THE NORMAL DISTRIBUTION

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• THE NORMAL DISTRIBUTION

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NORMAL DISTRIBUTION

• Frequently called the bell shaped curve
• Important because
• Many phenomena naturally have a bell shaped
  distribution
• Normal distribution is the limiting
  distribution for many statistical tests

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Characterizing a Normal Distribution

• To completely characterize a normal distribution,
  we need to know only 2 parameters
• Its mean (?)
• Its standard deviation (?)
• A normal distribution curve can be now drawn as
  follows

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NORMAL PROBABILITY DENSITY CURVE
s
µ
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NORMAL PROBABILITY DENSITY FUNCTION

• The density function for a normally distributed
  random variable with mean µ and standard
  deviation s is

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CALCULATING NORMAL PROBABILITIES

• There is no easy formula for integrating f(x)
• However, tables have been created for a
  standard normal random variable, Z, which has ?
  0 and s 1
• Probabilities for any normal random variable, X,
  can then be found by converting the value of x to
  z where

z the number of standard deviations x is from
its mean, ?
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Standardizing a Normal Random Variable X to Z
Standardization preserves probabilities
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Representing X and Z on the Same Graph

• Both the x-scale and the corresponding z-scale
  can be represented on the same graph

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10
2
s 2
10
0
Z
2
Example Suppose µ 10, s 2. Illustrate P(X gt
14).
Thus P(X gt 14) P(Zgt2)
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Facts About theNormal Distribution

• mean median mode
• Distribution is symmetric
• 50 of the probability is on each side of the
  mean
• Almost all of the probability lies within 3
  standard deviations from the mean
• On the z-scale this means that almost all the
  probability lies in the interval from
  z -3 to z 3

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Using the Cumulative Normal (Cumulative z) Table

• The cumulative z-table
• Gives the probability of getting a value of z or
  less
• P(Z lt z)
• Left-tail probabilities
• Excel gives left-tail probabilities
• To find any probability from a z-table
• Convert the problem into one involving only
  left-tail probabilities

P(Z lt a) z-table value for a
P(Z gt a) 1-(z-table value for a)
P(altZltb) (z-table value for b) (z-table value
for a)
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Using the Normal Table

• Look up the z value to the first decimal place
  down the first column
• Look up the second decimal place of the z-value
  in the first row
• The number in the table gives the probability
  P(Zltz)

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Example

• X is normally distributed with ? 244, ? 25
• Find P(X lt 200)
• For x 200, z (200-244)/25 -1.76

200
-1.76
0
Z
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Using Cumulative Normal Tables

• z .00 .01 .02 .03 .04
  .05 .06 .07 .08 .09
• -3.0 .0013 .0013 .0013 .0012 .0160 .0012
  .0011 .0011 .0010 .0010
• -2.9 .0019 .0018 .0018 .0017 .0016 .0016
  .0015 .0015 .0014 .0014
• -2.8 .0026 .0025 .0024 .0023 .0023 .0022
  .0021 .0021 .0020 .0019
• -2.7 .0035 .0034 .0033 .0032 .0031 .0030
  .0029 .0028 .0027 .0026
• -2.6 .0047 .0045 .0044 .0043 .0041 .0040
  .0039 .0038 .0037 .0036
• -2.5 .0062 .0060 .0059 .0057 .0055 .0054
  .0052 .0051 .0049 .0048
• -2.4 .0082 .0080 .0078 .0075 .0073 .0071
  .0069 .0068 .0066 .0064
• -2.3 .0107 .0104 .0102 .0099 .0096 .0094
  .0091 .0089 .0087 .0084
• -2.2 .0139 .0136 .0132 .0129 .0125 .0122
  .0119 .0116 .0113 .0110
• -2.1 .0179 .0174 .0170 .0166 .0162 .0158
  .0154 .0150 .0146 .0143
• -2.0 .0228 .0222 .0217 .0212 .0207 .0202
  .0197 .0192 .0188 .0183
• -1.9 .0287 .0281 .0274 .0268 .0262 .0256
  .0250 .0244 .0239 .0233
• -1.8 .0359 .0351 .0344 .0336 .0329 .0322
  .0314 .0307 .0301 .0294
• -1.7 .0446 .0436 .0427 .0418 .0409 .0401
  .0392 .0384 .0375 .0367
• -1.6 .0548 .0537 .0526 .0516 .0505 .0495
  .0485 .0475 .0465 .0455
• -1.5 .0668 .0655 .0643 .0630 .0618 .0606
  .0594 .0582 .0571 .0559

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EXAMPLE

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

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Probability a flight takes exactly 315 minutes

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

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Probability a Flight Takes Less Than 300 Minutes
300
-1.00
0
Z

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Probability a Flight Takes Longer Than 335 Minutes
.75
0
Z
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Probability a Flight Takes Between 320 and 350
Minutes
1.50
0
Z
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Probability a Flight Takes Between 325 and 355
Minutes
0.25
0
Z
1.75
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Probability a Flight Takes Between 308 and 347
Minutes
1.35
-0.60
0
Z
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Probability a Flight Takes Between 275 and 285
Minutes
-2.25
-1.75
0 Z
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Using Excel to Calculate Normal Probabilities

• Given values for µ and s, cumulative
  probabilities P(X lt x0) are given by
• Note that NORMDIST(x0,µ,s,FALSE) returns the
  value of the density function at x0, not a
  probability.
• If the value of z is given, then the cumulative
  probabilities P(Zltz) are given by

NORMDIST(x0, µ, s, TRUE)
NORMSDIST(z)
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Calculating x and z Values From Normal
Probabilities

• Basic Approach
• Convert to a cumulative probabiltity
• Locate that probability (or the closest to it) in
  the Cumulative Standard Normal Probability table
• This gives the z value
• This is the number of standard deviations x is
  from the mean

x µ zs
Note z can be a negative value
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90 of the Flights Take At Least How Long?

• .9000 of the probability lies above the x value
• .1000 lies below the x value

Look up .1000 in middle of z table
0 Z
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The Middle 75 of the Flight Times Lie Between
What Two Values?

• Required to find xL and xU such that .7500 lies
  between xL and xU -- this means .1250 lies below
  xL and .1250 lies above xU (.8750 lies below xU)

1.15
0 Z
-1.15
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Using Excel to Calculate x and z Values From
Normal Probabilities

• Given values for µ and s, the value of x0 such
  that P(X lt x0) p is given by
• NORMINV(p, µ, s)
  
• The value of z such that P(Zltz) p is given by
• NORMSINV(p)

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What Would the Mean Have to Be So That 80 of the
Flights Take Less Than 330 Minutes?

• Since x µ zs, then µ x - z s

Look up .8000 in the middle of the z-table
0 Z
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REVIEW

• Normal Distribution
• Importance and Properties
• Converting X to Z
• Use of Tables to Calculate Probabilities
• Use of Excel to Calculate Probabilities