Review normal distribution
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REVIEW Normal Distribution. Characterizing a Normal Distribution. To completely characterize a normal distribution, we need to know only 2 things: The mean ---  The standard deviation --- . HAND CALCULATIONS FOR THE NORMAL DISTRIBUTION.

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REVIEW Normal Distribution

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Review normal distribution

REVIEW

Normal Distribution


Characterizing a normal distribution

Characterizing a Normal Distribution

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

  • The mean --- 

  • The standard deviation --- 


Hand calculations for the normal distribution

HAND CALCULATIONS FOR THE 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.


Types of normal tables

TYPES 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


Review normal distribution

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

 a X

Normal Curve with X and Z ScalesProbabilities from 0 to z

µ a X

0 z Z


Review normal distribution

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

 a X

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

µ a X

0 z Z


Calculating normal probabilities using cumulative probabilities left tail probabilities

Calculating Normal Probabilities Using Cumulative 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:

    =NORMDIST(a,µ,σ,TRUE) or

    =NORMSDIST(z)


Calculating normal probabilities using cumulative probabilities right tail probabilities

Calculating Normal Probabilities Using Cumulative 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:

    =1-NORMDIST(a,µ,σ,TRUE) or

    =1-NORMSDIST(z)


Calculating normal probabilities using cumulative probabilities in between probabilities

Calculating Normal Probabilities Using Cumulative 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:

    =NORMDIST(b,µ,σ,TRUE) - NORMDIST(a,µ,σ,TRUE)or =NORMSDIST(zb) - NORMSDIST(za)


Calculating x values from cumulative normal probabilities

Calculating x Values from Cumulative 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 leftof x) and note the corresponding z value

  • x = µ + zσ

    EXCEL:

    = NORMINV(p,µ,σ) or

    = µ + NORMSINV(p)*σ


Example

EXAMPLE

  • 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 exactly 315 minutes

Probability a flight takes exactly 315 minutes

  • P(X = 315 ) = 0

    • Since X is a continuous random variable


Probability a flight takes less than 335 minutes

Probability a flight takes less than 335 minutes

FROM TABLE

.7734

.4332

335

σ = 20

320 X

0.75

0 Z

EXCEL

=NORMDIST(335,320,20,TRUE)

OR =NORMSDIST(.75)


Probability a flight takes more than 325 minutes

Probability a flight takes more than 325 minutes

FROM TABLE

.5987

1 - .5987 =

.4013

.4332

325

σ = 20

320 X

0.25

0 Z

EXCEL

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

OR =1-NORMSDIST(.25)


Probability a flight takes between 303 and 348 minutes

Probability a flight takes between 303 and 348 minutes

.9192 - .1977 =

.7215

FROM TABLE

.9192

FROM TABLE

.1977

.4332

303 348

σ = 20

320 X

-0.85

1.40

0 Z

EXCEL

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

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


75 of the flights arrive within how many minutes

75% of the flights arrive within how many minutes?

.7500 is to the

left of x

x = 320 + .67(20)

.4332

The closest value is .7486 which

corresponds to a z-value of 0.67.

x

333.4

0.67

σ = 20

Try to find .7500 in the middle

of the cumulative normal table.

320 X

0 Z

EXCEL

=NORMINV(.75,320,20)

OR =320 + NORMSINV(.75)*20


85 of the flights take at least how many minutes

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

Thus,

1-.8500 = .1500

is to the left of x

.8500 is to the

right of x

x = 320 + (-1.04)(20)

.4332

The closest value is .1492

which corresponds to a

z-value of -1.04

299.2

x

-1.04

σ = 20

Try to find .1500 in the middle

of the cumulative normal table.

320 X

0 Z

EXCEL

=NORMINV(.15,320,20)

OR =320 + NORMSINV(.15)*20


Excel calculatons using x

EXCEL CALCULATONS USING x


Excel calculatons using z

EXCEL CALCULATONS USING z


Review

REVIEW

  • 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


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