&)*"(+,*"-*.+&/01()/2"+("&)+&"3*&4**1"&)*",*+1"+15"6+-/+17*" confidence interval is the probability that this procedure produces an interval that includes the parameter. Most often, >-%&!2!1'?)2('!>'!&)<'!%#!?&##('!-!60.,)0!?#,,'?%-./!%&'!()*+,':!E"! confidence intervals& have coverage 0.95 and are known as 95% confidence intervals. The margin of error is the half-length of (!8!B356*+--1!5869*7!C8518D-3!*:!A0353!E85"*%!F!'"#$'%&! '()%& " the 95% confidence interval. The one-sample z-interval for the mean of a population is   y ± 2 s/   n. This confidence interval >'!&)<'!.#!-0')!#"! # =!-%!>#2,0!1'!)!/##0!-0')!%#!#1%)-.!)!(*),,! presumes a large sample. This same interval applies to proportions of large samples, with s2 estimated as ˆ p (1- ˆ p ). The ! #$"!12!/03!7386!*(!8!=*-+76!*(!6+7D352!=*939!82!#!38=0!/173!/03! " be rounded to presentation precision by applying the 3-to-30 Rule to the standard endpoints of a confidence interval should ()*+,'!%#!'(%-*)%'!"#:!A!'+(-.*/.012!?#,,'?%(!)!(*),,!()*+,'!#"=! error. C36/!*(!16/3532/!08,,362!869!=*939!82!G!*/035A123&!! |t| 0.3802 o Upper CL Dif 1.3583 2#()8'(06'!%,-(!,0'("$(6*"6"*)%"#0:(<'(&"#3)(#''&(,(6%-")(0,/6-'1( 0.8,1"/(1'&"58,39"":0-3!-1@3-)!/03!6873!=*732!(5*7!/03!(8=/!/08/! o Lower CL Dif -0.5200 Confidence 0.95 ;&'!$',)%-#.(&-+!1'%>''.!#o!).0!8!),,#>(!6#2!%#!'(%-*)%'!-! 03!9+77)!C8518D-3!12!9351C39!(5*7!86*/035:!7*53!+23(+--)!=*939! o >-%%!F.#>-./!).6%&-./!)1#2%!8:!E"!6#2!,##F!)%!.)%-#.),! *-+76&!!'6!/012!3487,-3:!/03!9+77)!C8518D-3!1691=8/32!A03/035! o 18. Which of the following is INcorrect? o A. The mean difference of about 0.4 is statistically insignificant at the 5% level. 04*+'50:(5"43--(#")%!'()8,)(/"0)("$()8'/(8,+'(,."4)(G:GHH(6'"6-'1( 03!=+2/*735!8==3,/2!86!8,,-1=8/1*6;! o B. The null hypothesis that the population means are the same cannot be rejected at the 5% significance level. o C. The values between –0.52 and 1.36 cannot be rejected as possible population mean differences at the 5% I"43--(,-0"(#")%!'(68*,0'0(-%E'(BJ8'0'(*'04-)0(,*'(,!!4*,)'()"(K(L( 62&F!#! 1(!8,,-1=8/1*6!12!53/+5639! significance level, and this holds true in particular for the value zero. 6"%#)01C(J8'(KLM(%0()8'(/,*9%#("$('**"*1(2)()4*#0("4)()8,)()8'0'()<"( 62&F!G! 1(!8,,-1=8/1*6!12!6*/!53/+5639! o The value of |t| says that the observed mean difference 0.419 is 0.879 standarderror estimates away from zero. o The p-value says that the!12!#:!A08/!A3! probability of observing a value of |t| > 0.879 in other datasets is about 0.38, assuming +$#+'$%-'(=!J=JKK!$'(+#.0'.%(!).0!)!L9!*)$/-.!#"!'$$#$=!?#*'! &$!12!/03!,5*,*5/1*6!*(!/1732!/08/!8!6 03!7386! 2 the null hypothesis of equal population means is correct. !"&!H*!233!/08/! !"theorem • The central limit#$ states that means across datasets are&$ ever more normally distributed as N → ∞. &$!!F! %#/'%&'$:! #$ 2+8--)!-8D3-! :!A51/3!/03!6+7358/*5!*(! !82!/03! • The standard deviation of proportions across datasets shrinks at the rate 1/N ½ $ ! ! !"3!4# +7!*(!/03!62&!!
"

%

'#

%

'#

P(A|B)*P(B)$ $ "& & &$ !" P(B|A) ="----------------------------------% & ( & #$ ( P(A|B)*P(B) + P(A|not B)*P(not" B)

(%"

$

"

$

(%"

Proof: " numerator = P(A and B)$ $ denominator and#$ not #$ B) !"B) +(P(A and"& !" ! % = P(A %&

'#

&( % %

$

'#

$

& ( %"

!"$ ( %" #"& #$ !"$ % !% & %" # #$

$

!"!F!7#&!'(!A3!,-+.!16!/012! 03!=*+6/!7#!12!/03!6+7D35!*(!*632:!2*!7 #$ 4,53221*6!(*5!7#:!A3!.3/! % " $ !"# #$ !"$ ' #& ( & &$$$ % %!"& #$!"# #$!"$ ( %#$"& (%" " !" !"$!"& #$( !" #$ !"# ! % %#$"& # #$

! !

! !

52)/-2;+/2.3!.0!/'(!/()/!)/,/2)/24! $% !2)! !""#"$%&'('"')*'%+#,"*'%)#-%(#,)./"#$%$! !""#"$%&'('"')*'%+#,"*'%)#-%(#,)./"#$%$!

&! , )! ! -$ +!! !"#$%&#'(&)&*"+&,)-."%&*/'&"!&'(&!()*!(++,-*.!/0!#12!34!"!5!1267!8&'*!9:))*;&! $% # ) (! '! & *! )(&*!,4!<=(>?!&'*;!&'*!=),/(/@+@&0!,4!(!<(>=+*!,4!"11!>*<<(A*!@>!'@8./'()2)!/(>>)!+)!-:!;+/!),@)!3./'23@! ,A!B.-!/()/)!.0!-:!D(!3((5!/.!()/26 ! '! *! -./0 & -.45 /'(!5,/,A! F igure 15-6. Sa m for !" under H0,! . $pling !)!%distribution % ! -.451/& -.452 3/-(! +! ! ()*+*%&",-"./0#.1&"2).#2*3"+*45,2*", #"!;@!/'(!)/,35,-5!5(12,/ !"#$%&'()*+"&,*#&-,.&'/+&#-"&,0%"+1"2&%/345"&3"/6& !&'()!*+,-!./(! ),68>(:!$A!!L'(3!D(!()/26,/(! , !0-.6!/'(!5,/,!/.!0235!/'(!)/, ! % % &4./ " # /01,./()'2(3!-(45!"67!8/(!).4534+3!(++,+!1+,9'3()!./(!):4&(7!!8/(! $ -.-/6 ! 6*"5,/*5"#$"6#$)","5.6*+"2,&*"7$%8! )4-1&(!-(45!453!./(!/01,./()'2(3!-(45!&'(! z-statistic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t-statistic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
! 3,441"56%7)'0+2.'.%8'+-%#(%1%9'1)% ! -.+/0(%&.*!+(1('#%#1!

"!!

!

2/00!34+.%3#$&$!

"6J!"!=&"6!

"6J!"!7&"6!

50%#1*(%&6#!34+.%3#$&$!

"4J!"!K!"6!

"4J!"!L!"6! #"!

7#$%!$%(%&$%&8! ! F igure 15-2. Sa mpling distributions for p = 0.20, 0.22, or 0.24.

9%(*)(1)!#11.1! $#= !" >!M!$>?*! F'*!9'(;9*!4,)!(!4(+<*!=,<@&@E*!*)),)!@=+@;A! :#;#8%!"N$#= 1D!&'*!/)*(I$*E*;!E(+:*!12612!34!"!5! ! !" >!K!,&!! = !$!"6>N$#= !" >!L!$,&!! :7;:<"'"$%&)&-)'="&35)63"&*/'&="##26=&)&%)40-")#&*//-%"&#"6#)#2,"& .*9@<@,;!):+*!@;&,!)*J*9&@;A!#12!34!"!5!1261D!&'*!9'(;9*!4,)!(!4(+<*!=,<@&@E*! 8F0=*!3?!*)),)!@
!

!

!

! "#$%!

"#$"%!