Sampling Distribution of the Sample Proportion and ... · Sampling Distribution of a Sample Proportion • Sampling Distribution: for!categorical!data,!the!distribution!of! all!sample!proportions!for!a!given!sample!size!!and!given!

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: ________________


• 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: ________________


• 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: _______________________________


• 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

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 = ________; " = __________

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:!_______________________________________________



• Question: Is!a!normal!model!appropriate?

• Answer: ________• Independence: _________________________________________________________

• Randomization: ________________________________________________________

• 10% Condition: __________________________________________________________________________________

• Success/Failure Condition: _________________• _____________________________

• _____________________________



• 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!___________________


• 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:!_____________________________

_________________


• 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_______

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


• 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!________________________________

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


• Question: What!is!a!95%!confidence!interval!for!the!true!proportion!of!all!college!students!who!binge!drink?

• Answer:• Statistic: ________________

• Standard Error: _________________________________

• Multiplier: _____________________________________________________________________!_______________________________________


• To!find!middle!95%,!leave!out!_______!of!the!area!in!each!tail• Need!________!and!________!percentile

Note: Symmetry gives us the

upper multiplier of ________.

______ ______



• 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%.

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


• 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:

__________________________________________________________________________________

Example: Interpreting the Confidence Interval


• 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


• 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


Note: Because we leave out

half of the area in each tail,

use:

• .95 for 90% confidence



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

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!______________________

Documents

Sampling Distribution of the Sample Proportion and ... · Sampling Distribution of a Sample Proportion • Sampling Distribution: for!categorical!data,!the!distribution!of! all!sample!proportions!for!a!given!sample!size!!and!given!