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Page 1: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Point Estimation

mth_%&'

Page 2: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Data Distributions vs Probability Distributions

Page 3: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Confidence Interval of the mean

Page 4: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Taking Samples

Page 5: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

!"#$!%!&!'!&(&!"!$!$#$()!!!!#(!'!"!!('('#!('!"(!!"!(#!!"!!!#

%,'),'*,'+,'&,',,'-,'.,'/,'

Population Size: /01

MU : /4.4/

SIGMA : 0./9990

VAR : /<.<4<9

Number of intervals:

D1/1

-> /4

-> D/

.4

/4 ->np ->

1.0EEE/F

1.9GE1<G

//

D11

E1

)))%)%))))))))

Number of Samples:

Sample. Size:

Mean of Sampling Distribution:

Standard Deviation ofSampling Distribution:

Standard Error: (Calculated)

S A M P L E

Sampling Distribution (Means)

G

Page 6: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

#'.)"("#"#.')!%)%").!+)&#%"+.#+&+&"!.&(%"$%"!.")+%&%#&.'$'+%%"'.('##'%"!.%'+"&%"(."#&%+"".'&"+#"#.$(##(%"+.#$!+'%"(.))"'&"#.$+'+#%$.#%+"+"'.!++"%$."("%!%"$.''!"'"$.)"(!!").%!$#)"$.)!$%+%""."!!&"%"!.#$'%#%(.''###%!.%)'+&").&"%+&%#".+((#)%#&.)%'&'%""."%)$+%

%,'),'*,'+,'&,',,%-,%.,)/,)

Population Size: /01

MU : G9.111D0/

SIGMA : G.FGE4F/

VAR : /G.GE90E0

Number of intervals:

D1/1

-> G0

-> 0

.4

/4 ->np ->

D.D19GFG

<.G9/10<

/9./9D11D

D111

/1

)/.&**&+,+*.-&%*%%%,.*/'%'+),.,+/&*-%..'-*)')),.%--'++%-..+%'-

Number of Samples:

Sample. Size:

Mean of Sampling Distribution:

Standard Deviation ofSampling Distribution:

Standard Error: (Calculated)

S A M P L E

Sampling Distribution (Variances)

G

Page 7: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

#'.)"("#"#.')!%)%").!+)&#%"+.#+&+&"!.&(%"$%"!.")+%&%#&.'$'+%%"'.('##'%"!.%'+"&%"(."#&%+"".'&"+#"#.$(##(%"+.#$!+'%"(.))"'&"#.$+'+#%$.#%+"+"'.!++"%$."("%!%"$.''!"'"$.)"(!!").%!$#)"$.)!$%+%""."!!&"%"!.#$'%#%(.''###%!.%)'+&").&"%+&%#".+((#)%#&.)%'&'%""."%)$+%

%,-),+/*,%*/+,)%&&,)%/,,%/'-,%%&.,+%/,)%

Population Size: /01

MU : G9.111D0/

SIGMA : G.FGE4F/

VAR : /G.GE90E0

Number of intervals:

D1/1

-> G0

-> 0

.4

/4 ->np ->

/./D/FEE

D.9GF0/E

G.F9D<F9

D111

0

,.&,%-*.-.//)'.,*../*.%*%.+,+'*-.+-/,.+.)-)&).+.*,+%')

Number of Samples:

Sample. Size:

Mean of Sampling Distribution:

Standard Deviation ofSampling Distribution:

Standard Error: (Calculated)

S A M P L E

Sampling Distribution (Standard Deviations)

G

Page 8: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

$.(&&#$+$.)&&%'%$.#)"''%$.!$$&++$.%$&&!'$.!))%)'$."+&&'($.$&'!+$."%!##!$.$++"(+$.$"%(#%$.(!+#%#$.)+$'$.#""&+)$.!%($!'$.!!%%!&$.$(#%''$.((%&'%$.$%"!''$.('!++($.#&+"+#$.$')%%)$.$(%$+'$.$#'&&)$.")&&&%$.$"($('$.$$'+'%$.$&"!#+$.$)#%)%$.(%)#)!

,%,%,%,),),),%,%,%,%,%,%,%,%,),%,*,%,%,%,),),%,%,),),%,%,%,%,%,%,%,),%,%,*,),%,),%,%,%,%,%,%,%,%,%,%,%,),%,%,%,),%,),%,%,%,%,%,%,%,%,%,%,%,%,),%,+,),%,%,%,%,%,*,),*,%,%,%,%,%,%,),),%,),%,%,%,%,),),%,%,%,%,%,%,),+,%,%,%,),%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,),%,%,%,%,%,%,%,%,%,%,%,),%,%,%,%,),),%,%,%,%,%,),%,),),),%,%,%,%,),),%,%,),%,%,),%,%,%,%,%,%,%,%,),*,*,%,%,%,%,%,%,%,%,%,%,%,*,),%,%,),),%,%,,,),%,*,%,%,%,%,%,%,%,%,%,+,%,%,*,),%,+,),%,*,%,%,%,%,%,),%,*,%,%,%,%,%,),%,%,%,),%,%,),%,%,%,),%,%,%,%,*,),%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,),),%,%,%,*,%,%,),%,%,*,%,+,),*,%,%,%,%,%,),),%,%,%,%,%,%,),%,*,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,%,),%,%,%,%,%,%,%,%,%,%,&,%,%,%,%,%,%,),%,%,%,%,%,%,),%,%,%,%,%,%,%,%,%,%,%,),%,),),%,%,%,%,%,%,%,),%,%,%,%,%,),),%,),%,%,%,%,%,%,%,%,),),%,%,),%,&,%,%,%,),*,%,%,),*,%,%,*,%,%,%,%,%,%,),%,),%,&,%,%,%,%,%,%,%,*,%,%,%,%,%,%,),%,%,%,%,%,),%,*,%,%,%,%,%,),),%,%,*,%,%,%,%,%,%,),),%,%,),%,%,%,),%,),%,%,%,%,*,%,%,%,%,%,),%,%,),%,%,),),%,%,),%,%,%,%,%,%,),%,%,%,%,),%,),%,%,%,%,%,%,%,%,),%,),%,%,%,%,*,%,%,*,),%,%,),%,%,%,),%,%,%,%,),%,%,%,%,%,%,%,%,%,%,%,),%,),),%,%,%,%,%,%,%,),%,),%,%,),),%,%,%,%,),%,%,%,%,%,%,%,%,%,%,%,%,%,%,),),%,),%,%,%,%,%,%,*,),%,),%,%,%,%,%,),%,),%,%,%,%,%,),%,%,%,),%,%,),%,%,%,*,*,+,%,),),%,%,%,%,),%,%,%,%,%,%,),%,%,),%,%,*,%,+,%,),%,%,%,%,%,%,),%,%,%,%,%,%,*,%,),),%,),*,*,+,),%,%,*,%,%,%,%,%,%,%,),%,%,),%,),%,%,),%,%,%,%,%,%,%,%,%,%,%,%,%,),),%,%,),%,%,%,%,*,%,),%,%,),%,),%,%,%,+,%,%,%,*,%,%,),%,%,%,%,%,%,%,%,%,%,%,%,),%,%,%,%,%,%),'),'*,'+,'&,',,'-,'.,'/,'

Population Size: /11

MU : 1./GFG/F

SIGMA : 1.//<F4/

VAR : 1.10DF04

Number of intervals:

D1/1

-> D4

-> D/

.<

/0 ->np ->

1.D1DF4G

1./D94<F

1.G/</90

D111

0

'.%.&).,'.,-+%*.'.%.-//.'.)'%*'+'.))/-.-'.+*/+&-'.)++-%&

Number of Samples:

Sample. Size:

Mean of Sampling Distribution:

Standard Deviation ofSampling Distribution:

Standard Error: (Calculated)

S A M P L E

Sampling Distribution (Ranges)

RANGE : D.4D9F9F

G

Page 9: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

!"#$!%!&!'!&(&!"!$!$#$()!!!!#(!'!"!!('('#!('!"(!!"!(#!!"!!!#

%,'),'*,'+,'&,',,'-,'.,'/,'

Population Size: /01

MU : /4.4/

SIGMA : 0./9990

VAR : /<.<4<9

Number of intervals:

D1/1

-> /4

-> D/

.4

/4 ->np ->

1.0EEE/F

1.9GE1<G

//

D11

E1

)))%)%))))))))

Number of Samples:

Sample. Size:

Mean of Sampling Distribution:

Standard Deviation ofSampling Distribution:

Standard Error: (Calculated)

S A M P L E

Sampling Distribution (Medians)

G

MEDIAN: /<.<4<9

Page 10: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

#+.%(("'%+."&#$)+"".$'"&+'%$.&"#"##.&%$$$'%+.%!(#!"#).$+!!#!!'."'$)#!#&.%)$+$"#(.#'%!'"%%.%%!$(!"+.+&!)(!%+.&&#"$"%".')'%"'"&.)(("'"&.(#+'!"#+."(&"''##.$''%(+#&.'"'!#$."'"+"'!'.+""!&!"(.#)"!%!"'.#"'((!#!.#'$#&+%(.)%'')!"".%#$(&!"+.#$$#+%+.)#$#%!().#%&%$""%.!%)#+'

%,'),'*,'+,'&,',,'-,'.,'/,'

Population Size: /111

MU : G4.DD4EF9

SIGMA : D/.19D4GG

VAR : DG0.G<91DF

Number of intervals:

D1/1

-> G4

-> D/

.4

/4 ->np ->

1.F0/

0.4/G<1E

G/.00F

D111

0

+'*.+.*.+++'+%

Number of Samples:

Sample. Size:

Mean of Sampling Distribution:

Standard Deviation ofSampling Distribution:

Standard Error: (Calculated)

S A M P L E

0.4F4FF<

F1%F0%FE%FF%

G

-D.F9 D.F91

Page 11: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Normality ?

Page 12: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Estimating the mean

-D.9G0/DD D.9G0/DD

R E S E T

z t

( using standard normal and Student's t )

Page 13: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Student's t Distribution

->

->

Page 14: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Estimating the mean( using standard normal and Student's t )

Page 15: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Estimating the mean

-D.F914F0 D.F914F0

R E S E T

z

( using population proportions )

Page 16: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Estimating the variance

1.00G4 D0.1E94

->n

R E S E T

Page 17: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Chi-Square Distribution

->

->

Page 18: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Calculating the sample size

R E S E T

/.4/9<E0

Page 19: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Estimating the difference between two means

-D.<G9/F< D.<G9/F<

z t

( Using standard normal and Student's t )

R E S E T

equalunequal

Page 20: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Estimating the difference between two proportions

z

( Using standard normal distrinution )

R E S E T

Page 21: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

Estimating the difference between two means

-/.DG04/< /.DG04/<

t

( interactive or "matched" pairs )

R E S E T

!!'!$(()&'&'!("!)!(%!&!(++!("()"(!"

("'(#!(%%("$(("(!!(!!(&#((!($"(('(+$($'

'$+&!"-%(-#!!$-%$-"(-!$+!&+("(+

Clear Data Tables

Page 22: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

!."()"$((.!%"%&&-(."$!#&!$.$%'$%!-$.")&")!-(.(+'%'+$.)""!$'$.'&%(%)-$.$#+$#&(.&%#'&-(.$#)!$%-(.#!'%$)-$.#'&('%$.(%%'!#(.+()$))-$.!!%&!#$."+)+)'(.$!%'''(.+"!"('-!.$)+))$.+%"+#&(.")"&('(.(%))(&$.$'$('$.%))+#$."'&)$#-!.")%'#%-(.!%$)%!#.+!%)%+$.'"""!(

-,./,//+/-+./,.//)-*./*/.&--*./%'&&+-*.-)*.-)-*.-''/&*-*.&,%/---*.*/,+%/-*.).+*'%

Population Size: 011

MU : 1.1140DE

SIGMA : D.G/140F

VAR : /.1D<GDF

Number of intervals:

D1/1

-> /4

-> D/

.4

/4 ->np ->

1.940/1G

1.4F4E1D

1.1EF

D111

0

'''''%'

Number of Samples:

Sample. Size:

Mean of Sampling Distribution:

Standard Deviation ofSampling Distribution:

Standard Error: (Calculated)

S A M P L E

Page 23: Point Estimation - Wilkes Universitymathcs.wilkes.edu/~rpryor/mth150/CI.pdfEstimating the difference between two means-

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

',%,),*,+,&,,,-,.,/,%',%%,%),%*,%+,%&,%,,%-,%.,%/,)',)),)+,),,).,*',*&,+'*,*+.%%,,*'..%-,)/.+&-,)..-%',)..)*-,)-./%%,)-.,-),)-.+./,)-.*+&,)-.))/,)-.%**,)-.'&),),./.*,),./)+,),..-),),..)-,),.-.-,),.-&%,),.-%/,),.,/',),.,+',),.&/.,),.&,),),.&*%,),.&'&,),.+&%,),.+%%+,)%.%/.,%..''',%,.,/+,%&./--,%&.&)),%&.)'-,%+./-,,%+.-//,%+.,&/,%+.&+,,%+.+&),%+.*-+,%+.*'-,%+.)+/,%+.%/.,%+.%&+,%+.%%&,%+.'.',%+.'+.,%+.')',%*./-',%*./)/,%*../+,%*..,+,%*..*.,%*.-.&,%*.-+&&,%,.)&.,%*.)-+,%).',',%%.*/),%'./,-,%'.,-),%'.+&,,%'.)./,%'.%&.,%'.'&%,/./,*,/....,/..)&,/.--',/.-)),/.,.',/.,+*,/.,%',/.&.',/.&&*,/.&',,/.+,,,/.+**,/.+'+,/.*-/,/.*)/,/.)/%,,%*.-+&,%'./)&,/.-.',/.%+.,..-+,,..+,,,..),',..%'),-./-,,-..-+,-.-/',-.-%.,-.,&-,-.,'&,-.&&/,-.&%/,-.+.*,-.+&%,-.+)),-.*/,,-.*&%,-.*%*,-.).',-.)&*,-.))/,-.%.',-.%+*-,%).)+,,/.&+-,..+&%,-..+-,-.+,',-.%/%,,.//*,,..+',,.-%/,,.,)',,.&*.,,.+,/,,.+%',,.*&/,,.*%+,,.)-&,,.)+',,.)'/,,.%.%,,.%&&,,.%%%,,.'-+,,.'+*,,.'%,,&.//),&./++,&./'..,%%.)&/,..,+/,-.&/%,-.'',,,.,*),,.*-%,,.%-.,,.')/,&./%%,&..%+,&.-*+,&.,,-,&.,'/,&.&&/,&.&%&,&.+--,&.++),&.+%),&.*.+,&.*&/,&.*%,,&.)-/,&.)+.,&.))%,&.%/.,&.%&%,&.%%,

',%,),*,+,&,,,-,.,/,%',%%,%),%*,%+,%&,%,,%-,%.,%/,)',)),)+,),,).,*',*&,+'*,%-.++*,%,.'++,%&.+*/,%&.%'%,%+...&,%+.-*&,%+.,)+,%+.&+',%+.+-*,%+.+%/,%+.*-+,%+.**-,%+.*'+,%+.)--,%+.)&*,%+.)*),%+.)%*,%+.%/,,%+.%.%,%+.%,-,%+.%++,%+.%)+,%+.%'-,%+.'/*,%+.'.%,%+.'&&,%+.'*-+,%).)%.,%'.,+/,/./-/,/.,'&,/.*,+,/.%/-,/.'-+,../.',../'&,...++,..-/+,..-&%,..-%&,..,.+,..,&-,..,**,..,%%,..&/),..&-&,..&,',..&**,..&%%,..+/),..+-,,..+,%,..+**,..+%%&,%'.''-,..+*+,-.-,+,-.*..,-.%+,,,./-.,,..&*,,.-&-,,.,.%,,.,%/,,.&,.,,.&)&,,.+..,,.+&,,,.+).,,.+'*,,.*.%,,.*,),,.*++,,.*)/,,.*'%,,.)-.,,.)&.,,.)+),,.))-,,.%/-,,.%-&,,...%*,-.),',,.&//,,.))-,&./..,&..)',&.,/&,&.,'',&.&)*,&.+,%,&.+%',&.*,,,&.*)/,&.)/-,&.),/,&.)++,&.))),&.)'),&.%.+,&.%,.,&.%+%,&.%%-,&.'/-,&.'.',&.',&,&.'*&,&.'%)-,..'-*,,.&+),&../',&.&)*,&.).&,&.%%/,+.//&,+..//,+..)*,+.-,%,+.-'/,+.,,,,+.,).,+.&/,,+.&,.,+.&+*,+.&)%,+.&'%,+.+.*,+.+,-,+.+*/,+.+%&,+.*/&,+.*-.,+.*,),+.**),+.*'/.,-.&-%,,.'&/,&.+%,,&.'&*,+..%-,+.,&),+.&)/,+.+**,+.*&-,+.)/&,+.)+*,+.)'',+.%,),+.%*',+.%'%,+.'-,,+.'&+,+.'*+,+.'%,,*.///,*./-%,*./+-,*./)-,*./'/,*../+,*..,*,*..+'

',%,),*,+,&,,,-,.,/,%',%%,%),%*,%+,%&,%,,%-,%.,%/,)',)),)+,),,).,*',*&,+'*,%'.%).,/.&&),/.)--,/.%%-,/.'%*,../+%,....-,...+&,...%),..-.,,..-,*,..-+&,..-)/,..-%&,..-'*,..,/),..,.*,..,-&,..,,-,..,,',..,+.,..,*/,..,*',..,)*,..,%-,..,'+,..&/++,-.-'/,,./++,,.&/%,,.*..,,.)&,,,.%,*,,.'/+,,.'+%,&.///,&./,+,&./*,,&./%),&../%,&..-*,&..&.,&..++,&..*),&..)%,&..%%,&..'*,&.-.-,&.--+,&.-,*,&.-&+,&.-+,,&.-)/,&.-%-&,,.,'.,&.-.,,&.+'/,&.%/),&.'&',+./&',+..-,,+..%.,+.--),+.-*&,+.-'+,+.,-.,+.,&&,+.,*,,+.,%/,+.,'+,+.&/',+.&-/,+.&,.,+.&&.,+.&+%,+.&)-,+.&%&,+.&'&,+.+/,,+.+-.,+.+,+,,&./.-,&.%+*,+.-&-,+.&*+,+.*.-,+.).+,+.)'-,+.%+-,+.'//,+.',',+.')-,+.''',*./-,,*./&,,*./*.,*./)),*./'.,*../,,*...+,*..-+,*..&,,*..+%,*..)/,*..%.,*..'.,*.-./,*.--+-,&.&/%,+.-*-,+.*+-,+.%)',*./-),*..,,,*.-.-,*.-),,*.,--,*.,*-,*.,'*,*.&-&,*.&&',*.&)/,*.&%%,*.+/+,*.+.',*.+,-,*.+&&,*.++&,*.+),,*.+%',*.*/-,*.*.,,*.*-,,*.*&,,*.*+'.,&.*%.,+.+&/,+.',,,*..*.,*.,.-,*.&.%,*.&'',*.+*.,*.*..,*.*+-,*.*%*,*.).+,*.)&/,*.)*-,*.)%.,*.)'),*.%.-,*.%-*,*.%,%,*.%&',*.%*%,*.%%&,*.%'),*.'/',*.'-/,*.'&/,*.'+*

',%,),*,+,&,,,-,.,/,%',%%,%),%*,%+,%&,%,,%-,%.,%/,)',)),)+,),,).,*',*&,+'*,&.&*.,&.+,),&.*/%,&.*+*,&.*'/,&.).&,&.),,,&.)&),&.)+',&.)*',&.))),&.)%,,&.)%',&.)'&,&.)'',&.%/,,&.%/*,&.%/',&.%.-,&.%.+,&.%.',&.%-,,&.%-*,&.%-',&.%,.,&.%,*,&.%,'+,+.&+&,+.*)&,+.%/%,+.%'-,+.'&%,+.'%',*./-/,*./&&,*./*,,*./)',*./'-,*../,,*...,,*..-.,*..-',*..,+,*..&.,*..&*,*..+/,*..++,*..*-,*..*%,*..),,*..)%,*..%-,*..%',*..'+&,+.',',*.-.',*.,%/,*.&)',*.+&*,*.+'&,*.*,.,*.**/,*.*%,,*.)/-,*.).),*.),.,*.)&-,*.)+-,*.)*.,*.)*',*.))*,*.)%-,*.)%),*.)'-,*.%/.,*.%/%,*.%.+,*.%-/,*.%-+,*.%,&,*.%&-,,*.--,,*.+,*,*.)./,*.%.%,*.%'.,*.'&&,*.'%+,)./.*,)./&.,)./*-,)./)',)./'&,)../),)...%,)..-%,)..,*,)..&&,)..+.,)..+),)..*,,)..)-,)..%.,)..%%,)..'&,)..'',).-./,).-.%-,*.&./,*.)&-,*.'-+,)./,%,)...*,)..)-,).-.&,).-&),).-)&,).-'*,).,.+,).,,.,).,&+,).,+*,).,*),).,)*,).,%&,).,'-,).,'%,).&/&,).&.+,).&-&,).&,.,).&,%,).&&&,).&++,).&*&.,*.+&.,*.%%*,)./)+,)..',,).-),,).,,.,).,)+,).&./,).&,%,).&*.,).&%/,).&'),).+..,).+-&,).+,+,).+&&,).++,,).+*.,).+*%,).+)&,).+%*,).+'+,).*/,,).*./,).*.*,).*-%,).*,%


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