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Normal DistributionNormal Distribution

Characterizing a Normal Characterizing a Normal DistributionDistribution

To completely characterize a normal distribution, we need to know only 2 things:

– The mean --- – The standard deviation ---

HAND CALCULATIONS FOR HAND CALCULATIONS FOR THE NORMAL DISTRIBUTIONTHE NORMAL DISTRIBUTION

• Probability tables have been created for the normal distribution expressed in terms of z, where

• z = the number of standard deviations x is from its mean, , i.e.

x

z

TYPES OF NORMAL TABLESTYPES OF NORMAL TABLES

• Two types of normal tables

– Tables giving probabilities from z = 0 to a positive value of z

– Cumulative normal tables giving probabilities from z = -∞ to any value of z• Excel uses this approach

0 z Z

Normal Curve with X and Z ScalesProbabilities from 0 to z

a X

Some tables give probability of falling between 0 and a positive z value

µ a X

0 z Z

Curve with X and Z ScalesCumulative Probabilities from -∞ to z

a X µ a X

A cumulative normal table gives the probability of falling between -∞ and any z value

Calculating Normal Probabilities Calculating Normal Probabilities Using Cumulative ProbabilitiesUsing Cumulative Probabilities

LEFT TAIL PROBABILITIESLEFT TAIL PROBABILITIES

P(X<a) or P(Z<z) = area between -∞ and a (or z)– Probability to the left – Cumulative normal table value

EXCEL:EXCEL: =NORMDIST(a,=NORMDIST(a,µ,µ,σσ,TRUE) ,TRUE) or

==NORMSDIST(z)NORMSDIST(z)

Calculating Normal Probabilities Calculating Normal Probabilities Using Cumulative ProbabilitiesUsing Cumulative Probabilities

RIGHT TAIL PROBABILITIESRIGHT TAIL PROBABILITIES

P(X>a) or P(Z>z) = area between a (or z) and +∞• Probability to the right• 1 - (Cumulative normal table value)

EXCEL:EXCEL: =1-NORMDIST(a,=1-NORMDIST(a,µ,µ,σσ,TRUE) ,TRUE) or

=1-=1-NORMSDIST(z)NORMSDIST(z)

Calculating Normal Probabilities Calculating Normal Probabilities Using Cumulative ProbabilitiesUsing Cumulative ProbabilitiesIN BETWEEN PROBABILITIESIN BETWEEN PROBABILITIES

P(a<X<b) or P(za<Z<zb)

– Probability between a and b on the X scale or between za and zb on the Z scale

– (Cumulative normal table value for zb) - (Cumulative normal table value for za)

EXCEL:EXCEL: =NORMDIST(b,=NORMDIST(b,µ,µ,σσ,TRUE) - N,TRUE) - NORMDIST(a,ORMDIST(a,µ,µ,σσ,TRUE),TRUE) or ==NORMSDIST(zNORMSDIST(zbb) - NORMSDIST(z) - NORMSDIST(zaa))

Calculating x Values from Calculating x Values from Cumulative Normal ProbabilitiesCumulative Normal Probabilities

Determining the x value such that the probability of getting a value less than x is p

– Find the cumulative normal probability, p, (approximately) in the table (to the leftleft of x) and note the corresponding z value

– x = µ + zσ

EXCEL:EXCEL: = NORMINV(p,= NORMINV(p,µ,µ,σσ) ) or = µ + = µ + NORMSINV(p)*NORMSINV(p)*σσ

EXAMPLEEXAMPLE

• Flight times from LAX to New York:– Are distributed normal– The average flight time is 320 minutes– The standard deviation is 20 minutes

Probability a flight takes Probability a flight takes exactly 315 minutesexactly 315 minutes

• P(X = 315 ) = 0– Since X is a continuous random variable

Probability a flight takes less Probability a flight takes less than 335 minutesthan 335 minutes

.4332

σ = 20

320 X335

0 Z20

320335 0.75

EXCEL=NORMDIST(335,320,20,TRUE)

OR =NORMSDIST(.75)

FROM TABLE

.7734

Probability a flight takes more Probability a flight takes more than 325 minutesthan 325 minutes

.4332

σ = 20

320 X

0 Z20

320325 0.25

EXCEL=1-NORMDIST(325,320,20,TRUE)

OR =1-NORMSDIST(.25)

325

FROM TABLE

.5987

1 - .5987 =.4013

Probability a flight takes Probability a flight takes between 303 and 348 minutesbetween 303 and 348 minutes

.4332

σ = 20

320 X

20

320303 303 348

-0.85 0 Z20

320348 1.40

EXCEL=NORMDIST(348,320,20,TRUE)-NORMDIST(303,320,20,TRUE)

OR =NORMSDIST(1.40)-NORMSDIST(-0.85)

.9192 - .1977 =.7215

FROM TABLE

.9192

FROM TABLE

.1977

75% of the flights arrive within 75% of the flights arrive within how many minutes?how many minutes?

.4332

σ = 20

320 X

0 ZEXCEL=NORMINV(.75,320,20)

OR =320 + NORMSINV(.75)*20

x

.7500 is to theleft of x

Try to find .7500 in the middleof the cumulative normal table.

0.67

The closest value is .7486 whichcorresponds to a z-value of 0.67.

x = 320 + .67(20)x = 320 + .67(20)

333.4333.4

85% of the flights take at least 85% of the flights take at least how many minutes?how many minutes?

.4332

σ = 20

320 X 0 Z

EXCEL=NORMINV(.15,320,20)

OR =320 + NORMSINV(.15)*20

x

.8500 is to theright of x

Thus,1-.8500 = .1500

is to the left of x

Try to find .1500 in the middleof the cumulative normal table.

-1.04

The closest value is .1492which corresponds to a z-value of -1.04

x = 320 + (-1.04)(20)x = 320 + (-1.04)(20)

299.2299.2

EXCEL CALCULATONS USING xEXCEL CALCULATONS USING x

EXCEL CALCULATONS USING zEXCEL CALCULATONS USING z

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• Normal distribution is completely characterized by µ and σ

• Calculation of:– “<” probabilities, “>” probabilities, and “in

between” probabilities using:• Cumulative probability table• NORMDIST and NORMSDIST functions

– “x values” and “z values” corresponding to a cumulative probability using:

• Cumulative probability table• NORMINV and NORMSINV functions