Download Section 2.2: The Normal Distributions Normal Distributions

Section 2.2 Notes - Almost Done
Section2.2:TheNormalDistributions
NormalDistributions
Aclassofdistributionswhosedensitycurvesaresymmetric,
uni-modal,andbell-shaped.
NormaldistributionsareVERYimportantinstatistics.
Whichnumericalsummarywouldweusetodescribethe
centerandspreadofaNormaldistribution?
Notation:
Section 2.2 Notes - Almost Done
CalculatingσusingtheNormaldensitycurve
Section 2.2 Notes - Almost Done
The68-95-99.7Rule-IntheNormaldistributionwith
meanμadstandarddeviationσ:
• 68%ofalltheobservationsfallwithinonestandard
deviation(σ)ofthemeanμ(inbothdirections)
• 95%ofalltheobservationsfallwithintwostandard
deviations(2σ)ofthemeanμ(inbothdirections)
• 99.7%ofalltheobservationsfallwithinthreestandard
deviations(3σ)ofthemeanμ(inbothdirections)
Section 2.2 Notes - Almost Done
Thedistributionofheightsofwomenaged20to29isapproximately
Normalwithmean64inchesandstandarddeviation2.7inches.Use
the68-95-99.7ruletoanswerthefollowingquestions.
(a)Betweenwhatheightsdothemiddle95%ofyoungwomenfall?
(b)Whatpercentofyoungwomenaretallerthan61.3inches?
Youtry:
Thelengthofhumanpregnanciesfromconceptiontobirthvaries
accordingtoadistributionthatisapproximatelyNormalwithmean266
daysandstandarddeviation16days.Usethe68-95-99.7ruleto
answerthefollowingquestions.
(a)Betweenwhatvaluesdothelengthsofthemiddle68%ofall
pregnanciesfall?
(b)Howshortaretheshortest2.5%ofallpregnancies?
(c)Whatpercentofpregnanciesarelongerthan314days?
Section 2.2 Notes - Almost Done
ThestandardNormaldistribution
• InXinitelymanyNormaldistributions
Oneforeverypossiblecombinationofmeansandstandard
deviations
• StandardNormaldistribution-N(0,1)
• Wecanstandardizeanyvalueofavariable,x.Thisstandardized
valueiscalledthez-score,orz.Ifweactuallywanttodo
calculationsusingthisstandardizedscoreweneedtoknowthe
distributionoftheoriginalvariable.Iftheoriginalvariableis
Normalthenthez-scorecomesfromastandardNormal
distribution.
•Az-scoretellsushowmanystandarddeviationstheoriginal
observationfallsawayfromitsmeanANDinwhich
direction.
Section 2.2 Notes - Almost Done
YOUTRY:
Theheightsofwomenaged20to29areapproximatelyNormal
withmean64inchesandstandarddeviation2.7inches:N(64,2.7).
Menthesameagehavemeanheight69.3incheswithstandard
deviation2.8inchesandfollowanapproximatelyNormal
distribution:N(69.3,2.8).Whatarethez-scoresforawoman6feet
tallandaman6feettall?Sayinsimplelanguagewhatinformation
thez-scoresgivetheactualheightsdonot.
Section 2.2 Notes - Almost Done
UsingtheNormalcurvetodetermineproportions/percentiles
(Beyondthe68-95-99.7rule)
• TheareaunderanyNormalcurve(ordensitycurveforthatmatter)
isequalto1.
• Ifwewanttoknowtheproportionofobservationsthatliewithina
certainrangeofobservationvalueswelookfortheareaofthe
densitycurvebetweenthosetwovalues(forANYdensitycurve-
notjustNormal)
• WehaveatablethatgivesusthesevaluesforONLYthestandard
Normaldistribution.
UsetableAto+indtheproportionofobservationsfroma
standardNormaldistributionthatsatis+ieseachofthe
followingstatements.Ineachcase,sketchastandard
Normalcurveandshadetheareaunderthecurvethatis
theanswertothequestion.
(a)z<2.66
(b)z>-1.45
(c)-0.58< z<1.93
SincewecanstandardizeANYNormaldistributionwecanusethistableforANYNormaldistribution.
Section 2.2 Notes - Almost Done
FOREXAMPLE:
SupposethattheheightsofyoungwomenhaveaNormaldistribution,
N(64,2.7).Whatproportionorpercentageofallyoungwomenare
lessthan70inchestall?
Usingthesamedistributionfromthelastexample,whatproportionof
womenaregreaterthan60inchestall?
Section 2.2 Notes - Almost Done
Usingthesamedistributionfromthelastexample,whatproportionof
womenarebetweentheheightsof62and68inchestall?
YOUTRY:
UsetableAtoXindtheproportionofobservationsfroma
standardNormaldistributionthatsatisXieseachofthe
followingstatements.Ineachcase,sketchastandard
Normalcurveandshadetheareaunderthecurvethatis
theanswertothequestion.
(a)z<2.85
(b)z>2.85
(c)z>-1.66
(d)-1.66<z<2.85
Section 2.2 Notes - Almost Done
Inthe2008Wimbledontennistournament,RafaelNadalaveraged115
milesperhour(mph)onhisXirstserves.Assumethatthedistributionof
hisXirstservespeedsisNormalwithameanof115mphandastandard
deviationof6mph.
a)AboutwhatproportionofhisXirstserveswouldyouexpecttoexceed
120mph?
b)WhatpercentofRafaelNadal’sXirstservesarebetween100and110
mph?
x
z
TableA%
Section 2.2 Notes - Almost Done
Sometimeswearegivenaparticularproportionofobservationsthatlie
aboveorbelowsomeobservedvalueandwewantto7indthatobserved
value.
FOREXAMPLE:
UsetableAto+indthevalueof zofastandardNormalvariablethat
satis2ieseachofthefollowingconditions.
(a)Thepointzwith34%oftheobservationsfallingbelowit.
(b)Thepointzwith12%oftheobservationsfallingaboveit.
FOREXAMPLE:
SupposethattheheightsofyoungwomenhaveaNormaldistribution,
N(64,2.7).Whatheightsare75%ofyoungwomenlessthan?
Section 2.2 Notes - Almost Done
YOUTRY:
UsetableAtoXindthevaluezofastandardNormalvariablethatsatisXies
eachofthefollowingconditions.(UsethevalueofzfromTableAthat
comesclosesttosatisfyingthecondition.)Ineachcase,sketchastandard
Normalcurvewithyourvalueofzmarkedontheaxis.
(a)Thepointzwith25%oftheobservationsfallingbelowit.
(b)Thepointzwith40%oftheobservationsfallingaboveit.
Section 2.2 Notes - Almost Done
ScoresontheWechslerAdultIntelligenceScaleareapproximately
Normallydistributedwithμ=100andσ=15.
(a)WhatIQscoresfallinthelowest25%ofthedistribution?
(b)HowhighanIQscoreisneededtobeinthehighest5%?
Normal Calculations using your Calculator
Taking an observation - x - and converting it to a percentile:
1. Press 2nd VARS to get the DISTR menu
2. select option 2:normalcdf
3. enter the lower bound, upper bound, µ, σ
4. If you are interested in the values LESS than a certain xvalue then use -10^99 as your lower bound
5. If you are interested in the values GREATER than a
certain x-value then use 10^99 as your upper bound.
6. IF you have already converted your x-value to its
corresponding z-score, don't enter the µ and σ
Section 2.2 Notes - Almost Done
Taking a percentile (percent to the left of a value) and
converting it to a value of the original variable (x):
1. Press 2nd VARS to get the DISTR menu
2. select option invnorm
3. enter the percentile (percent to the left of a value), µ, σ
4. Remember, if you are given the percent GREATER than
a value, subtract from 1 (100%) to get the percentile.
5. IF you just want the correspondingz-score (not x-value)
don't enter µ and σ
Assessing Normality
As we've seen, Normal models provide good models for some
distributions of real data.
However, some common variables are usually skewed and therefore
distinctly non-Normal.
It is risky to assume that a distribution is Normal without inspecting the
data or even if the data are uni-modal and roughly symmetric.
We an check to see if the distribution of the data follow the 68-95-99.7
rule.
We an also use a Normal Probability Plot - a plot of each observation
against the corresponding z-score for the percentile it represents. If there
is a strong linear pattern, the distribution is close to Normal.
Section 2.2 Notes - Almost Done
Normal Probability Plots on the Calculator
1. Enter the data into a single list
2. Go to STATPLOT (2nd Y=)
3. Turn a single plot on
4. Select the LAST of the graphs - bottom right
5. Select the correct Data List
6. Select X as your Data Axis
7. Choose the mark you would like to see in your graph for
the points
8. Go to your graph and use Zoom option 9:zoomstat
Section 2.2 Notes - Almost Done
Themeasurementslistedbelowdescribetheuseablecapacity(incubicfeet)of
asampleof36side-by-siderefrigerators.ArethedataclosetoNormal?
12.913.714.114.214.514.514.614.715.115.215.315.3
15.315.315.515.615.615.816.016.016.216.216.316.4
16.516.616.616.616.817.017.017.217.417.417.918.4
Section 2.2 Notes - Almost Done
Homework:p.131#s41-59odd,63,65,66,
68,69-74all