# STAT 100 Study Guide - Final Guide: Central Limit Theorem, Blood Pressure, Process State

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STAT 100

Chance and Data Analysis

Winter 2018

Final

Professor: Gamage Perera

Exam Guide

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Confidence Interval

Confidence$interval$

•A!statistic!is!a!number!that!summarizes!something!about!a!sample.

•A!parameter!is!a!number!that!summarizes!something!about!a!population.

•The!sample!proportion!(denoted!pˆ),!0.17,!estimates!the!population!proportion

Sampling$distributions$

•For!every!random!sample!taken!the!p^!value!will!be!slightly!different

•The!sampling!distribution!of!pˆ!describes!the!pattern!of!those!many!sample

proportion!values.

•We!know!that!when!the!sample!size!n!is!reasonably!large,!the!sampling

distribution!of!pˆ!is!approximately!normal.

•The!sampling!distribution!of!pˆ!has!a!mean!of!p,!the!true!population!proportion.

•The!sampling!distribution!of!pˆ!has!a!standard!deviation!of:

o

Using$the$68,$94,$99.7$rule$

•Since!the!sampling!distribution!of!pˆ!is!approximately!normal,!the!empirical!rule

tells!us!about!95%!of!all!possible!samples!will!produce!a!value!of!pˆ!within!2

standard!deviations!of!the!true!p!(which!is!unknown).

•So!in!about!95%!of!samples,!pˆ!will!be!between!p!−!2!×!(sd)!and!p!+!2!×!(sd).

•Then!logically,!in!about!95%!of!samples,!the!true!p!will!be!within!2!standard

deviations!of!whatever!pˆ!we!got!from!that!sample.

•In!other!words,!in!about!95%!of!samples,!the!unknown!p!will!be!between!pˆ!−

2×(sd)!and!pˆ!+!2!×!(sd)

•The!population!proportion!p!does!not!change!from!sample!to!sample.

•It!is!the!sample!proportion!pˆ!that!changes!across!different!samples

•Using!the!confidence!interval!formula!you!must!have!the!standard!deviation

before!calculation!your!interval

o

n

pp )1( −

n

pp

p)1(

2

⌢⌢

⌢−

×±

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Different$confidence$intervals$

•To!change!the!level!of!confidence,!we!just!adjust!the!number!of!standard

deviations!in!the!“margin!of!error”!part!of!the!formula.

•Actually!for!99%!confidence,!we!would!use!2.58!rather!than!2!or!3.

•So!99%!confidence!is!better!than!95%!confidence,!right?

•In!some!ways,!it!is:!We!have!better!odds!that!the!interval!we!get!will!contain!the

true!parameter!value.

•But!the!interval!will!also!be!wider!–!less!informative

•For!90%!confidence,!we!would!use!1.64!rather!than!2.

•This!is!more!precise!(more!informative)!than!the!95%!interval,!but!there’s!more

of!a!chance!that!a!90%!interval!will!miss!the!true!parameter!value,!across

repeated!samples.

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