Ø Law!of!Large!Numbers
Ø Sampling!Distribution!Idea
Ø Central!Limit!Theorem
ØDetermining!the!Sampling!Distribution
ØUsing!the!Sampling!Distribution
Ø Standard!Normal!vs.!t-Distribution
Sampling Distribution of the Sample Mean and t-Distribution
Lecture!12
Section!11.1�11.4
Motivation: Law of Large Numbers
• Scenario: SAT!scores!are!normally!distributed!with!! = 1000 and!" = 140.!!Ask!a!random!college!freshman!for!their!SAT!score.
• Question: How!unusual!would!it!be!for!their!score!to!be!greater!than!1080?
• Answer: ________________________• 1080!is!_________________________________________________
______________
Motivation: Law of Large Numbers
• Scenario: SAT!scores!are!normally!distributed!with!! = 1000 and!" = 140.!!Ask!9!random!college!freshmen!for!their!SAT!scores.
• Question: How!unusual!would!it!be!for!their!sample!mean!score!to!be!greater!than!1080?
• Answer: ________________________• Requires!each!to!do!____________________________________________
Example Data: #$ = %&'&860,!930,!1010,!1050,!1090,!1120,!1180,!1200,!1280
Motivation: Law of Large Numbers
• Scenario: SAT!scores!are!normally!distributed!with!! = 1000 and!" = 140.!!Ask!100!random!college!freshmen!for!their!SAT!scores.
• Question: How!unusual!would!it!be!for!their!sample!mean!score!to!be!greater!than!1080?
• Answer: ________________________• We!would!expect!50!to!score!________!1000!and!50!to!score!_________,!but�
• A!sample!mean!of!1080!would!require!______________________!to!score!above!1000!with!______________________
• Law of Large Numbers: as!the!sample!size!increases,!the!statistic!gets!closer!and!closer!to!the!true!value!of!the!parameter
Motivation: Sampling Distribution
• Scenario: SAT!scores!are!normally!distributed!with!! = 1000 and!" = 140.!!Ask!a!random!college!freshman!for!their!SAT!score.
• Question: What!is!the!probability!that!a!single!student!scores!above!1080?
• Answer: ( ) > 1080 = ________________________________________________
Distribution!of!
1!Observation
1080
______________
Motivation: Sampling Distribution
• Scenario: SAT!scores!are!normally!distributed!with!! = 1000 and!" = 140.!!Ask!9!random!college!freshmen!for!their!SAT!scores.
• Question: What!is!the!probability!the!average!of!these!9!scores!is!above!1080?
• Answer: ( *) > 1080 = _________________________________
Distribution!of!
9!Observations
1080
______________
Motivation: Sampling Distribution
• Scenario: SAT!scores!are!normally!distributed!with!! = 1000 and!" = 140.!!Ask!100!random!college!freshmen!for!their!SAT!scores.
• Question: What!is!the!probability!the!average!of!these!100!scores!is!above!1080?
• Answer: ( *) > 1080 = _________________________________
Distribution!of!
100!Observations
1080
_________________
Motivation: Sampling Distribution
• Question: What!do!you!notice!about!how!the!probabilities!change!as!the!sample!size!increases?
• Answer:• Shape!________________________
• More!area!condensed!__________________!+! = 1000,,!less!__________________• Probability!mean!exceeds!1080!___________________
• 1!person!à ____________
• 9!people!à ____________
• 100!people!à_______________
• Question: What!are!these!distributions?
• Answer: ____________________________________________________
Sampling Distribution of a Sample Mean
• Sampling Distribution: for!quantitative!data,!the!distribution!of!all!sample!means!for!a!given!sample!size!-,!population!mean!!,!and!population!standard!deviation!"
• Standard Error: standard!deviation!of!a!sampling!distribution• Measure!of!how!spread!out!sample!means!are!from!one!another
• How!much!we!expect!sample!means!to!deviate!from!the!population!mean
• Dependent!upon!sample!size
Mean and Standard Error of Sampling Distribution
Suppose!we!are!sampling!from!a!population!with!quantitative data!that!has!mean!! and!standard!deviation!".!!Then:
1. Mean:!! *. = !• Mean!of!the!sampling!distribution!of! *) equals!the!population!mean!from!the!original!population
2. Standard!Error:!/2+*), =3
5• Standard!error!equals!the!standard!deviation!of!the!original!population!divided!by!the!square!root!of!the!sample!size
Motivation: Central Limit Theorem
• Scenario: Roll!a!single!fair!die.
• Question: What!does!the!probability!distribution!look!like?
• Answer: __________________________!(__________________)• All!outcomes!_______________________
Motivation: Central Limit Theorem
• Scenario: Roll!2!fair!dice!and!average!the!rolls.
• Question: What!does!the!sampling!distribution!look!like?
• Answer: _____________!(_______________________)• Means!between!______!and!______!are!___________________
• Means!of!___!and!___!are!_________________________
Motivation: Central Limit Theorem
• Scenario: Roll!25!fair!dice!and!average!the!rolls.
• Question: What!does!the!sampling!distribution!look!like?
• Answer: _____________________________• Most!means!between!___!and!____
• Means!________________!or!___________________!will!likely!_______________________
Central Limit Theorem
• Central Limit Theorem: The!mean!of!a!random!sample!has!a!sampling!distribution!whose!shape!can!be!approximated!by!a!normal!model.!!The!larger!than!sample,!the!better!the!approximation!will!be.
• Rules of Thumb: Take a random sample of size - from some distribution ) of quantitative values. Then the distribution of the sample mean *) is normal if any of the following are true:
1. The original distribution of ) is normal. (Sample size does not matter.)
2. The distribution of ) is unimodal and somewhat symmetric and - 6 17.
3. The distribution of ) is skewed and - 6 90.
Conditions to Use Normal Model
To!use!a!normal!model!to!describe!sample!means,!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
• Nearly Normal: Shape!of! *)must!be!approximately!normal!using!one!of!the!Rules!of!Thumb
Example: Finding Sampling Distribution
• Scenario: SAT!scores!follow!normal!distribution!with!! = 1000and!" = 140.!!Ask!9!random!college!freshmen!what!their!SAT!scores!were.
• Question: What!is!the!sampling!distribution!of!the!sample!mean!SAT!score?
• Answer:1. Mean:!! *. = ____________
2. Standard!Error:!/2 *) = _____________________
Example: Finding Sampling Distribution
• Scenario: SAT!scores!follow!normal!distribution!with!! = 1000and!" = 140.!!Ask!9!random!college!freshmen!what!their!SAT!scores!were.
• Question: Is!a!normal!model!appropriate?
• Answer: ________• Independence: Students�!scores!likely!_____________________________________
• Randomization: Representative!sample!taken!from!___________________________________
• Nearly Normal: Original!population!is________________________________________________
Example: Finding Sampling Distribution
• Scenario: Die!roll!follows!a!uniform!distribution!with!a!mean!of!3.5!and!a!standard!deviation!of!1.72.!!Roll!25!fair!dice!and!average!the!rolls.
• Question: What!is!the!sampling!distribution!of!the!sample!mean!of!25!die!rolls?
• Answer:1. Mean:!! *. = ____________
2. Standard!Error:!/2 *) = _____________________
Example: Finding Sampling Distribution
• Scenario: Die!roll!follows!a!uniform!distribution!with!a!mean!of!3.5!and!a!standard!deviation!of!1.72.!!Roll!25!fair!dice!and!average!the!rolls.
• Question: Is!a!normal!model!appropriate?
• Answer: Yes• Independence: One!die!roll!_______________________________
• Randomization: Sample!is!representative!of
• Nearly Normal: Original!population!is__________________________!,!but!__________so!shape!of! *) is!___________
Example: Finding Sampling Distribution
• Scenario: Batting!averages!in!baseball!follow!a!beta!distribution!with!mean!.250!and!standard!deviation!.08.!!Randomly!sample!4!batters!and!average!their!batting!averages.
• Question: What!is!the!sampling!distribution!of!the!sample!mean!batting!average?
• Answer:1. Mean:!! *. = ____________
2. Standard!Error:!/2 *) = _____________________
Example: Finding Sampling Distribution
• Scenario: Batting!averages!in!baseball!follow!a!beta!distribution!with!mean!.250!and!standard!deviation!.08.!!Randomly!sample!4!batters!and!average!their!batting!averages.
• Question: Is!a!normal!model!appropriate?
• Answer: ________• Independence: Batters�!at-bats!and!averages!______________________________
• Randomization: Batters!sampled!from_____________________________________________
• Nearly Normal: Original!population!is___________!and!________!so!shape!of! *) is!__________________
Standardizing the Sample Mean
• If!a!normal!model!is!appropriate!to!model!quantitative!data,!then!the!sample!mean!can!be!standardized!using:
: =*; < !
/2+*;,
where!! and!" are!the!mean!and!standard!deviation!of!) and
/2 *; =3
5
Example: Calculating Probabilities
• Scenario: SAT!scores!follow!normal!distribution!with!! = 1000and!" = 140.!!Ask!9!random!college!freshmen!what!their!SAT!scores!were.
• Question: What!is!the!probability!the!sample!mean!is!greater!than!1080?
• Answer:
( *) > 1080 = ___________________________________
= ________________________= __________
0 1.71
Sampling Distributions
• If!" is!known!and!the!sample!mean! *? is!normally!distributed,!then! *?can!be!standardized!using:
: =@A < !
B" -• Because!it!is!a!parameter,!" is!usually!an!unknown!value.!!Instead,!we!estimate!" using!the!sample!standard!deviation!C.!!However:
: D@A < !
BC -• Instead�
E =@A < !
BC -where!E stands!for!the!Student�s!t-distribution.
Student’s t-Distribution
• Student’s t-Distribution: continuous!probability!distribution!similar!to!the!standard!normal!in!that!it!is:
• Symmetric!and!bell-shaped
• Centered!at!0
but!differs!from!the!standard!normal!because!it:• Is!a!family!of!distributions!whose!shape!changes!depending!on!the!
degrees of freedom
• Has!fatter!tails!and!is!shorter!in!the!middle
Standard Normal vs. t-Distribution
___________________!
_____________ ________________!
____________________
_____________
Degrees of Freedom
• Degrees of Freedom:measure!of!how!much!information!is!contained!in!a!sample!that!determines!the!shape!of!the!distribution!that!is!appropriate!for!the!situation
• Occurs!in!the!t-distribution,!chi-square!distribution,!and!F-distribution
• Range!from!1!to!infinity!and!cause!the!distributions!to!change!shape
• Often!denoted!by!the!letter!F and!is!placed!in!the!subscript!of!the!statistic!(i.e.!EG, HG
I,!JGKL GM)• More!on!the!chi-square!distribution!and!F-distribution!later!in!the!semester��
• In!the!t-distribution,!degrees!of!freedom!are!dependent!upon!the!sample!size.
• Larger!sample!àMore!information!àMore!degrees!of!freedom
Examples of t-Distributions
Note: Because of the change in shape,
calculating probabilities in the tails
now brings an additional challenge
that will require us to use software.
t-Distribution Table
Table!Continues
Two-Tail Probability: Total!
area!in!both!tails!beyond!
positive!and!negative!t-statistic
One-Tail Probability: Total!
area!in!a!single!tail!beyond!
either!positive!t-statistic!or!
negative!t-statistic
Note: Complete t-distribution
table posted on CourseWeb.
Example: Critical Value
• Question: What!critical!value!should!be!used!for!a!95%!confidence!interval!with!10!degrees!of!freedom?
• Answer: _____________________• Leaves!out!____________________
• Note: : = _______!for!a!95%!CI
____________