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REVIEW Normal DistributionPowerPoint Presentation

REVIEW Normal Distribution

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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

- 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

- 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

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

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 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 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 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 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 Probabilities

- 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 Probabilities

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

Probability a flight takes less than 335 minutes Probabilities

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 Probabilities

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 Probabilities

.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? Probabilities

.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? Probabilities

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 Probabilities

EXCEL CALCULATONS USING z Probabilities

REVIEW Probabilities

- 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

- “<” probabilities, “>” probabilities, and “in between” probabilities using:

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