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ØWhat!is!a!sampling!distribution?
Ø Standard!Error
Ø Conditions!for!Using!the!Normal!Model
Ø Standardizing!the!Sample!Proportion
Sampling Distribution of the Sample Proportion and Confidence Intervals
Lecture!10
Sections!9.1!� 9.4
Motivation: Sampling Distribution
• Scenario: Flip!a!fair!coin!once.!!Let!�Heads�!be!a!success.!!Calculate!the!proportion!of!heads!flipped.
• Question: What!does!the!probability!distribution!of!the!proportion!of!heads!for!one!flip!of!a!fair!coin!look!like?
• Answer: ________________
Motivation: Sampling Distribution
• Scenario: Flip!a!fair!coin!10!times.!!Let!�Heads�!be!a!success.!!Calculate!the!proportion!of!heads!flipped.
• Question: What!does!the!probability!distribution!of!the!proportion!of!heads!for!10!flips!of!a!fair!coin!look!like?
• Answer: ________________
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Motivation: Sampling Distribution
• Scenario: Flip!a!fair!coin!50!times.!!Let!�Heads�!be!a!success.!!Calculate!the!proportion!of!heads!flipped.
• Question: What!does!the!probability!distribution!of!the!proportion!of!heads!for!50!flips!of!a!fair!coin!look!like?
• Answer: _______________________________
Motivation: Sampling Distribution
• Question: What!do!you!notice!about!the!changing!shape!of!the!distributions?
• Answer: As!sample!size!increases:• Shape!changed!from!______________________________
• More!likely!to!get!sample!proportion!_____________!(________!of!distribution)
• Less!likely!to!get!_____________!sample!proportion!close!to!____________
• Question: What!are!these!distributions?
• Answer: ________________________________________________________________
Sampling Distribution of a Sample Proportion
• Sampling Distribution: for!categorical!data,!the!distribution!of!all!sample!proportions!for!a!given!sample!size!! and!given!probability!of!success!" on!any!individual!trial
• Idea:• Repeat!a!random!experiment!! times
• Count!the!number!of!successes!#
• Calculate!the!sample!proportion!of!successes! $" =%
&
• Expect! $" to!be!close!to!",!but!in!unusual!samples,! $"may!be!in!a!tail
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Standard Error
• Standard Error: standard!deviation!of!a!sampling!distribution• Measure!of!how!spread!out!sample!proportions!are!from!one!another
• How!much!we!expect!sample!proportion!to!deviate!from!population!proportion
• Dependent!upon!sample!size
! = 10
Standard!Error!=!.158
! = 50
Standard!Error!=!.0707
Example: Unusual Results
• Scenario: Flip!a!fair!coin!10!times!(left)!or!50!times!(right).
• Question: How!unusual!would!it!be!to!get!a!sample!proportion!of!.60!or!greater!in!each!situation?
• Answer:• 10!flips:!_________________________à Z = ________; " = __________
• 50!flips:!________________________!à Z = ________; " = __________
Example: Unusual Results
• Scenario: Flip!a!fair!coin!10!times!(left)!or!50!times!(right).
• Question: How!unusual!would!it!be!to!get!a!sample!proportion!of!.80!or!greater!in!each!situation?
• Answer:• 10!flips:!_________________________à Z = ________; " = __________
• 50!flips:!________________________!à Z = ________; " = __________
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Mean and Standard Error of Sampling Distribution
Suppose!we!are!sampling!from!a!population!with!categorical data!that!has!probability!of!success!".!!Then:
1. Mean:!' () = "
• Mean!of!the!sampling!distribution!of! $" equals!the!population!proportion!from!the!original!population
2. Standard!Error:!*+, ("- =).
&
• Standard!error!equals!the!square!root!of!the!success!probability!times!the!failure!probability!divided!by!the!sample!size
Conditions to Use Normal Model
To!use!a!normal!model!to!describe!sample!proportions,!the!following!assumptions!and!conditions!must!be!satisfied:
• Independence: Sampled!observations!must!be!independent
• Randomization: Sampling!method!must!be!unbiased!and!sample!must!be!representative!of!population
• 10% Condition: If!sampling!is!done!without!replacement!from!a!finite!population,!sample!size!must!be!less!than!10%!of!the!size!of!the!population
• Success/Failure Condition: Expected!number!of!successes!and!failures!must!both!be!at!least!10;!that!is,!!" / 10 and!!2 / 10
Example: Determining the Sampling Distribution
• Scenario: Flip!a!fair!coin!50!times.!Let!�Heads�!be!a!success.!!Calculate!the!proportion!of!heads!flipped.
• Question: What!is!the!sampling!distribution?
• Answer:1. Mean:!________________
2. Standard!Error:!_______________________________________________
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Example: Determining the Sampling Distribution
• Scenario: Flip!a!fair!coin!50!times.!Let!�Heads�!be!a!success.!!Calculate!the!proportion!of!heads!flipped.
• Question: Is!a!normal!model!appropriate?
• Answer: ________• Independence: _________________________________________________________
• Randomization: ________________________________________________________
• 10% Condition: __________________________________________________________________________________
• Success/Failure Condition: _________________• _____________________________
• _____________________________
Example: Determining the Sampling Distribution
• Scenario: Flip!a!fair!coin!50!times.!Let!�Heads�!be!a!success.!!Calculate!the!proportion!of!heads!flipped.
• Question: What!does!the!sampling!distribution!tell!us?
• Answer:• Expect!half!of!coin!flips!to!be!__________!and!half!to!be!__________
• Proportion!of!heads!tend!to!deviate!from.50!by!about!______!in!each!direction
• For!sample!sizes!of!50,!most!proportions!will!be!between!___________________
Example: Determining the Sampling Distribution
• Scenario: Basketball!player!makes!85%!of!his!free!throws.!!!He!plans!on!taking!30!shots!during!practice!one!day?
• Question: What!is!the!sampling!distribution!of!the!proportion!of!shots!he!will!make!during!this!practice!session?
• Answer:1. Mean:!________________
2. Standard!Error:!_____________________________
_________________
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Example: Determining the Sampling Distribution
• Scenario: Basketball!player!makes!85%!of!his!free!throws.!!!He!plans!on!taking!30!shots!during!practice!one!day?
• Question: Is!a!normal!model!appropriate?
• Answer: ________• Independence: ________________________________________________________________!___________________________________________________
• Randomization: __________________________________________________________________________________
• 10% Condition: _________________________________________________________
• Success/Failure Condition: _________________• _____________________________
• _____________________________
Standardizing the Sample Proportion
• If!a!normal!model!is!appropriate!to!model!categorical!data,!then!the!sample!proportion!can!be!standardized!using:
3 =(" 4 "
"2!
where!" is!the!probability!of!success!on!any!individual!trial
Example: Calculating Probabilities
• Scenario: Flip!a!fair!coin!50!times.!Let!�Heads�!be!a!success.
• Question: What!is!the!probability!that!the!sample!proportion!of!heads!is!greater!than!.60?
• Answer:
__________________!= ___________________________
= _____________________
= _____________________
= ____________
0 1.41_______
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Motivation: Confidence Intervals
• Scenario: Survey!of!796!college!students!found!288!(or!36.2%)!reported!binge!drinking!at!some!point!in!the!past!month
• Question: Is!one-third!a!plausible!value!for!the!proportion!of!all!college!students!who!binge!drink?
• Thoughts:• .362!is!_____________________________________
• Only!sampled!_____________________
• Different!sample!would!_______________________________________________________
• Conclusion: ____________
Point Estimate
• Point Estimate: best!individual!guess!for!an!unknown!population!parameter• Equal!to!the!sample!statistic• 6# is!a!point!estimate!for!'
• 7 is!a!point!estimate!for!8
• $" is!a!point!estimate!for!"
• Problems:• Never!equal!to!the!exact!value!of!the!parameter
• Cannot!display!the!effect!of!taking!larger!samples
• Do!not!give!us!an!idea!of!the!spread!of!the!population
Example: Point Estimates and Parameters
• Scenario: Survey!of!796!college!students!found!288!(or!36.2%)!reported!binge!drinking!at!some!point!in!the!past!month
• Question: What!notation!and!value!should!be!used!for!a!point!estimate!of!the!proportion!of!college!students!who!binge!drink?
• Answer: _______________
• Question: What!notation!and!value!should!be!used!for!the!population!proportion!of!all!college!students!who!binge!drink?
• Answer: ___________________________________
• Question: How!sure!are!we!that!.362!is!the!true!value!of!"?
• Answer: ___________________________________• .362!probably!________________________________
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Confidence Interval
• Confidence Interval: interval!of!plausible!values!for!an!unknown!parameter!that!is!calculated!based!on!the!responses!from!a!sample• Provides!us!with!a!range!of!values!that!could be!the!true!parameter
• Confidence Level:measure!of!how!certain!we!are!that!the!confidence!interval!contains!the!true!population!parameter
•Most!confidence!intervals!have!the!form:
Statistic ± Multiplier 9 Standard Error
Depends!on!
confidence!levelMargin of Error:!maximum!expected!
difference!between!statistic!and!parameter
Point!
Estimate
One Proportion Z-Interval
• To!estimate!a!population!proportion!" using!a!confidence!interval:
(" ± :("(2
!
where:• ::!Multiplier!(or!critical!value)!corresponding!to!desired!level!of!confidence
• $":!Sample!proportion!of!successes
• (2:!Sample!proportion!of!failures
• !:!Sample!size
Example: Confidence Interval
• Scenario: Survey!of!796!college!students!found!288!(or!36.2%)!reported!binge!drinking!at!some!point!in!the!past!month
• Question: What!is!a!95%!confidence!interval!for!the!true!proportion!of!all!college!students!who!binge!drink?
• Answer:• Statistic: ________________
• Standard Error: _________________________________
• Multiplier: _____________________________________________________________________!_______________________________________
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Example: Confidence Interval
• To!find!middle!95%,!leave!out!_______!of!the!area!in!each!tail• Need!________!and!________!percentile
Note: Symmetry gives us the
upper multiplier of ________.
______ ______
Example: Confidence Interval
• Scenario: Survey!of!796!college!students!found!288!(or!36.2%)!reported!binge!drinking!at!some!point!in!the!past!month
• Question: What!is!a!95%!confidence!interval!for!the!true!proportion!of!all!college!students!who!binge!drink?
• Answer: ____________________• Lower Bound:
____________________________________
• Question: Is!one-third!a!plausible!value!for!the!proportion!of!all!college!students!who!binge!drink?
• Answer: ________• 95%!C.I.!___________________________!so!.333!is!__________________________________
Upper Bound:
___________________________________________
Critical Values
• Critical Value:multiplier!in!a!confidence!interval!that!tells!how!many!standard!error!to!extend!in!each!direction!from!the!statistic• When!calculating!a!confidence!interval!for!a!proportion,!use!the!standard!normal!distribution!to!find!critical!values
• Critical!values!change!depending!on!the!confidence!level
Confidence Level Critical Value
90% 1<>?5
95% 1<@>
99% A<5B>
Note: Any level of confidence can be used. These are the most common.
However, it doesn’t make much sense to use a confidence level below 80%.
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Conditions and Assumptions
To!use!a!normal!model!to!estimate!a!proportion!using!a!confidence!interval,!the!following!assumptions!and!conditions!must!be!satisfied:
• Independence: Sampled!observations!must!be!independent
• Randomization: Sampling!method!must!be!unbiased!and!sample!must!be!representative!of!population
• 10% Condition: If!sampling!is!done!without!replacement!from!a!finite!population,!sample!size!must!be!less!than!10%!of!the!size!of!the!population
• Success/Failure Condition: Expected!number!of!successes!and!failures!must!both!be!at!least!10;!that!is,!! (" / 10 and!!(2 / 10
Example: Confidence Interval Conditions
• Scenario: Poll!of!870!Americans!asked!�Do!you!believe!there!is!intelligent!life!on!another!planet?�!!503!responded!that!they!did.
• Question: Are!the!conditions!for!calculating!a!99%!confidence!interval!satisfied?
• Answer: Yes• Independence: One!person�s!belief!in!intelligent!life!______________________!__________________________
• Randomization: Responses!from!a!poll!tend!to!be!a!_______________________
• 10% Condition: _______!is!less!than!10%!of!the!_____________________________
• Success/Failure Condition:
• ! $" = ____________________________
• !(2 = ____________________________
Example: Calculating a 99% Confidence Interval
• Scenario: Poll!of!870!Americans!asked!�Do!you!believe!there!is!intelligent!life!on!another!planet?�!!503!responded!that!they!did.
• Question: What!is!a!99%!confidence!interval!for!the!proportion!of!Americans!who!think!there!is!intelligent!life!on!another!planet?
• Answer:• Sample Proportions: $" = ________________;! (2 = ________________
• Multiplier: 99%!confidence!à _______________
• Confidence Interval:
__________________________________________________________________________________
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Example: Interpreting the Confidence Interval
• Scenario: Poll!of!870!Americans!asked!�Do!you!believe!there!is!intelligent!life!on!another!planet?�!!503!responded!that!they!did.
• Question: What!does!this!confidence!interval!mean!in!context?
• Answer: Couple!legitimate!interpretations�1. We!are!99%!confident!that!the!_____________________________________________!
______________________________________________________!is!between!________!and!________.
2. Proportions!between!________!and!________!are!plausible!values!for!the!_________________________________________________________________________________!_____________________________________
Example: Using the Confidence Interval
• Scenario: Poll!of!870!Americans!asked!�Do!you!believe!there!is!intelligent!life!on!another!planet?�!!503!responded!that!they!did.
• Question: Does!it!appear!that!a!majority!of!Americans!believe!there!is!intelligent!life!on!another!planet?
• Answer: ________• Confidence!interval!___________________________________
• .50!is!_______________________________
• Question: Can!we!conclude!that!a!majority!of!people!worldwide!believe!there!is!intelligent!life!on!another!planet?
• Answer: ________• Not!the!____________________________________
• Opinions!may!be!________________________________!in!other!countries
Using Excel
• Scenario: Poll!of!870!Americans!asked!�Do!you!believe!there!is!intelligent!life!on!another!planet?�!!503!responded!that!they!did.
Note: Because we leave out
half of the area in each tail,
use:
• .95 for 90% confidence
• .975 for 95% confidence
• .995 for 99% confidence
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Example: Measuring Accuracy of Confidence Intervals
• Scenario: Suppose!we!have!a!fair!coin!("C = <50).!!Flip!the!coin!100!times,!count!the!number!of!heads,!and!calculate!a!90%!confidence!interval!for!each.!!Repeat!the!experiment!10!times.
• Question: How!many!of!these!10!confidence!intervals!would!we!expect!to!contain!the!true!proportion!of!.50?
• Answer: ____• Bad!samples!________________________
• 90%!confidence!literally!means!______!of!the!intervals!will!not!contain!the!___________________________________________
• Takeaway:While!confidence!intervals!help!us!understand!what!a!parameter�s!value!might!be,!they!are!___________________
Example: Measuring Accuracy of Confidence Intervals
• Scenario: Simulated!results!of!100!coin!flips!and!their!90%!confidence!intervals.
• Observations:• 9!confidence!intervals!contained!true!proportion!______________!(_____)
• One!sample!(___)!was!a!bad!sample!whose!interval!__________________________
• Overall!90%!of!the!intervals!____________________
Example: Adjusting the Width of the Confidence Interval
• Scenario: Based!on!a!sample!of!796!students,!a!95%!confidence!interval!for!the!proportion!of!college!students!who!binge!drink!was!,<DA@F <D@5-.
• Question: Would!a!99%!confidence!interval!be!wider!or!narrower?
• Answer: ____________• Multiplier!would!have!been!_________!instead!of!1.96
• < D>A ± __________<GHI <HGJ
KLH= ____________________
• Question: What!are!the!ramifications!of!a!wider!interval?
• Answer: ______________________!about!where!true!proportion!lies!but!with!a!____________________!of!plausible!values
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Example: Adjusting the Width of the Confidence Interval
• Scenario: Based!on!a!sample!of!796!students,!a!95%!confidence!interval!for!the!proportion!of!college!students!who!binge!drink!was!,<DA@F <D@5-.
• Question: Assuming!the!same!sample!proportion!of!.362,!would!a!95%!confidence!interval!have!been!wider!or!narrower!with!a!sample!size!of!1592?
• Answer: ________________
• Standard!error!gets!smaller:!SE $" =<GHI <HGJ
NOLI= ______!instead!of!.017
• < D>A ± 1<@>,______- = ___________________
Sample Size Calculations
• To!determine!how!large!a!sample!is!needed!to!attain!a!margin!of!error!of!P:
! =: ("(2
P
I
• Round!up!if!! is!a!decimal
• Problem: Need!__________________,!but!we!need!a!___________!to!find! ("
• Solutions:1. Use!a!value!of! $" from!a!________________________________
2. Use! $" = <50 as!the!___________________________________________!if!no!previous!information!exists
Example: Sample Size Calculations
• Scenario: Company!making!a!new!computer!processor!wants!to!estimate!defect!rate.!!Defect!rate!on!old!processors!was!2%.
• Question: How!large!a!sample!is!needed!to!estimate!the!defect!rate!to!within!1%!using!95%!confidence?
• Answer: __________________________________!à Round!up!to!__________
• Question: How!would!the!sample!size!have!changed!if!we!had!used!the!most!conservative!estimate!of! (" = <50?
• Answer: __________________________________
• Takeaway: Sample!sizes!get!_________!as! (" gets!______________________