Sampling Distribution of the Sample Mean and t …...distribution )of quantitative values. Then the distribution of the sample mean )*is normal if any of the following are true: 1

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!_________________________________________________

______________


• 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



• 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

______________



• Question: What!is!the!probability!the!average!of!these!9!scores!is!above!1080?

• Answer: ( *) > 1080 = _________________________________

Distribution!of!

9!Observations

1080

______________



• Question: What!is!the!probability!the!average!of!these!100!scores!is!above!1080?

• Answer: ( *) > 1080 = _________________________________

Distribution!of!

100!Observations

1080

_________________


• 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!_______________________


• 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!_________________________


• 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 *) = _____________________



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

• Answer: ________• Independence: Students�!scores!likely!_____________________________________

• Randomization: Representative!sample!taken!from!___________________________________

• Nearly Normal: Original!population!is________________________________________________


• 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 *) = _____________________


• 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.


• Answer: Yes• Independence: One!die!roll!_______________________________

• Randomization: Sample!is!representative!of

• Nearly Normal: Original!population!is__________________________!,!but!__________so!shape!of! *) is!___________


• 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 *) = _____________________


• 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.


• 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


• 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

____________

Documents

Sampling Distribution of the Sample Mean and t …...distribution )of quantitative values. Then the distribution of the sample mean )*is normal if any of the following are true: 1